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APEX Calculus: for University of Lethbridge

Appendix A Answers to Selected Exercises

I Math 1560: Calculus I
1 Limits
1.1 An Introduction To Limits
1.1.3 Exercises

Terms and Concepts

1.1.3.2.
Answer.
\(\text{an indeterminate form}\)
1.1.3.3.
Answer.
\(\text{False}\)
1.1.3.6.
Answer.
\(1\)

Problems

1.1.3.7.
Answer.
\(5\)
1.1.3.8.
Answer.
\(3\)
1.1.3.9.
Answer.
\(\text{DNE}\)
1.1.3.10.
Answer.
\({\frac{2}{3}}\)
1.1.3.11.
Answer.
\(-4\)
1.1.3.12.
Answer.
\(\text{DNE}\hbox{ or }\infty \)
1.1.3.13.
Answer.
\(\text{DNE}\)
1.1.3.14.
Answer.
\(6\)
1.1.3.15.
Answer.
\(1\)
1.1.3.16.
Answer.
\(\text{DNE}\)
1.1.3.17.
Answer.
\(1\)
1.1.3.18.
Answer.
\(\text{DNE}\)
1.1.3.19.
Answer.
\(\text{DNE}\)
1.1.3.20.
Answer.
\(1\)
1.1.3.21.
Answer.
\(-7\)
1.1.3.22.
Answer.
\(9\)
1.1.3.23.
Answer.
\(5\)
1.1.3.24.
Answer.
\(-0.111111\)
1.1.3.25.
Answer.
\(29\)
1.1.3.26.
Answer.
\(0.2\)
1.1.3.27.
Answer.
\(-1\)
1.1.3.28.
Answer.
\(0\)

1.2 Epsilon-Delta Definition of a Limit

Exercises

Terms and Concepts
1.2.2.
Answer.
\(\text{y-tolerance}\)
1.2.3.
Answer.
\(\text{True}\)
1.2.4.
Answer.
\(\text{True}\)

1.3 Finding Limits Analytically

Exercises

Terms and Concepts
1.3.6.
Answer.
\(\text{True}\)
Problems
1.3.7.
Answer.
\(9\)
1.3.8.
Answer.
\(6\)
1.3.9.
Answer.
\(0\)
1.3.10.
Answer.
\(\text{DNE}\)
1.3.11.
Answer.
\(3\)
1.3.12.
Answer.
\(\text{not possible to know}\)
1.3.13.
Answer.
\(3\)
1.3.14.
Answer.
\(-45\)
1.3.15.
Answer.
\(0\)
1.3.16.
Answer.
\(\cos\mathopen{}\left(3.14159\right)\)
1.3.17.
Answer.
\(\pi \)
1.3.18.
Answer.
\(1\)
1.3.19.
Answer.
\(23\)
1.3.20.
Answer.
\(\left(\frac{\pi -5}{\pi -8}\right)^{4}\)
1.3.21.
Answer.
\(\frac{\sqrt{3}}{4}\)
1.3.22.
Answer.
\(-{\frac{16}{5}}\)
1.3.23.
Answer.
\(\text{DNE}\)
1.3.24.
Answer.
\(256\)
1.3.25.
Answer.
\(\frac{2\sqrt{3}}{3}\)
1.3.26.
Answer.
\(\ln\mathopen{}\left(4\right)\)
1.3.27.
Answer.
\(\frac{\pi ^{2}-4\pi -2}{2\pi ^{2}-2\pi +1}\)
1.3.28.
Answer.
\(\frac{2\pi -4}{5\pi -5}\)
1.3.29.
Answer.
\({\frac{1}{4}}\)
1.3.30.
Answer.
\(-{\frac{7}{2}}\)
1.3.31.
Answer.
\({\frac{17}{4}}\)
1.3.32.
Answer.
\({\frac{13}{3}}\)
1.3.33.
Answer.
\({\frac{4}{9}}\)
1.3.34.
Answer.
\({\frac{5}{4}}\)
1.3.35.
Answer.
\(0\)
1.3.36.
Answer.
\(0\)
1.3.37.
Answer.
\(1\)
1.3.38.
Answer.
\(9\)
1.3.39.
Answer.
\(8\)
1.3.40.
Answer.
\({\frac{9}{8}}\)
1.3.41.
Answer.
\(1\)
1.3.42.
Answer.
\(\frac{\pi }{180}\)

1.4 One-Sided Limits

Exercises

Terms and Concepts
1.4.2.
Answer.
\(\text{False}\)
1.4.3.
Answer.
\(\text{False}\)
1.4.4.
Answer.
\(\text{True}\)
Problems
1.4.5.
1.4.5.a
Answer.
\(2\)
1.4.5.b
Answer.
\(2\)
1.4.5.c
Answer.
\(2\)
1.4.5.d
Answer.
\(1\)
1.4.5.e
Answer.
\(\text{DNE}\)
1.4.5.f
Answer.
\(4\)
1.4.6.
1.4.6.a
Answer.
\(0\)
1.4.6.b
Answer.
\(4\)
1.4.6.c
Answer.
\(\text{DNE}\)
1.4.6.d
Answer.
\(4\)
1.4.6.e
Answer.
\(\text{DNE}\)
1.4.6.f
Answer.
\(1\)
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
Answer.
\(\text{DNE}\)
1.4.7.e
Answer.
\(5\)
1.4.7.f
Answer.
\(4\)
1.4.8.
1.4.8.a
Answer.
\(2\)
1.4.8.b
Answer.
\(3\)
1.4.8.c
Answer.
\(\text{DNE}\)
1.4.8.d
Answer.
\(4\)
1.4.9.
1.4.9.a
Answer.
\(1\)
1.4.9.b
Answer.
\(1\)
1.4.9.c
Answer.
\(1\)
1.4.9.d
Answer.
\(1\)
1.4.10.
1.4.10.a
Answer.
\(-5\)
1.4.10.b
Answer.
\(1\)
1.4.10.c
Answer.
\(\text{DNE}\)
1.4.10.d
Answer.
\(3\)
1.4.11.
1.4.11.a
Answer.
\(2\)
1.4.11.b
Answer.
\(2\)
1.4.11.c
Answer.
\(2\)
1.4.11.d
Answer.
\(0\)
1.4.11.e
Answer.
\(2\)
1.4.11.f
Answer.
\(2\)
1.4.11.g
Answer.
\(2\)
1.4.11.h
Answer.
\(\text{DNE}\)
1.4.12.
1.4.12.a
Answer.
\(a-1\)
1.4.12.b
Answer.
\(a\)
1.4.12.c
Answer.
\(\text{DNE}\)
1.4.12.d
Answer.
\(a\)
1.4.13.
1.4.13.a
Answer.
\(2\)
1.4.13.b
Answer.
\(6\)
1.4.13.c
Answer.
\(\text{DNE}\)
1.4.13.d
Answer.
\(2\)
1.4.14.
1.4.14.a
Answer.
\(-17\)
1.4.14.b
Answer.
\(0\)
1.4.14.c
Answer.
\(\text{DNE}\)
1.4.14.d
Answer.
\(0\)
1.4.15.
1.4.15.a
Answer.
\(9\)
1.4.15.b
Answer.
\(9\)
1.4.15.c
Answer.
\(9\)
1.4.15.d
Answer.
\(9\)
1.4.15.e
Answer.
\(126\)
1.4.15.f
Answer.
\(126\)
1.4.15.g
Answer.
\(126\)
1.4.15.h
Answer.
\(126\)
1.4.16.
1.4.16.a
Answer.
\(-1\)
1.4.16.b
Answer.
\(0\)
1.4.16.c
Answer.
\(\text{DNE}\)
1.4.16.d
Answer.
\(0\)
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.
\(\sin^{2}\mathopen{}\left(a\right)\)
1.4.18.
1.4.18.a
Answer.
\(0\)
1.4.18.b
Answer.
\(1\)
1.4.18.c
Answer.
\(\text{DNE}\)
1.4.18.d
Answer.
\(-2\)
1.4.19.
1.4.19.a
Answer.
\(-4\)
1.4.19.b
Answer.
\(-4\)
1.4.19.c
Answer.
\(-4\)
1.4.19.d
Answer.
\(-2\)
1.4.20.
1.4.20.a
Answer.
\(c\)
1.4.20.b
Answer.
\(c\)
1.4.20.c
Answer.
\(c\)
1.4.20.d
Answer.
\(c\)
1.4.21.
1.4.21.a
Answer.
\(-1\)
1.4.21.b
Answer.
\(1\)
1.4.21.c
Answer.
\(\text{DNE}\)
1.4.21.d
Answer.
\(0\)

1.5 Continuity

Exercises

Terms and Concepts
1.5.5.
Answer.
\(\text{False}\)
1.5.6.
Answer.
\(\text{True}\)
1.5.7.
Answer.
\(\text{True}\)
1.5.8.
Answer.
\(\text{False}\)
1.5.9.
Answer.
\(\text{False}\)
1.5.10.
Answer.
\(\text{True}\)
Problems
1.5.11.
Answer.
\(\text{No.}\)
1.5.12.
Answer.
\(\text{No.}\)
1.5.13.
Answer.
\(\text{No.}\)
1.5.14.
Answer.
\(\text{Yes.}\)
1.5.15.
Answer.
\(\text{Yes.}\)
1.5.16.
Answer.
\(\text{Yes.}\)
1.5.17.
Answer 1.
\(5\)
Answer 2.
Undefined
Answer 3.
\(\text{No.}\)
1.5.18.
Answer.
\(\text{Yes.}\)
1.5.19.
1.5.19.a
Answer.
\(5\)
1.5.19.b
Answer.
Undefined
1.5.20.
1.5.20.a
Answer.
\(5\)
1.5.20.b
Answer.
Undefined
1.5.21.
1.5.21.a
Answer.
\(5\)
1.5.21.b
Answer.
Undefined
1.5.22.
1.5.22.a
Answer.
\(5\)
1.5.22.b
Answer.
Undefined
1.5.23.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.24.
Answer.
\(\left(-\infty ,-2\right], \left[2,\infty \right)\)
1.5.25.
Answer.
\(\left[-2,2\right]\)
1.5.26.
Answer.
\(\left[-3,3\right]\)
1.5.27.
Answer.
\(\left(-\infty ,-1.73205\right], \left[1.73205,\infty \right)\)
1.5.28.
Answer.
\(\left(-7,7\right)\)
1.5.29.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.30.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.31.
Answer.
\(\left(0,\infty \right)\)
1.5.32.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.33.
Answer.
\(\left(-\infty ,1.09861\right]\)
1.5.34.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.39.
Answer.
\(1.23633\)
1.5.40.
Answer.
\(0.523633\)
1.5.41.
Answer.
\(0.693164\)
1.5.42.
Answer.
\(0.785547\)

1.6 Limits Involving Infinity
1.6.4 Exercises

Terms and Concepts

1.6.4.1.
Answer.
\(\text{False}\)
1.6.4.2.
Answer.
\(\text{True}\)
1.6.4.3.
Answer.
\(\text{False}\)
1.6.4.4.
Answer.
\(\text{True}\)
1.6.4.5.
Answer.
\(\text{True}\)

Problems

1.6.4.9.
1.6.4.9.a
Answer.
\(-\infty \)
1.6.4.9.b
Answer.
\(\infty \)
1.6.4.10.
1.6.4.10.a
Answer.
\(-\infty \)
1.6.4.10.b
Answer.
\(\infty \)
1.6.4.10.c
Answer.
\(\text{DNE}\)
1.6.4.10.d
Answer.
\(\infty \)
1.6.4.10.e
Answer.
\(\infty \)
1.6.4.10.f
Answer.
\(\infty \)
1.6.4.11.
1.6.4.11.a
Answer.
\(0\)
1.6.4.11.b
Answer.
\(3\)
1.6.4.11.c
Answer.
\(1.5\)
1.6.4.11.d
Answer.
\(1.5\)
1.6.4.12.
1.6.4.12.a
Answer.
\(\text{DNE}\)
1.6.4.12.b
Answer.
\(\text{DNE}\)
1.6.4.12.c
Answer.
\(0\)
1.6.4.12.d
Answer.
\(0\)
1.6.4.13.
1.6.4.13.a
Answer.
\(\text{DNE}\)
1.6.4.13.b
Answer.
\(\text{DNE}\)
1.6.4.14.
1.6.4.14.a
Answer.
\(-9\)
1.6.4.14.b
Answer.
\(\infty \)
1.6.4.15.
1.6.4.15.a
Answer.
\(-\infty \)
1.6.4.15.b
Answer.
\(\infty \)
1.6.4.15.c
Answer.
\(\text{DNE}\)
1.6.4.16.
1.6.4.16.a
Answer.
\(-\infty \)
1.6.4.16.b
Answer.
\(-\infty \)
1.6.4.16.c
Answer.
\(-\infty \)
1.6.4.17.
1.6.4.17.a
Answer.
\(\infty \)
1.6.4.17.b
Answer.
\(\infty \)
1.6.4.17.c
Answer.
\(\infty \)
1.6.4.18.
1.6.4.18.a
Answer.
\(1.8\)
1.6.4.18.b
Answer.
\(1.8\)
1.6.4.18.c
Answer.
\(1.8\)
1.6.4.19.
Answer.
\(y = 2, x = -2, x = 9\)
1.6.4.20.
Answer.
\(y = \frac{5}{-2}, x = -9\)
1.6.4.21.
Answer.
\(y = 0, x = 0, x = 4\)
1.6.4.22.
Answer.
\(x = -3\)
1.6.4.23.
Answer.
\(\text{NONE}\)
1.6.4.24.
Answer.
\(y = \frac{4}{-1}\)
1.6.4.25.
Answer.
\(\infty \)
1.6.4.26.
Answer.
\(\infty \)
1.6.4.27.
Answer.
\(\infty \)
1.6.4.28.
Answer.
\(\infty \)

2 Derivatives
2.1 Instantaneous Rates of Change: The Derivative
2.1.3 Exercises

Terms and Concepts

2.1.3.1.
Answer.
\(\text{True}\)
2.1.3.2.
Answer.
\(\text{True}\)

Problems

2.1.3.7.
Answer.
\(0\)
2.1.3.8.
Answer.
\(2\)
2.1.3.9.
Answer.
\(-3\)
2.1.3.10.
Answer.
\(2x\)
2.1.3.11.
Answer.
\(3x^{2}\)
2.1.3.12.
Answer.
\(6x-1\)
2.1.3.13.
Answer.
\(\frac{-1}{x^{2}}\)
2.1.3.14.
Answer.
\(\frac{-1}{\left(s-2\right)^{2}}\)
2.1.3.15.
Answer 1.
\(y = 6\)
Answer 2.
\(x = -2\)
2.1.3.16.
Answer 1.
\(y-2x = 0\)
Answer 2.
\(0.5x+y = 7.5\)
2.1.3.17.
Answer 1.
\(3x+y = 4\)
Answer 2.
\(y-0.333333x = -19.3333\)
2.1.3.18.
Answer 1.
\(y-4x = -4\)
Answer 2.
\(0.25x+y = 4.5\)
2.1.3.19.
Answer 1.
\(y-48x = -128\)
Answer 2.
\(0.0208333x+y = 64.0833\)
2.1.3.20.
Answer 1.
\(7x+y = 1\)
Answer 2.
\(y-0.142857x = 8.14286\)
2.1.3.21.
Answer 1.
\(0.25x+y = -1\)
Answer 2.
\(y-4x = 7.5\)
2.1.3.22.
Answer 1.
\(x+y = 4\)
Answer 2.
\(y-x = -2\)
2.1.3.23.
Answer.
\(5.9x+y = 1.2\)
2.1.3.24.
Answer.
\(y-11.1111x = 110\)
2.1.3.25.
Answer.
\(y-0.0192627x = 0.0953664\)
2.1.3.26.
Answer.
\(0.04996x+y = 1\)
2.1.3.27.
2.1.3.27.a
Answer.
\(-2, 0, 4\)
2.1.3.27.b
Answer.
\(2x\)
2.1.3.27.c
Answer.
Undefined
2.1.3.28.
2.1.3.28.a
Answer.
\(-1, -0.25\)
2.1.3.28.b
Answer.
\(\frac{-1}{\left(x+1\right)^{2}}\)
2.1.3.28.c
Answer.
Undefined
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.
\(\left(-1,1\right)\)
Answer 5.
\(\left(-\infty ,-1\right)\cup \left(1,\infty \right)\)
Answer 6.
\(\left\{-1,1\right\}\)
2.1.3.34.
Answer 1.
\(\left(-2,2\right)\)
Answer 2.
\(\left(-\infty ,-2\right)\cup \left(2,\infty \right)\)
Answer 3.
\(\left\{-2,2\right\}\)
Answer 4.
\(\left(-1,0\right)\cup \left(1,\infty \right)\)
Answer 5.
\(\left(-\infty ,-1\right)\cup \left(0,1\right)\)
Answer 6.
\(\left\{-1,0,1\right\}\)
2.1.3.35.
Answer.
\(\text{no}\)
2.1.3.36.
Answer.
\(\text{yes}\)

2.2 Interpretations of the Derivative
2.2.5 Exercises

Terms and Concepts

2.2.5.1.
Answer.
\(\text{velocity}\)
2.2.5.3.
Answer.
\(\text{linear functions}\)

Problems

2.2.5.4.
Answer.
\(20\)
2.2.5.5.
Answer.
\(-89\)
2.2.5.6.
Answer.
\(91\)
2.2.5.7.
Answer.
\(\text{f(10.1)}\)
2.2.5.8.
Answer.
\(-2\)
2.2.5.9.
Answer.
\(7\)
2.2.5.10.
Answer.
\(\text{decibels per customer}\)
2.2.5.11.
Answer.
\(\text{foot per second squared}\)
2.2.5.12.
Answer.
\(\text{foot per hour}\)
2.2.5.15.
Answer.
\(\text{Choice 1}\)
2.2.5.16.
Answer.
\(\text{Choice 2}\)
2.2.5.17.
Answer.
\(\text{Choice 2}\)
2.2.5.18.
Answer.
\(\text{Choice 2}\)

2.3 Basic Differentiation Rules
2.3.3 Exercises

Terms and Concepts

2.3.3.1.
Answer.
\(\text{the power rule}\)
2.3.3.2.
Answer.
\(\frac{1}{x}\)
2.3.3.3.
Answer.
\(e^{x}\)
2.3.3.4.
Answer.
\(10\)
2.3.3.5.
Answer.
\(\text{Choice 1, Choice 2, Choice 5, Choice 6}\)
2.3.3.7.
Answer.
\(17x-205\)
2.3.3.9.
Answer 1.
\(\text{a velocity function}\)
Answer 2.
\(\text{an acceleration function}\)
2.3.3.10.
Answer.
\(\text{pound per foot squared}\)

Problems

2.3.3.11.
Answer.
\(-\left(14x+8\right)\)
2.3.3.12.
Answer.
\(28x-48x^{2}+5\)
2.3.3.13.
Answer.
\(9-\left(20t^{4}+{\frac{3}{4}}t^{2}\right)\)
2.3.3.14.
Answer.
\(19\sin\mathopen{}\left(\theta\right)-3\cos\mathopen{}\left(\theta\right)\)
2.3.3.15.
Answer.
\(3e^{r}\)
2.3.3.16.
Answer.
\(21t^{2}+5\sin\mathopen{}\left(t\right)-2\cos\mathopen{}\left(t\right)\)
2.3.3.17.
Answer.
\(\frac{6}{x}+9\)
2.3.3.18.
Answer.
\(s^{3}+s^{2}+s+1\)
2.3.3.19.
Answer.
\(\sin\mathopen{}\left(t\right)-\left(e^{t}+\cos\mathopen{}\left(t\right)\right)\)
2.3.3.20.
Answer.
\(\frac{8}{x}\)
2.3.3.21.
Answer.
\(0\)
2.3.3.22.
Answer.
\(18t+24\)
2.3.3.23.
Answer.
\(24x^{2}+96x+96\)
2.3.3.24.
Answer.
\(3x^{2}+18x+27\)
2.3.3.25.
Answer.
\(8x+28\)
2.3.3.27.
Answer 1.
\(9x^{8}\)
Answer 2.
\(9\cdot 8x^{7}\)
Answer 3.
\(9\cdot 8\cdot 7x^{6}\)
Answer 4.
\(9\cdot 8\cdot 7\cdot 6x^{5}\)
2.3.3.28.
Answer 1.
\(-8\sin\mathopen{}\left(x\right)\)
Answer 2.
\(-\left(8\cos\mathopen{}\left(x\right)\right)\)
Answer 3.
\(8\sin\mathopen{}\left(x\right)\)
Answer 4.
\(8\cos\mathopen{}\left(x\right)\)
2.3.3.29.
Answer 1.
\(-\left(4\cdot 2t+3+e^{t}\right)\)
Answer 2.
\(-\left(8+e^{t}\right)\)
Answer 3.
\(-e^{t}\)
Answer 4.
\(-e^{t}\)
2.3.3.30.
Answer 1.
\(2\theta+8\theta^{7}\)
Answer 2.
\(2+8\cdot 7\theta^{6}\)
Answer 3.
\(8\cdot 7\cdot 6\theta^{5}\)
Answer 4.
\(8\cdot 7\cdot 6\cdot 5\theta^{4}\)
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.
\(-\left(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\right)\)
2.3.3.32.
Answer 1.
\(0\)
Answer 2.
\(0\)
Answer 3.
\(0\)
Answer 4.
\(0\)
2.3.3.33.
Answer 1.
\(y = 20\mathopen{}\left(x-2\right)+24\)
Answer 2.
\(y = -{\frac{1}{20}}\mathopen{}\left(x-2\right)+24\)
2.3.3.34.
Answer 1.
\(y = e^{0}\ln\mathopen{}\left(e\right)\mathopen{}\left(t-0\right)+e^{0}-2\)
Answer 2.
\(y = \frac{-1}{e^{0}\ln\mathopen{}\left(e\right)}\mathopen{}\left(t-0\right)+e^{0}-2\)
2.3.3.35.
Answer 1.
\(y = x-1\)
Answer 2.
\(y = -\left(x-1\right)\)
2.3.3.36.
Answer 1.
\(y = \frac{4\sqrt{3}}{2}\mathopen{}\left(x-\frac{\pi }{6}\right)+\frac{4\cdot 1}{2}\)
Answer 2.
\(y = -\left({\frac{1}{4}}\frac{2\sqrt{3}}{3}\right)\mathopen{}\left(x-\frac{\pi }{6}\right)+\frac{4\cdot 1}{2}\)
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.
\(y = -\left({\frac{1}{2}}\cdot 2\right)\mathopen{}\left(x-\frac{\pi }{6}\right)+\frac{-2\sqrt{3}}{2}\)
2.3.3.38.
Answer 1.
\(9-9x\)
Answer 2.
\(y = \frac{-1}{-9}\mathopen{}\left(x-\left(-9\right)\right)+90\)

2.4 The Product and Quotient Rules

Exercises

Terms and Concepts
2.4.1.
Answer.
\(\text{False}\)
2.4.2.
Answer.
\(\text{False}\)
2.4.3.
Answer.
\(\text{True}\)
2.4.4.
Answer.
\(\text{the quotient rule}\)
2.4.5.
Answer.
\(\text{False}\)
Problems
2.4.15.
Answer.
\(\sin\mathopen{}\left(y\right)+y\cos\mathopen{}\left(y\right)\)
2.4.16.
Answer.
\(3t^{2}\cos\mathopen{}\left(t\right)-t^{3}\sin\mathopen{}\left(t\right)\)
2.4.17.
Answer.
\(e^{q}\ln\mathopen{}\left(q\right)+e^{q}\frac{1}{q}\)
2.4.18.
Answer.
\(-\left(\frac{6y^{5}}{\left(y^{6}\right)^{2}}\mathopen{}\left(\csc\mathopen{}\left(y\right)-5\right)+\frac{1}{y^{6}}\csc\mathopen{}\left(y\right)\cot\mathopen{}\left(y\right)\right)\)
2.4.19.
Answer.
\(\frac{t-4-\left(t+8\right)}{\left(t-4\right)^{2}}\)
2.4.20.
Answer.
\(\frac{3q^{2}\mathopen{}\left(\sin\mathopen{}\left(q\right)-8q^{2}\right)-q^{3}\mathopen{}\left(\cos\mathopen{}\left(q\right)-8\cdot 2q\right)}{\left(\sin\mathopen{}\left(q\right)-8q^{2}\right)^{2}}\)
2.4.21.
Answer.
\(-\left(\csc\mathopen{}\left(y\right)\cot\mathopen{}\left(y\right)+e^{y}\right)\)
2.4.22.
Answer.
\(\sec^{2}\mathopen{}\left(t\right)\ln\mathopen{}\left(t\right)+\frac{1}{t}\tan\mathopen{}\left(t\right)\)
2.4.23.
Answer.
\(7\cdot 2q-6\)
2.4.24.
Answer.
\(5y^{4}\)
2.4.25.
Answer.
\(\left(5r^{2}+17r+10\right)e^{r}\)
2.4.26.
Answer.
\(\frac{9z^{8}-z^{9}-z^{5}+5z^{4}}{e^{z}}\)
2.4.27.
Answer.
\(3\)
2.4.28.
Answer.
\(5r^{4}\mathopen{}\left(\tan\mathopen{}\left(r\right)+e^{r}\right)+r^{5}\mathopen{}\left(\sec^{2}\mathopen{}\left(r\right)+e^{r}\right)\)
2.4.29.
Answer.
\(\frac{\csc\mathopen{}\left(z\right)\sin\mathopen{}\left(z\right)-\csc\mathopen{}\left(z\right)\cot\mathopen{}\left(z\right)\mathopen{}\left(\cos\mathopen{}\left(z\right)+2\right)}{\left(\cos\mathopen{}\left(z\right)+2\right)^{2}}\)
2.4.30.
Answer.
\(4\theta^{3}\sec\mathopen{}\left(\theta\right)+\theta^{4}\sec\mathopen{}\left(\theta\right)\tan\mathopen{}\left(\theta\right)+\frac{\sec\mathopen{}\left(\theta\right)\tan\mathopen{}\left(\theta\right)\theta^{4}-4\theta^{3}\sec\mathopen{}\left(\theta\right)}{\left(\theta^{4}\right)^{2}}\)
2.4.31.
Answer.
\(\frac{\tan\mathopen{}\left(r\right)-r\sec^{2}\mathopen{}\left(r\right)}{\tan^{2}\mathopen{}\left(r\right)}-\frac{\csc^{2}\mathopen{}\left(r\right)r+\cot\mathopen{}\left(r\right)}{r^{2}}\)
2.4.32.
Answer.
\(0\)
2.4.33.
Answer.
\(7\cdot 5x^{4}e^{x}+7x^{5}e^{x}-\left(\cos\mathopen{}\left(x\right)\cos\mathopen{}\left(x\right)-\sin\mathopen{}\left(x\right)\sin\mathopen{}\left(x\right)\right)\)
2.4.34.
Answer.
\(\frac{\left(2r\sin\mathopen{}\left(r\right)+r^{2}\cos\mathopen{}\left(r\right)\right)\mathopen{}\left(r^{2}\cos\mathopen{}\left(r\right)-9\right)-\left(r^{2}\sin\mathopen{}\left(r\right)-7\right)\mathopen{}\left(2r\cos\mathopen{}\left(r\right)-r^{2}\sin\mathopen{}\left(r\right)\right)}{\left(r^{2}\cos\mathopen{}\left(r\right)-9\right)^{2}}\)
2.4.35.
Answer.
\(\left(4z^{3}\ln\mathopen{}\left(z\right)+z^{4}\frac{1}{z}\right)\cos\mathopen{}\left(z\right)-z^{4}\ln\mathopen{}\left(z\right)\sin\mathopen{}\left(z\right)\)
2.4.36.
Answer.
\(\left(9\cos\mathopen{}\left(x\right)-9x\sin\mathopen{}\left(x\right)\right)\tan\mathopen{}\left(x\right)+9x\cos\mathopen{}\left(x\right)\sec^{2}\mathopen{}\left(x\right)\)
2.4.37.
Answer 1.
\(y = -\left(7x+7\right)\)
Answer 2.
\(y = \left({\frac{1}{7}}\right)x-7\)
2.4.38.
Answer 1.
\(y = 5.0345\mathopen{}\left(x-\frac{5\pi }{3}\right)+\frac{5\pi }{6}\)
Answer 2.
\(y = \frac{5\pi }{6}-\left({\frac{12837432}{64630031}}\right)\mathopen{}\left(x-\frac{5\pi }{3}\right)\)
2.4.39.
Answer 1.
\(y = -\left(15\mathopen{}\left(x+5\right)+25\right)\)
Answer 2.
\(y = \left({\frac{1}{15}}\right)\mathopen{}\left(x+5\right)-25\)
2.4.40.
Answer 1.
\(y = \left({\frac{1}{8}}\right)x\)
Answer 2.
\(y = -8x\)
2.4.41.
Answer.
\({\frac{17}{2}}\)
2.4.42.
Answer.
\(0\)
2.4.43.
Answer.
\(\text{NONE}\)
2.4.44.
Answer.
\(0, 4\)
2.4.45.
Answer.
\(2\cos\mathopen{}\left(x\right)-x\sin\mathopen{}\left(x\right)\)
2.4.46.
Answer.
\(-4\cos\mathopen{}\left(x\right)+x\sin\mathopen{}\left(x\right)\)
2.4.47.
Answer.
\(\csc\mathopen{}\left(x\right)\cot\mathopen{}\left(x\right)\cot\mathopen{}\left(x\right)+\csc^{2}\mathopen{}\left(x\right)\csc\mathopen{}\left(x\right)\)
2.4.48.
Answer.
\(0\)

2.5 The Chain Rule

Exercises

Terms and Concepts
2.5.1.
Answer.
\(\text{True}\)
2.5.2.
Answer.
\(\text{False}\)
2.5.3.
Answer.
\(\text{False}\)
2.5.4.
Answer.
\(\text{True}\)
2.5.5.
Answer.
\(\text{True}\)
2.5.6.
Answer.
\(\text{True}\)
Problems
2.5.7.
Answer.
\(10\mathopen{}\left(4x^{3}-x\right)^{9}\mathopen{}\left(12x^{2}-1\right)\)
2.5.8.
Answer.
\(15\mathopen{}\left(3t-2\right)^{4}\)
2.5.9.
Answer.
\(3\mathopen{}\left(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\right)^{2}\mathopen{}\left(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\right)\)
2.5.10.
Answer.
\(\left(6t+1\right)e^{3t^{2}+t-1}\)
2.5.11.
Answer.
\(4\mathopen{}\left(\ln\mathopen{}\left(x\right)-x^{4}\right)^{3}\mathopen{}\left(\frac{1}{x}-4x^{3}\right)\)
2.5.12.
Answer.
\(0.693147\cdot 2^{q^{5}+4q}\mathopen{}\left(5q^{4}+4\right)\)
2.5.13.
Answer.
\(5\mathopen{}\left(y+\frac{1}{y}\right)^{4}\mathopen{}\left(1-\frac{1}{y^{2}}\right)\)
2.5.14.
Answer.
\(-5\sin\mathopen{}\left(5t\right)\)
2.5.15.
Answer.
\(2\sec^{2}\mathopen{}\left(2q\right)\)
2.5.16.
Answer.
\(-\csc^{2}\mathopen{}\left(\theta^{2}+3\right)\cdot 2\theta\)
2.5.17.
Answer.
\(\left(6t^{5}-\frac{3t^{2}}{\left(t^{3}\right)^{2}}\right)\cos\mathopen{}\left(t^{6}+\frac{1}{t^{3}}\right)\)
2.5.18.
Answer.
\(-5\cos^{4}\mathopen{}\left(7q\right)\cdot 7\sin\mathopen{}\left(7q\right)\)
2.5.19.
Answer.
\(-3\cos^{2}\mathopen{}\left(y^{2}+3y-3\right)\mathopen{}\left(2y+3\right)\sin\mathopen{}\left(y^{2}+3y-3\right)\)
2.5.20.
Answer.
\(-\frac{1}{\cos\mathopen{}\left(t\right)}\sin\mathopen{}\left(t\right)\)
2.5.21.
Answer.
\(\frac{1}{q^{8}}\cdot 8q^{7}\)
2.5.22.
Answer.
\(3\frac{1}{y}\)
2.5.23.
Answer.
\(1.79176\cdot 6^{t}\)
2.5.24.
Answer.
\(-0.693147\cdot 2^{\csc\mathopen{}\left(z\right)}\csc\mathopen{}\left(z\right)\cot\mathopen{}\left(z\right)\)
2.5.25.
Answer.
\(0\)
2.5.26.
Answer.
\(\frac{1.38629\cdot 4^{t}\cdot 9^{t}-4^{t}\cdot 2.19722\cdot 9^{t}}{\left(9^{t}\right)^{2}}\)
2.5.27.
Answer.
\(\frac{1.79176\cdot 6^{w}\mathopen{}\left(5^{w}+6\right)-\left(6^{w}+5\right)\cdot 1.60944\cdot 5^{w}}{\left(5^{w}+6\right)^{2}}\)
2.5.28.
Answer.
\(\frac{1.94591\cdot 7^{y}\cdot 5^{y}-\left(7^{y}+8\right)\cdot 1.60944\cdot 5^{y}}{\left(5^{y}\right)^{2}}\)
2.5.29.
Answer.
\(\frac{\left(1.60944\cdot 5^{r^{2}}\cdot 2r-1\right)\cdot 6^{r^{2}}-\left(5^{r^{2}}-r\right)\cdot 1.79176\cdot 6^{r^{2}}\cdot 2r}{\left(6^{r^{2}}\right)^{2}}\)
2.5.30.
Answer.
\(3w^{2}\cot\mathopen{}\left(5w\right)-w^{3}\cdot 5\csc^{2}\mathopen{}\left(5w\right)\)
2.5.31.
Answer.
\(6\mathopen{}\left(x^{2}+4x\right)^{5}\mathopen{}\left(2x+4\right)\mathopen{}\left(7x^{4}+x\right)^{3}+\left(x^{2}+4x\right)^{6}\cdot 3\mathopen{}\left(7x^{4}+x\right)^{2}\mathopen{}\left(7\cdot 4x^{3}+1\right)\)
2.5.32.
Answer.
\(-\left(4\cos\mathopen{}\left(8-4r\right)\cos\mathopen{}\left(6r+r^{2}\right)+\left(6+2r\right)\sin\mathopen{}\left(6r+r^{2}\right)\sin\mathopen{}\left(8-4r\right)\right)\)
2.5.33.
Answer.
\(7\cos\mathopen{}\left(9+7w\right)\cos\mathopen{}\left(4w-5\right)-4\sin\mathopen{}\left(4w-5\right)\sin\mathopen{}\left(9+7w\right)\)
2.5.34.
Answer.
\(e^{8x^{2}}\cdot 8\cdot 2x\sin\mathopen{}\left(\frac{1}{x}\right)-e^{8x^{2}}\frac{1}{x^{2}}\cos\mathopen{}\left(\frac{1}{x}\right)\)
2.5.35.
Answer.
\(-\frac{6\sin\mathopen{}\left(6r+4\right)\mathopen{}\left(3r+1\right)^{3}+3\cdot 3\mathopen{}\left(3r+1\right)^{2}\cos\mathopen{}\left(6r+4\right)}{\left(\left(3r+1\right)^{3}\right)^{2}}\)
2.5.36.
Answer.
\(\frac{3\cdot 2\mathopen{}\left(3z+5\right)\sin\mathopen{}\left(9z\right)-\left(3z+5\right)^{2}\cdot 9\cos\mathopen{}\left(9z\right)}{\sin^{2}\mathopen{}\left(9z\right)}\)
2.5.37.
Answer 1.
\(y = 0\)
Answer 2.
\(x = 0\)
2.5.38.
Answer 1.
\(y = 15\mathopen{}\left(x-1\right)+1\)
Answer 2.
\(y = \frac{-1}{15}\mathopen{}\left(x-1\right)+1\)
2.5.39.
Answer 1.
\(y = -3\mathopen{}\left(x-\frac{\pi }{2}\right)+1\)
Answer 2.
\(y = \frac{1}{3}\mathopen{}\left(x-\frac{\pi }{2}\right)+1\)
2.5.40.
Answer 1.
\(y = -5e\mathopen{}\left(x+1\right)+e\)
Answer 2.
\(y = \frac{1}{5e}\mathopen{}\left(x+1\right)+e\)
2.5.41.
Answer.
\(\frac{1}{x}\)
2.5.42.
Answer.
\(\frac{k}{x}\)

2.6 Implicit Differentiation
2.6.4 Exercises

Terms and Concepts

2.6.4.2.
Answer.
\(\text{the chain rule}\)
2.6.4.3.
Answer.
\(\text{True}\)
2.6.4.4.
Answer.
\(\text{True}\)

Problems

2.6.4.5.
Answer.
\(\frac{1}{2\sqrt{w}}+\frac{\frac{1}{2\sqrt{w}}}{\left(\sqrt{w}\right)^{2}}\)
2.6.4.6.
Answer.
\({\frac{1}{6}}\frac{1}{\left(\sqrt[6]{y}\right)^{5}}+\left({\frac{5}{6}}\right)\frac{1}{y^{0.166667}}\)
2.6.4.7.
Answer.
\(\frac{1}{2\sqrt{9+t^{2}}}\cdot 2t\)
2.6.4.8.
Answer.
\(\frac{1}{2\sqrt{w}}\tan\mathopen{}\left(w\right)+\sec^{2}\mathopen{}\left(w\right)\sqrt{w}\)
2.6.4.9.
Answer.
\(1.2y^{0.2}\)
2.6.4.10.
Answer.
\(\pi r^{\pi -1}+3.8r^{2.8}\)
2.6.4.11.
Answer.
\(\frac{\sqrt{w}-\left(w-8\right)\frac{1}{2\sqrt{w}}}{\left(\sqrt{w}\right)^{2}}\)
2.6.4.12.
Answer.
\({\frac{1}{6}}\frac{1}{\left(\sqrt[6]{x}\right)^{5}}\mathopen{}\left(\cos\mathopen{}\left(x\right)+e^{x}\right)+\left(e^{x}-\sin\mathopen{}\left(x\right)\right)\sqrt[6]{x}\)
2.6.4.13.
Answer.
\(\frac{-4x^{3}}{2y+1}\)
2.6.4.14.
Answer.
\(\frac{-y^{0.6}}{x^{0.6}}\)
2.6.4.15.
Answer.
\(\sin\mathopen{}\left(x\right)\sec\mathopen{}\left(y\right)\)
2.6.4.16.
Answer.
\(\frac{y}{x}\)
2.6.4.17.
Answer.
\(\frac{y}{x}\)
2.6.4.18.
Answer.
\(\frac{-\left(e^{x}x\mathopen{}\left(x+2\right)\cdot 2^{-y}\right)}{\ln\mathopen{}\left(2\right)}\)
2.6.4.19.
Answer.
\(\frac{-2\sin\mathopen{}\left(y\right)\cos\mathopen{}\left(y\right)}{x}\)
2.6.4.20.
Answer.
\(-\frac{x}{y^{2}}\)
2.6.4.21.
Answer.
\(\frac{1}{2y+2}\)
2.6.4.22.
Answer.
\(\frac{y-x^{2}-2xy^{2}}{x-y^{2}-2x^{2}y}\)
2.6.4.23.
Answer.
\(\frac{1-\cos\mathopen{}\left(x\right)}{\sin\mathopen{}\left(y\right)+1}\)
2.6.4.24.
Answer.
\(\frac{-x}{y}\)
2.6.4.25.
Answer.
\(\frac{-\left(2x+y\right)}{2y+x}\)
2.6.4.27.
2.6.4.27.a
Answer.
\(y = 0\)
2.6.4.27.b
Answer.
\(y = -1.859\mathopen{}\left(x-0.1\right)+0.2811\)
2.6.4.28.
2.6.4.28.a
Answer.
\(x = 1\)
2.6.4.28.b
Answer.
\(y = \frac{-3\sqrt{3}}{8}\mathopen{}\left(x-\sqrt{0.6}\right)+\sqrt{0.8}\)
2.6.4.29.
2.6.4.29.a
Answer.
\(y = 4\)
2.6.4.29.b
Answer.
\(y = \frac{3}{108^{\frac{1}{4}}}\mathopen{}\left(x-2\right)-108^{\frac{1}{4}}\)
2.6.4.30.
2.6.4.30.a
Answer.
\(y = -x+1\)
2.6.4.30.b
Answer.
\(y = \frac{3\sqrt{3}}{4}\)
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.
\(y = \frac{\sqrt{3}\mathopen{}\left(x-\left(4+3\sqrt{3}\right)\right)}{2}+\frac{3}{2}\)
2.6.4.32.
2.6.4.32.a
Answer.
\(y = 1\)
2.6.4.32.b
Answer.
\(y = \frac{-2}{\sqrt{5}}\mathopen{}\left(x+1\right)+\frac{1}{2}\mathopen{}\left(-1+\sqrt{5}\right)\)
2.6.4.32.c
Answer.
\(y = \frac{2}{\sqrt{5}}\mathopen{}\left(x+1\right)+\frac{1}{2}\mathopen{}\left(-1-\sqrt{5}\right)\)
2.6.4.33.
Answer.
\(\frac{-\left(\left(2y+1\right)\cdot 12x^{2}-4x^{3}\frac{2\mathopen{}\left(-\left(4x^{3}\right)\right)}{2y+1}\right)}{\left(2y+1\right)^{2}}\)
2.6.4.34.
Answer.
\(\frac{-\left(\frac{x^{0.6}\cdot 3}{5}y^{-0.4}\frac{-y^{0.6}}{x^{0.6}}-\frac{y^{0.6}\cdot 3}{5}x^{-0.4}\right)}{x^{1.2}}\)
2.6.4.35.
Answer.
\(\sin^{2}\mathopen{}\left(x\right)\sec^{2}\mathopen{}\left(y\right)\tan\mathopen{}\left(y\right)+\cos\mathopen{}\left(x\right)\sec\mathopen{}\left(y\right)\)
2.6.4.36.
Answer.
\(0\)
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.
\(y = \left(1-2\ln\mathopen{}\left(2\right)\right)\mathopen{}\left(x-1\right)+2\)
2.6.4.38.
Answer 1.
\(\left(2x\right)^{x^{2}}\mathopen{}\left(2x\ln\mathopen{}\left(2x\right)+x\right)\)
Answer 2.
\(y = \left(2+4\ln\mathopen{}\left(2\right)\right)\mathopen{}\left(x-1\right)+2\)
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.
\(y = \frac{1}{4}\mathopen{}\left(x-1\right)+\frac{1}{2}\)
2.6.4.40.
Answer 1.
\(x^{\sin\mathopen{}\left(x\right)+2}\mathopen{}\left(\cos\mathopen{}\left(x\right)\ln\mathopen{}\left(x\right)+\frac{\sin\mathopen{}\left(x\right)+2}{x}\right)\)
Answer 2.
\(y = \frac{3\pi ^{2}}{4}\mathopen{}\left(x-\frac{\pi }{2}\right)+\left(\frac{\pi }{2}\right)^{3}\)
2.6.4.41.
Answer 1.
\(\frac{x+1}{x+2}\mathopen{}\left(\frac{1}{x+1}-\frac{1}{x+2}\right)\)
Answer 2.
\(y = \frac{1}{9}\mathopen{}\left(x-1\right)+\frac{2}{3}\)
2.6.4.42.
Answer 1.
\(\frac{\left(x+1\right)\mathopen{}\left(x+2\right)}{\left(x+3\right)\mathopen{}\left(x+4\right)}\mathopen{}\left(\frac{1}{x+1}+\frac{1}{x+2}-\frac{1}{x+3}-\frac{1}{x+4}\right)\)
Answer 2.
\(y = \frac{11}{72}x+\frac{1}{6}\)

2.7 Derivatives of Inverse Functions

Exercises

Terms and Concepts
2.7.1.
Answer.
\(\text{False}\)
Problems
2.7.9.
Answer.
\({\frac{1}{7}}\)
2.7.10.
Answer.
\(-{\frac{1}{14}}\)
2.7.11.
Answer.
\(-0.5\)
2.7.12.
Answer.
\({\frac{1}{132}}\)
2.7.13.
Answer.
\(-{\frac{25}{4}}\)
2.7.14.
Answer.
\({\frac{1}{12}}\)
2.7.15.
Answer.
\(-\frac{1}{\sqrt{1-\left(4w\right)^{2}}}\cdot 4\)
2.7.16.
Answer.
\(-\frac{1}{\left|7x\right|\sqrt{\left(7x\right)^{2}-1}}\cdot 7\)
2.7.17.
Answer.
\(\frac{1}{1+\left(2r\right)^{2}}\cdot 2\)
2.7.18.
Answer.
\(\cos^{-1}\mathopen{}\left(w\right)-w\frac{1}{\sqrt{1-w^{2}}}\)
2.7.19.
Answer.
\(\left(\sec\mathopen{}\left(x\right)\right)^{2}\cos^{-1}\mathopen{}\left(x\right)-\frac{1}{\sqrt{1-x^{2}}}\tan\mathopen{}\left(x\right)\)
2.7.20.
Answer.
\(\frac{e^{t}}{t}+\ln\mathopen{}\left(t\right)e^{t}\)
2.7.21.
Answer.
\(\frac{\frac{1}{1+z^{2}}\sin^{-1}\mathopen{}\left(z\right)-\frac{1}{\sqrt{1-z^{2}}}\tan^{-1}\mathopen{}\left(z\right)}{\left(\sin^{-1}\mathopen{}\left(z\right)\right)^{2}}\)
2.7.22.
Answer.
\(\left(\sec\mathopen{}\left(\sqrt[4]{x}\right)\right)^{2}{\frac{1}{4}}\frac{1}{\left(\sqrt[4]{x}\right)^{3}}\)
2.7.23.
Answer.
\(\csc\mathopen{}\left(\frac{1}{q^{3}}\right)\cot\mathopen{}\left(\frac{1}{q^{3}}\right)\frac{3q^{2}}{\left(q^{3}\right)^{2}}\)
2.7.24.
Answer.
\(1\)
2.7.29.
Answer.
\(y = 2\mathopen{}\left(x-\frac{-\sqrt{3}}{2}\right)+\left(-\frac{\pi }{3}\right)\)
2.7.30.
Answer.
\(y = -4\mathopen{}\left(x-\frac{\sqrt{3}}{4}\right)+\frac{\pi }{6}\)

3 The Graphical Behavior of Functions
3.1 Extreme Values

Exercises

Terms and Concepts
3.1.2.
Answer.
Answers will vary.
3.1.4.
Answer.
Answers will vary.
3.1.5.
Answer.
\(\text{False}\)
3.1.6.
Answer 1.
\(0\)
Answer 2.
\(\text{undefined}\)
Problems
3.1.7.
Answer 1.
\(\text{B}\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\text{B}, \text{G}\)
Answer 4.
\(\text{C}, \text{F}\)
3.1.8.
Answer 1.
\(\text{C}\)
Answer 2.
\(\text{A}\)
Answer 3.
\(\text{C}\)
Answer 4.
\(\text{A}\)
3.1.9.
Answer.
\(0\)
3.1.10.
Answer 1.
\(0\)
Answer 2.
\(0\)
3.1.11.
Answer 1.
\(0\)
Answer 2.
\(0\)
3.1.12.
Answer 1.
\(0\)
Answer 2.
\(0\)
Answer 3.
\(\text{DNE}\)
3.1.13.
Answer 1.
\(\text{DNE}\)
Answer 2.
\(0\)
3.1.14.
Answer 1.
\(\text{DNE}\)
Answer 2.
\(\text{DNE}\)
3.1.15.
Answer.
\(0\)
3.1.16.
Answer.
\(\text{DNE}\)
3.1.17.
Answer 1.
\(14\)
Answer 2.
\(-2\)
3.1.18.
Answer 1.
\(-6\)
Answer 2.
\(-28\)
3.1.19.
Answer 1.
\(-2.82843\)
Answer 2.
\(-4\)
3.1.20.
Answer 1.
\(30.4664\)
Answer 2.
\(0\)
3.1.21.
Answer 1.
\({\frac{9}{2}}\)
Answer 2.
\(2.82843\)
3.1.22.
Answer 1.
\({\frac{4}{11}}\)
Answer 2.
\(0\)
3.1.23.
Answer 1.
\(\frac{e^{\frac{\pi }{4}}}{\sqrt{2}}\)
Answer 2.
\(-e^{\pi }\)
3.1.24.
Answer 1.
\(\frac{e^{\frac{3\pi }{4}}}{\sqrt{2}}\)
Answer 2.
\(0\)
3.1.25.
Answer 1.
\(\frac{1}{2e}\)
Answer 2.
\(0\)
3.1.26.
Answer 1.
\(0.47247\)
Answer 2.
\(-6.31821\)

3.2 The Mean Value Theorem

Exercises

Problems
3.2.3.
Answer.
Undefined
3.2.4.
Answer.
Undefined
3.2.5.
Answer.
Undefined
3.2.6.
Answer.
Undefined
3.2.7.
Answer.
Undefined
3.2.8.
Answer.
Undefined
3.2.9.
Answer.
Undefined
3.2.10.
Answer.
Undefined
3.2.11.
Answer.
Undefined
3.2.12.
Answer.
Undefined
3.2.13.
Answer.
Undefined
3.2.14.
Answer.
Undefined
3.2.15.
Answer.
Undefined
3.2.16.
Answer.
Undefined
3.2.17.
Answer.
Undefined
3.2.18.
Answer.
Undefined
3.2.19.
Answer.
Undefined
3.2.20.
Answer.
Undefined

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 \(x=2\text{.}\)
3.3.5.
Answer.
\(\text{False}\)
Problems
3.3.15.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(-2\)
Answer 3.
\(\left[-2,\infty \right)\)
Answer 4.
\(\left(-\infty ,-2\right]\)
Answer 5.
\(\text{NONE}\)
Answer 6.
\(-2\)
3.3.16.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(-{\frac{4}{3}}, 0\)
Answer 3.
\(\left(-\infty ,-1.33333\right], \left[0,\infty \right)\)
Answer 4.
\(\left[-1.33333,0\right]\)
Answer 5.
\(-1.33333\)
Answer 6.
\(0\)
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.
\(-{\frac{5}{7}}\)
Answer 6.
\({\frac{7}{3}}\)
3.3.18.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(3\)
Answer 3.
\(\left(-\infty ,\infty \right)\)
Answer 4.
\(\text{NONE}\)
Answer 5.
\(\text{NONE}\)
Answer 6.
\(\text{NONE}\)
3.3.19.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(5\)
Answer 3.
\(\left(-\infty ,5\right]\)
Answer 4.
\(\left[5,\infty \right)\)
Answer 5.
\(5\)
Answer 6.
\(\text{NONE}\)
3.3.20.
Answer 1.
\(\left(-\infty ,-6\right)\cup \left(-6,6\right)\cup \left(6,\infty \right)\)
Answer 2.
\(0\)
Answer 3.
\(\left(-\infty ,-6\right), \left(-6,0\right]\)
Answer 4.
\(\left[0,6\right), \left(6,\infty \right)\)
Answer 5.
\(0\)
Answer 6.
\(\text{NONE}\)
3.3.21.
Answer 1.
\(\left(-\infty ,-7\right)\cup \left(-7,-5\right)\cup \left(-5,\infty \right)\)
Answer 2.
\(-5.91608, 5.91608\)
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.
\(5.91608\)
Answer 6.
\(-5.91608\)
3.3.22.
Answer 1.
\(\left(-\infty ,0\right)\cup \left(0,\infty \right)\)
Answer 2.
\(-5, -15\)
Answer 3.
\(\left[-15,-5\right]\)
Answer 4.
\(\left(-\infty ,-15\right], \left[-5,0\right), \left(0,\infty \right)\)
Answer 5.
\(-5\)
Answer 6.
\(-15\)
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.
\(-2.35619, 0.785398\)
Answer 6.
\(-0.785398, 2.35619\)
3.3.24.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(-2\)
Answer 3.
\(\left[-2,\infty \right)\)
Answer 4.
\(\left(-\infty ,-2\right]\)
Answer 5.
\(\text{NONE}\)
Answer 6.
\(-2\)

3.4 Concavity and the Second Derivative
3.4.3 Exercises

Terms and Concepts

3.4.3.1.
Answer.
Answers will vary.
3.4.3.2.
Answer.
Answers will vary.
3.4.3.3.
Answer.
Yes; Answers will vary.
3.4.3.4.
Answer.
No.

Problems

3.4.3.15.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\left(-\infty ,\infty \right)\)
Answer 3.
\(\text{NONE}\)
3.4.3.16.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\left(-\infty ,\infty \right)\)
3.4.3.17.
Answer 1.
\(0\)
Answer 2.
\(\left[0,\infty \right)\)
Answer 3.
\(\left(-\infty ,0\right]\)
3.4.3.18.
Answer 1.
\(-{\frac{1}{4}}\)
Answer 2.
\(\left[-0.25,\infty \right)\)
Answer 3.
\(\left(-\infty ,-0.25\right]\)
3.4.3.19.
Answer 1.
\(-{\frac{32}{3}}, 0\)
Answer 2.
\(\left(-\infty ,-10.6667\right], \left[0,\infty \right)\)
Answer 3.
\(\left[-10.6667,0\right]\)
3.4.3.20.
Answer 1.
\(4.42265, 5.57735\)
Answer 2.
\(\left(-\infty ,4.42265\right], \left[5.57735,\infty \right)\)
Answer 3.
\(\left[4.42265,5.57735\right]\)
3.4.3.21.
Answer 1.
\(-2\)
Answer 2.
\(\left(-\infty ,\infty \right)\)
Answer 3.
\(\text{NONE}\)
3.4.3.22.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\left(-1.5708,1.5708\right)\)
Answer 3.
\(\left(-4.71239,-1.5708\right), \left(1.5708,4.71239\right)\)
3.4.3.23.
Answer 1.
\(-0.57735, 0.57735\)
Answer 2.
\(\left(-\infty ,-0.57735\right], \left[0.57735,\infty \right)\)
Answer 3.
\(\left[-0.57735,0.57735\right]\)
3.4.3.24.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\left(-\infty ,2\right), \left(5,\infty \right)\)
Answer 3.
\(\left(2,5\right)\)
3.4.3.25.
Answer 1.
\(-0.785398, 2.35619\)
Answer 2.
\(\left(-3.14159,-0.785398\right], \left[2.35619,3.14159\right)\)
Answer 3.
\(\left[-0.785398,2.35619\right]\)
3.4.3.26.
Answer 1.
\(-0.585786, -3.41421\)
Answer 2.
\(\left(-\infty ,-3.41421\right], \left[-0.585786,\infty \right)\)
Answer 3.
\(\left[-3.41421,-0.585786\right]\)
3.4.3.27.
Answer 1.
\(0.22313\)
Answer 2.
\(\left[0.22313,\infty \right)\)
Answer 3.
\(\left(0,0.22313\right]\)
3.4.3.28.
Answer 1.
\(0.707107, -0.707107\)
Answer 2.
\(\left(-\infty ,-0.707107\right], \left[0.707107,\infty \right)\)
Answer 3.
\(\left[-0.707107,0.707107\right]\)
3.4.3.29.
Answer 1.
\(-7\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(-7\)
3.4.3.30.
Answer 1.
\(-{\frac{5}{2}}\)
Answer 2.
\(-{\frac{5}{2}}\)
Answer 3.
\(\text{NONE}\)
3.4.3.31.
Answer 1.
\(-1.1547, 1.1547\)
Answer 2.
\(-1.1547\)
Answer 3.
\(1.1547\)
3.4.3.32.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\text{NONE}\)
3.4.3.33.
Answer 1.
\(-4\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(-4\)
3.4.3.34.
Answer 1.
\(-3, -2, 2\)
Answer 2.
\(-2\)
Answer 3.
\(-3, 2\)
3.4.3.35.
Answer 1.
\(3\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\text{NONE}\)
3.4.3.36.
Answer 1.
\(-3.14159, 0, 3.14159\)
Answer 2.
\(-3.14159, 3.14159\)
Answer 3.
\(0\)
3.4.3.37.
Answer 1.
\(-9\)
Answer 2.
\(-9\)
Answer 3.
\(\text{NONE}\)
3.4.3.38.
Answer 1.
\(0\)
Answer 2.
\(0\)
Answer 3.
\(\text{NONE}\)
3.4.3.39.
Answer 1.
\(-2.35619, 0.785398\)
Answer 2.
\(0.785398\)
Answer 3.
\(-2.35619\)
3.4.3.40.
Answer 1.
\(-2, 0\)
Answer 2.
\(-2\)
Answer 3.
\(0\)
3.4.3.41.
Answer 1.
\(0.606531\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(0.606531\)
3.4.3.42.
Answer 1.
\(0\)
Answer 2.
\(0\)
Answer 3.
\(\text{NONE}\)
3.4.3.43.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
3.4.3.44.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
3.4.3.45.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(0\)
3.4.3.46.
Answer 1.
\(-{\frac{8}{27}}\)
Answer 2.
\(\text{NONE}\)
3.4.3.47.
Answer 1.
\(-{\frac{28}{3}}\)
Answer 2.
\(0\)
3.4.3.48.
Answer 1.
\(1.42265\)
Answer 2.
\(2.57735\)
3.4.3.49.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
3.4.3.50.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
3.4.3.51.
Answer 1.
\(0\)
Answer 2.
\(2\)
3.4.3.52.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
3.4.3.53.
Answer 1.
\(-0.785398\)
Answer 2.
\(2.35619\)
3.4.3.54.
Answer 1.
\(-3.41421\)
Answer 2.
\(-0.585786\)
3.4.3.55.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(0.22313\)
3.4.3.56.
Answer 1.
\(-0.707107\)
Answer 2.
\(0.707107\)

3.5 Curve Sketching

Exercises

Terms and Concepts
3.5.3.
Answer.
\(\text{True}\)
3.5.4.
Answer.
\(\text{True}\)
3.5.5.
Answer.
\(\text{True}\)

4 Applications of the Derivative
4.1 Newton’s Method

Exercises

Terms and Concepts
4.1.1.
Answer.
\(\text{False}\)
4.1.2.
Answer.
\(\text{False}\)
Problems
4.1.3.
Answer 1.
\(1.57091\)
Answer 2.
\(1.5708\)
Answer 3.
\(1.5708\)
Answer 4.
\(1.5708\)
Answer 5.
\(1.5708\)
4.1.4.
Answer 1.
\(-0.557408\)
Answer 2.
\(0.0659365\)
Answer 3.
\(-9.57219\times 10^{-5}\)
Answer 4.
\(0\)
Answer 5.
\(0\)
4.1.5.
Answer 1.
\(2\)
Answer 2.
\(1.2\)
Answer 3.
\(1.01176\)
Answer 4.
\(1.00005\)
Answer 5.
\(1\)
4.1.6.
Answer 1.
\(1.41667\)
Answer 2.
\(1.41422\)
Answer 3.
\(1.41421\)
Answer 4.
\(1.41421\)
Answer 5.
\(1.41421\)
4.1.7.
Answer 1.
\(0.613706\)
Answer 2.
\(0.913341\)
Answer 3.
\(0.996132\)
Answer 4.
\(0.999993\)
Answer 5.
\(1\)
4.1.8.
Answer 1.
\(1.44444\)
Answer 2.
\(1.13057\)
Answer 3.
\(1.01498\)
Answer 4.
\(1.00022\)
Answer 5.
\(1\)
4.1.9.
Answer.
\(\left\{-5.15633,-0.369102,0.525428\right\}\)
4.1.10.
Answer.
\(\left\{-3.71448,-0.856723,1,1.5712\right\}\)
4.1.11.
Answer.
\(\left\{-1.0134,0.988312,1.39341\right\}\)
4.1.12.
Answer.
\(\left\{-2.16477,0,0.524501,1.81328\right\}\)
4.1.13.
Answer.
\(\left\{-0.824132,0.824132\right\}\)
4.1.14.
Answer.
\(\left\{-0.636733,1.40962\right\}\)
4.1.15.
Answer.
\(\left\{0\right\}\)
4.1.16.
Answer.
\(\left\{-4.49341,0,4.49341\right\}\)

4.2 Related Rates

Exercises

Terms and Concepts
4.2.1.
Answer.
\(\text{True}\)
4.2.2.
Answer.
\(\text{False}\)
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.
\(0.00198944\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.4.
4.2.4.a
Answer.
\(0.397887\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.4.b
Answer.
\(0.00397887\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.4.c
Answer.
\(3.97887\times 10^{-5}\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.5.
Answer.
\(51.066\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut h}}\)
4.2.6.
4.2.6.a
Answer.
\(68.75\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut h}}\)
4.2.6.b
Answer.
\(75\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut h}}\)
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.
\(424\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
4.2.8.
4.2.8.a
Answer.
\(0.0225641\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut s}}\)
4.2.8.b
Answer.
\(0.553459\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut s}}\)
4.2.8.c
Answer.
\(7.33333\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut s}}\)
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
Answer.
\(\infty \)
4.2.10.
4.2.10.a
Answer.
\(30.5941\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut min}}\)
4.2.10.b
Answer.
\(36.0555\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut min}}\)
4.2.10.c
Answer.
\(301.496\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut min}}\)
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
Answer.
\(381.791\ {\rm s}\)
4.2.12.
4.2.12.a
Answer.
\(0.632456\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.12.b
Answer.
\(1.6\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.12.c
Answer.
\(51.9615\ {\rm ft}\)
4.2.13.
4.2.13.a
Answer.
\(80\ {\rm ft}\)
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
Answer.
\(74.162\ {\rm ft}\)
4.2.14.
4.2.14.a
Answer.
\(96\ {\rm ft}\)
4.2.14.b
Answer.
\(9.42478\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.15.
Answer.
\(0.00230973\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)

4.3 Optimization

Exercises

Terms and Concepts
4.3.1.
Answer.
\(\text{True}\)
4.3.2.
Answer.
\(\text{False}\)
Problems
4.3.3.
Answer.
\(5625\)
4.3.4.
Answer.
\(2\sqrt{560}\)
4.3.5.
Answer.
\(\text{DNE}\)
4.3.6.
Answer.
\({\frac{8450}{29}}\)
4.3.7.
Answer.
\(1\)
4.3.8.
Answer.
\(150\ {\rm ft};\,\left({\frac{225}{2}}\right)\ {\rm ft}\)
4.3.9.
Answer 1.
\(3.83722\ {\rm cm}\)
Answer 2.
\(7.67443\ {\rm cm}\)
4.3.10.
Answer 1.
\(3.20058\ {\rm in}\)
Answer 2.
\(6.40117\ {\rm in}\)
4.3.11.
Answer 1.
\(3.0456\ {\rm cm}\)
Answer 2.
\(12.1824\ {\rm cm}\)
4.3.12.
Answer.
\(11664\ {\rm in^{3}}\)
4.3.13.
Answer.
\(10.3923\ {\rm in};\,14.6969\ {\rm in}\)
4.3.14.
Answer 1.
\(0.535898\ {\rm mi}\)
Answer 2.
\(\$503{,}730.67\)
4.3.15.
Answer 1.
\(0\ {\rm mi}\)
Answer 2.
\(\$474{,}341.65\)
4.3.16.
Answer.
\(33.6239\ {\rm ft}\)
4.3.17.
Answer.
\(23.7599\ {\rm ft}\)
4.3.18.
Answer.
\(\sqrt{2};\,\sqrt{2}\)

4.4 Differentials

Exercises

Terms and Concepts
4.4.1.
Answer.
\(\text{True}\)
4.4.2.
Answer.
\(\text{True}\)
4.4.3.
Answer.
\(\text{False}\)
4.4.4.
Answer.
\(\text{True}\)
4.4.6.
Answer.
\(\text{True}\)
Problems
4.4.7.
Answer.
\(4.28\)
4.4.8.
Answer.
\(8.7\)
4.4.9.
Answer.
\(83.2\)
4.4.10.
Answer.
\(102.5\)
4.4.11.
Answer.
\(5.05\)
4.4.12.
Answer.
\(5.88333\)
4.4.13.
Answer.
\(4.98667\)
4.4.14.
Answer.
\(6.00556\)
4.4.15.
Answer.
\(0.141593\)
4.4.16.
Answer.
\(1.1\)
4.4.17.
Answer.
\(\left(2x-5\right)dx\)
4.4.18.
Answer.
\(\left(5x^{4}+9x^{8}\right)dx\)
4.4.19.
Answer.
\(-\frac{24x^{5}}{\left(4x^{6}\right)^{2}}dx\)
4.4.20.
Answer.
\(2\mathopen{}\left(6x+\sin\mathopen{}\left(x\right)\right)\mathopen{}\left(6+\cos\mathopen{}\left(x\right)\right)dx\)
4.4.21.
Answer.
\(\left(7x^{6}+8e^{8x}\right)dx\)
4.4.22.
Answer.
\(-\frac{40x^{4}}{\left(x^{5}\right)^{2}}dx\)
4.4.23.
Answer.
\(\frac{9\mathopen{}\left(\tan\mathopen{}\left(x\right)+2\right)-9x\sec^{2}\mathopen{}\left(x\right)}{\left(\tan\mathopen{}\left(x\right)+2\right)^{2}}dx\)
4.4.24.
Answer.
\(\frac{9}{9x}dx\)
4.4.25.
Answer.
\(\left(e^{x}\sin\mathopen{}\left(x\right)+e^{x}\cos\mathopen{}\left(x\right)\right)dx\)
4.4.26.
Answer.
\(-\sin\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)\cos\mathopen{}\left(x\right)dx\)
4.4.27.
Answer.
\(\frac{x+5-\left(x-4\right)}{\left(x+5\right)^{2}}dx\)
4.4.28.
Answer.
\(\left(1.60944\cdot 5^{x}\ln\mathopen{}\left(x\right)+\frac{5^{x}\cdot 1}{x}\right)dx\)
4.4.29.
Answer.
\(\tan^{-1}\mathopen{}\left(x\right)dx\)
4.4.30.
Answer.
\(\cot\mathopen{}\left(x\right)dx\)
4.4.31.
Answer.
\(5.02655\ {\rm cm^{3}}\)
4.4.32.
4.4.32.a
Answer.
\(51.2\)
4.4.32.b
Answer.
\(76.8\)
4.4.33.
Answer.
\(3.92699\)
4.4.34.
Answer.
\(-4\ {\rm ft^{2}}\)
4.4.35.
4.4.35.a
Answer.
\(297.717\ {\rm ft}\)
4.4.35.b
Answer.
\(62.3155\ {\rm ft}\)
4.4.35.c
Answer.
\(20.9\%\)
4.4.36.
4.4.36.a
Answer.
\(298.868\ {\rm ft}\)
4.4.36.b
Answer.
\(17.335\ {\rm ft}\)
4.4.36.c
Answer.
\(5.8\%\)
4.4.37.
4.4.37.a
Answer.
\(298.868\ {\rm ft}\)
4.4.37.b
Answer.
\(8.66751\ {\rm ft}\)
4.4.37.c
Answer.
\(2.9\%\)
4.4.38.
Answer.
\(\text{Isosceles ... feet}\)
4.4.39.
Answer.
\(1\%\)

4.5 Taylor Polynomials

Exercises

Terms and Concepts
4.5.2.
Answer.
\(\text{True}\)
4.5.3.
Answer.
\(6+3x-4x^{2}\)
4.5.4.
Answer.
\(30\)
Problems
4.5.5.
Answer.
\(1-x+0.5x^{2}-0.166667x^{3}\)
4.5.6.
Answer.
\(x-0.166667x^{3}+0.00833333x^{5}-0.000198413x^{7}\)
4.5.7.
Answer.
\(x+x^{2}+0.5x^{3}+0.166667x^{4}+0.0416667x^{5}\)
4.5.8.
Answer.
\(x+0.333333x^{3}+0.133333x^{5}\)
4.5.9.
Answer.
\(1+2x+2x^{2}+1.33333x^{3}+0.666667x^{4}\)
4.5.10.
Answer.
\(1+x+x^{2}+x^{3}+x^{4}\)
4.5.11.
Answer.
\(1-x+x^{2}-x^{3}+x^{4}\)
4.5.12.
Answer.
\(1-x+x^{2}-x^{3}+x^{4}-x^{5}+x^{6}-x^{7}\)
4.5.13.
Answer.
\(1+0.5\mathopen{}\left(x-1\right)-0.125\mathopen{}\left(x-1\right)^{2}+0.0625\mathopen{}\left(x-1\right)^{3}-0.0390625\mathopen{}\left(x-1\right)^{4}\)
4.5.14.
Answer.
\(0.693147+0.5\mathopen{}\left(x-1\right)-0.125\mathopen{}\left(x-1\right)^{2}+0.0416667\mathopen{}\left(x-1\right)^{3}-0.015625\mathopen{}\left(x-1\right)^{4}\)
4.5.15.
Answer.
\(0.707107-0.707107\mathopen{}\left(x-\frac{\pi }{4}\right)-0.353553\mathopen{}\left(x-\frac{\pi }{4}\right)^{2}+0.117851\mathopen{}\left(x-\frac{\pi }{4}\right)^{3}+0.0294628\mathopen{}\left(x-\frac{\pi }{4}\right)^{4}-0.00589256\mathopen{}\left(x-\frac{\pi }{4}\right)^{5}-0.000982093\mathopen{}\left(x-\frac{\pi }{4}\right)^{6}\)
4.5.16.
Answer.
\(0.5+0.866025\mathopen{}\left(x-\frac{\pi }{6}\right)-0.25\mathopen{}\left(x-\frac{\pi }{6}\right)^{2}-0.144338\mathopen{}\left(x-\frac{\pi }{6}\right)^{3}+0.0208333\mathopen{}\left(x-\frac{\pi }{6}\right)^{4}+0.00721688\mathopen{}\left(x-\frac{\pi }{6}\right)^{5}\)
4.5.17.
Answer.
\(0.5-0.25\mathopen{}\left(x-2\right)+0.125\mathopen{}\left(x-2\right)^{2}-0.0625\mathopen{}\left(x-2\right)^{3}+0.03125\mathopen{}\left(x-2\right)^{4}+0.015625\mathopen{}\left(x-2\right)^{5}\)
4.5.18.
Answer.
\(1-2\mathopen{}\left(x-1\right)+3\mathopen{}\left(x-1\right)^{2}-4\mathopen{}\left(x-1\right)^{3}+5\mathopen{}\left(x-1\right)^{4}-6\mathopen{}\left(x-1\right)^{5}+7\mathopen{}\left(x-1\right)^{6}-8\mathopen{}\left(x-1\right)^{7}+9\mathopen{}\left(x-1\right)^{8}\)
4.5.19.
Answer.
\(0.5+0.5\mathopen{}\left(x+1\right)+0.25\mathopen{}\left(x+1\right)^{2}\)
4.5.20.
Answer.
\(-\pi ^{2}-2\pi \mathopen{}\left(x-\pi \right)+\frac{\pi ^{2}-2}{2}\mathopen{}\left(x-\pi \right)^{2}\)
4.5.31.
Answer.
The \(n\)th term is: when \(n\) even, 0; when \(n\) is odd, \(\frac{(-1)^{(n-1)/2}}{n!}x^n\text{.}\)

5 Integration
5.1 Antiderivatives and Indefinite Integration

Exercises

Terms and Concepts
5.1.2.
Answer.
\(\text{an antiderivative}\)
5.1.4.
Answer 1.
\(\text{opposite}\)
Answer 2.
\(\text{opposite}\)
5.1.6.
Answer.
\(\text{velocity}\)
5.1.7.
Answer.
\(\text{velocity}\)
5.1.8.
Answer.
\(F\mathopen{}\left(x\right)+G\mathopen{}\left(x\right)\)
Problems
5.1.9.
Answer.
\(\left({\frac{4}{3}}\right)x^{6}+C\)
5.1.10.
Answer.
\({\frac{1}{10}}x^{10}+C\)
5.1.11.
Answer.
\(\left({\frac{5}{9}}\right)x^{9}-6x+C\)
5.1.12.
Answer.
\(t+C\)
5.1.13.
Answer.
\(s+C\)
5.1.14.
Answer.
\(C-\frac{1}{35t^{7}}\)
5.1.15.
Answer.
\(C-\frac{2}{t^{3}}\)
5.1.16.
Answer.
\(2\sqrt{x}+C\)
5.1.17.
Answer.
\(\sec\mathopen{}\left(\theta\right)+C\)
5.1.18.
Answer.
\(-\cos\mathopen{}\left(\theta\right)+C\)
5.1.19.
Answer.
\(\sec\mathopen{}\left(x\right)+\csc\mathopen{}\left(x\right)+C\)
5.1.20.
Answer.
\(2e^{\theta}+C\)
5.1.21.
Answer.
\(\frac{3^{t}}{\ln\mathopen{}\left(3\right)}+C\)
5.1.22.
Answer.
\(\frac{4^{t}}{9\ln\mathopen{}\left(4\right)}+C\)
5.1.23.
Answer.
\(\left({\frac{25}{3}}\right)t^{3}+10t^{2}+4t+\left({\frac{8}{15}}\right)+C\)
5.1.24.
Answer.
\(\frac{t^{10}}{10}-\frac{t^{6}}{2}-5t^{2}+C\)
5.1.25.
Answer.
\(\frac{x^{17}}{17}+C\)
5.1.26.
Answer.
\(1.41421^{e}x+C\)
5.1.27.
Answer.
\(rx+C\)
5.1.30.
Answer.
\(8-\cos\mathopen{}\left(x\right)\)
5.1.31.
Answer.
\(2e^{x}+6\)
5.1.32.
Answer.
\(3\frac{x^{4}}{4}-3x^{2}+9\)
5.1.33.
Answer.
\(\sec\mathopen{}\left(x\right)+4\)
5.1.34.
Answer.
\(\frac{5^{x}}{\ln\mathopen{}\left(5\right)}-\frac{25}{\ln\mathopen{}\left(5\right)}+5\)
5.1.35.
Answer.
\(3x^{2}+2x+5\)
5.1.36.
Answer.
\(\left({\frac{2}{3}}\right)x^{3}+7x+\left(-{\frac{5}{3}}\right)\)
5.1.37.
Answer.
\(7e^{x}-10x-15\)
5.1.38.
Answer.
\(6\theta-\cos\mathopen{}\left(\theta\right)+10\)
5.1.39.
Answer.
\(x^{6}+\frac{2^{x}}{0.480453}-\cos\mathopen{}\left(x\right)-1.4427x+0.918631\)
5.1.40.
Answer.
\(-\left(2x+11\right)\)

5.2 The Definite Integral

Exercises

Terms and Concepts
5.2.3.
Answer.
\(0\)
5.2.4.
Answer.
\(\int 0^2 (2x+3)\, dx\)
Problems
5.2.5.
5.2.5.a
Answer.
\(3\)
5.2.5.b
Answer.
\(4\)
5.2.5.c
Answer.
\(3\)
5.2.5.d
Answer.
\(0\)
5.2.5.e
Answer.
\(-4\)
5.2.5.f
Answer.
\(9\)
5.2.6.
5.2.6.a
Answer.
\(-4\)
5.2.6.b
Answer.
\(-5\)
5.2.6.c
Answer.
\(-3\)
5.2.6.d
Answer.
\(1\)
5.2.6.e
Answer.
\(-2\)
5.2.6.f
Answer.
\(10\)
5.2.7.
5.2.7.a
Answer.
\(4\)
5.2.7.b
Answer.
\(2\)
5.2.7.c
Answer.
\(4\)
5.2.7.d
Answer.
\(2\)
5.2.7.e
Answer.
\(1\)
5.2.7.f
Answer.
\(2\)
5.2.8.
5.2.8.a
Answer.
\(-{\frac{1}{2}}\)
5.2.8.b
Answer.
\(0\)
5.2.8.c
Answer.
\({\frac{3}{2}}\)
5.2.8.d
Answer.
\({\frac{3}{2}}\)
5.2.8.e
Answer.
\({\frac{9}{2}}\)
5.2.8.f
Answer.
\({\frac{15}{2}}\)
5.2.9.
5.2.9.a
Answer.
\(\pi \)
5.2.9.b
Answer.
\(\pi \)
5.2.9.c
Answer.
\(2\pi \)
5.2.9.d
Answer.
\(10\pi \)
5.2.10.
5.2.10.a
Answer.
\(15\)
5.2.10.b
Answer.
\(12\)
5.2.10.c
Answer.
\(0\)
5.2.10.d
Answer.
\(3\mathopen{}\left(b-a\right)\)
5.2.11.
5.2.11.a
Answer.
\(-59\)
5.2.11.b
Answer.
\(-48\)
5.2.11.c
Answer.
\(-27\)
5.2.11.d
Answer.
\(-33\)
5.2.12.
5.2.12.a
Answer.
\(\frac{4}{\pi }\)
5.2.12.b
Answer.
\(\frac{-4}{\pi }\)
5.2.12.c
Answer.
\(0\)
5.2.12.d
Answer.
\(\frac{2}{\pi }\)
5.2.13.
5.2.13.a
Answer.
\(4\)
5.2.13.b
Answer.
\(4\)
5.2.13.c
Answer.
\(-4\)
5.2.13.d
Answer.
\(-2\)
5.2.14.
5.2.14.a
Answer.
\({\frac{40}{3}}\)
5.2.14.b
Answer.
\({\frac{26}{3}}\)
5.2.14.c
Answer.
\({\frac{8}{3}}\)
5.2.14.d
Answer.
\({\frac{38}{3}}\)
5.2.15.
5.2.15.a
Answer.
\(2\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.2.15.b
Answer.
\(2\ {\rm ft}\)
5.2.15.c
Answer.
\(1.5\ {\rm ft}\)
5.2.16.
5.2.16.a
Answer.
\(3\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.2.16.b
Answer.
\(9.5\ {\rm ft}\)
5.2.16.c
Answer.
\(9.5\ {\rm ft}\)
5.2.17.
5.2.17.a
Answer.
\(64\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.2.17.b
Answer.
\(64\ {\rm ft}\)
5.2.17.c
Answer.
\(2\ {\rm s}\)
5.2.17.d
Answer.
\(4.64575\ {\rm s}\)
5.2.18.
5.2.18.a
Answer.
\(96\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.2.18.b
Answer.
\(6\ {\rm s}\)
5.2.18.c
Answer.
\(6\ {\rm s}\)
5.2.18.d
Answer.
\(208\ {\rm ft}\)
5.2.19.
Answer.
\(2\)
5.2.20.
Answer.
\(5\)
5.2.21.
Answer.
\(16\)
5.2.22.
Answer.
\(a = -{\frac{2}{7}}b\)
5.2.23.
Answer.
\(22\)
5.2.24.
Answer.
\(-7\)
5.2.25.
Answer.
\(0\)
5.2.26.
Answer.
\(a = -{\frac{18}{11}}b\)

5.3 Riemann Sums
5.3.4 Exercises

Terms and Concepts

5.3.4.1.
Answer.
\(\text{limits}\)
5.3.4.2.
Answer.
\(12\)
5.3.4.3.
Answer.
\(\text{rectangles}\)
5.3.4.4.
Answer.
\(\text{True}\)

Problems

5.3.4.5.
Answer 1.
\(9+16+25+36\)
Answer 2.
\(86\)
5.3.4.6.
Answer 1.
\(-4+\left(-1\right)+2+5+8\)
Answer 2.
\(10\)
5.3.4.7.
Answer 1.
\(0+\left(-1\right)+0+1\)
Answer 2.
\(0\)
5.3.4.8.
Answer 1.
\(9+9+9+9+9+9+9+9\)
Answer 2.
\(72\)
5.3.4.9.
Answer 1.
\(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\)
Answer 2.
\({\frac{49}{20}}\)
5.3.4.10.
Answer 1.
\(-1+2+\left(-3\right)+4+\left(-5\right)+6+\left(-7\right)+8\)
Answer 2.
\(4\)
5.3.4.11.
Answer 1.
\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}\)
Answer 2.
\({\frac{3}{4}}\)
5.3.4.12.
Answer 1.
\(1+1+1+1+1+1\)
Answer 2.
\(6\)
5.3.4.13.
Answer.
\(1;\,4;\,3i\)
5.3.4.14.
Answer.
\(0;\,6;\,i^{2}+2\)
5.3.4.15.
Answer.
\(1;\,5;\,\frac{i}{i+3}\)
5.3.4.16.
Answer.
\(1;\,5;\,-\left(-e\right)^{i}\)
5.3.4.17.
Answer.
\(72\)
5.3.4.18.
Answer.
\(435\)
5.3.4.19.
Answer.
\(1456\)
5.3.4.20.
Answer.
\(30336\)
5.3.4.21.
Answer.
\(-3220\)
5.3.4.22.
Answer.
\(-687\)
5.3.4.23.
Answer.
\(4560\)
5.3.4.24.
Answer.
\(4324\)
5.3.4.25.
Answer.
\(135\)
5.3.4.26.
Answer.
\(146340\)
5.3.4.27.
Answer.
\(21\)
5.3.4.28.
Answer.
\(106272\)
5.3.4.35.
Answer 1.
\(\frac{\left(n-1\right)^{2}}{4n^{2}}\)
Answer 2.
\(0.2025\)
Answer 3.
\(0.245025\)
Answer 4.
\(0.2495\)
Answer 5.
\({\frac{1}{4}}\)
5.3.4.36.
Answer 1.
\(6+\frac{9}{1n}+\frac{9}{1n^{2}}\)
Answer 2.
\(6.99\)
Answer 3.
\(6.0909\)
Answer 4.
\(6.00901\)
Answer 5.
\(6\)
5.3.4.37.
Answer 1.
\(36\)
Answer 2.
\(36\)
Answer 3.
\(36\)
Answer 4.
\(36\)
Answer 5.
\(36\)
5.3.4.38.
Answer 1.
\(\left({\frac{212}{3}}\right)+\frac{-48}{1n}+\frac{16}{3n^{2}}\)
Answer 2.
\(65.92\)
Answer 3.
\(70.1872\)
Answer 4.
\(70.6187\)
Answer 5.
\({\frac{212}{3}}\)
5.3.4.39.
Answer 1.
\(132-\frac{242}{n}\)
Answer 2.
\(107.8\)
Answer 3.
\(129.58\)
Answer 4.
\(131.758\)
Answer 5.
\(132\)
5.3.4.40.
Answer 1.
\(-{\frac{1}{12}}+\frac{1}{12n^{2}}\)
Answer 2.
\(-0.0825\)
Answer 3.
\(-0.083325\)
Answer 4.
\(-0.0833332\)
Answer 5.
\(-{\frac{1}{12}}\)

5.4 The Fundamental Theorem of Calculus
5.4.6 Exercises

Terms and Concepts

5.4.6.2.
Answer.
\(0\)
5.4.6.3.
Answer.
\(\text{True}\)

Problems

5.4.6.5.
Answer.
\(4\)
5.4.6.6.
Answer.
\({\frac{65}{3}}\)
5.4.6.7.
Answer.
\(0\)
5.4.6.8.
Answer.
\(1\)
5.4.6.9.
Answer.
\(2-\sqrt{2}\)
5.4.6.10.
Answer.
\(7\)
5.4.6.11.
Answer.
\(\frac{\left({\frac{32767}{512}}\right)}{\ln\mathopen{}\left(8\right)}\)
5.4.6.12.
Answer.
\(-2\)
5.4.6.13.
Answer.
\(-4\)
5.4.6.14.
Answer.
\(e^{2}-e^{1}\)
5.4.6.15.
Answer.
\(42\)
5.4.6.16.
Answer.
\(2\)
5.4.6.17.
Answer.
\({\frac{4096}{5}}\)
5.4.6.18.
Answer.
\(\ln\mathopen{}\left(6\right)\)
5.4.6.19.
Answer.
\({\frac{6}{7}}\)
5.4.6.20.
Answer.
\({\frac{59048}{295245}}\)
5.4.6.21.
Answer.
\({\frac{1}{2}}\)
5.4.6.22.
Answer.
\({\frac{1}{3}}\)
5.4.6.23.
Answer.
\({\frac{1}{4}}\)
5.4.6.24.
Answer.
\({\frac{1}{91}}\)
5.4.6.25.
Answer.
\(14\)
5.4.6.26.
Answer.
\(24\)
5.4.6.27.
Answer.
\(0\)
5.4.6.28.
Answer.
\(2-\sqrt{2}\)
5.4.6.31.
Answer.
\(1.1547\)
5.4.6.32.
Answer.
\(-4.6188, 4.6188\)
5.4.6.33.
Answer.
\(0.541325\)
5.4.6.34.
Answer.
\(4\)
5.4.6.35.
Answer.
\(\frac{\frac{1}{\pi -\frac{\pi }{2}}\cdot 3.14159}{\pi }\)
5.4.6.36.
Answer.
\(\frac{\frac{0}{\pi -0}\cdot 3.14159}{\pi }\)
5.4.6.37.
Answer.
\({\frac{7}{2}}\)
5.4.6.38.
Answer.
\({\frac{64}{3}}\)
5.4.6.39.
Answer.
\({\frac{729}{4}}\)
5.4.6.40.
Answer.
\(\frac{1}{e^{1}-1}\)
5.4.6.41.
Answer.
\(-168\ {\rm ft}\)
5.4.6.42.
Answer.
\(144\ {\rm ft}\)
5.4.6.43.
Answer.
\(76\ {\rm ft}\)
5.4.6.44.
Answer.
\(11.4965\ {\rm mi}\)
5.4.6.45.
Answer.
\(0\ {\rm ft}\)
5.4.6.46.
Answer.
\({\frac{10240}{3}}\ {\rm ft}\)
5.4.6.47.
Answer.
\(-256\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.48.
Answer.
\(72\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.49.
Answer.
\({\frac{1}{2}}\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.50.
Answer.
\(1\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.55.
Answer.
\(\frac{3x^{2}-7}{x^{3}-7x}\)
5.4.6.56.
Answer.
\(-3x^{11}\)
5.4.6.57.
Answer.
\(3x^{2}\mathopen{}\left(x^{3}-1\right)-\left(x-1\right)\)
5.4.6.58.
Answer.
\(e^{x}\cos\mathopen{}\left(e^{x}\right)-\cos\mathopen{}\left(x\right)\cos\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)\)
5.4.6.59.
Answer.
\(4x^{3}\sin\mathopen{}\left(4x^{8}\right)\)
5.4.6.60.
Answer.
\(\frac{1}{x}\sqrt{\ln^{4}\mathopen{}\left(x\right)+6\ln^{2}\mathopen{}\left(x\right)}-\cos\mathopen{}\left(x\right)\sqrt{\sin^{4}\mathopen{}\left(x\right)+6\sin^{2}\mathopen{}\left(x\right)}\)

5.5 Numerical Integration
5.5.6 Exercises

Terms and Concepts

5.5.6.1.
Answer.
\(\text{False}\)
5.5.6.4.
Answer.
A quadratic function (i.e., parabola)

Problems

5.5.6.5.
5.5.6.5.a
Answer.
\(0.75\)
5.5.6.5.b
Answer.
\(0.666667\)
5.5.6.5.c
Answer.
\(0.666667\)
5.5.6.6.
5.5.6.6.a
Answer.
\(250\)
5.5.6.6.b
Answer.
\(250\)
5.5.6.6.c
Answer.
\(250\)
5.5.6.7.
5.5.6.7.a
Answer.
\(1.89612\)
5.5.6.7.b
Answer.
\(2.00456\)
5.5.6.7.c
Answer.
\(2\)
5.5.6.8.
5.5.6.8.a
Answer.
\(5.14626\)
5.5.6.8.b
Answer.
\(5.25221\)
5.5.6.8.c
Answer.
\(5.33333\)
5.5.6.9.
5.5.6.9.a
Answer.
\(38.5781\)
5.5.6.9.b
Answer.
\(36.75\)
5.5.6.9.c
Answer.
\(36.75\)
5.5.6.10.
5.5.6.10.a
Answer.
\(0.220703\)
5.5.6.10.b
Answer.
\(0.200521\)
5.5.6.10.c
Answer.
\(0.2\)
5.5.6.11.
5.5.6.11.a
Answer.
\(0\)
5.5.6.11.b
Answer.
\(0\)
5.5.6.11.c
Answer.
\(0\)
5.5.6.12.
5.5.6.12.a
Answer.
\(12.2942\)
5.5.6.12.b
Answer.
\(13.3923\)
5.5.6.12.c
Answer.
\(14.1372\)
5.5.6.13.
5.5.6.13.a
Answer.
\(0.900628\)
5.5.6.13.b
Answer.
\(0.904523\)
5.5.6.14.
5.5.6.14.a
Answer.
\(3.02419\)
5.5.6.14.b
Answer.
\(2.93151\)
5.5.6.15.
5.5.6.15.a
Answer.
\(13.9604\)
5.5.6.15.b
Answer.
\(13.9066\)
5.5.6.16.
5.5.6.16.a
Answer.
\(3.06949\)
5.5.6.16.b
Answer.
\(3.14295\)
5.5.6.17.
5.5.6.17.a
Answer.
\(1.17029\)
5.5.6.17.b
Answer.
\(1.18728\)
5.5.6.18.
5.5.6.18.a
Answer.
\(2.52971\)
5.5.6.18.b
Answer.
\(2.54465\)
5.5.6.19.
5.5.6.19.a
Answer.
\(1.08025\)
5.5.6.19.b
Answer.
\(1.07699\)
5.5.6.20.
5.5.6.20.a
Answer.
\(3.46822\)
5.5.6.20.b
Answer.
\(3.4985\)
5.5.6.21.
5.5.6.21.a
Answer.
\(161\)
5.5.6.21.b
Answer.
\(12\)
5.5.6.22.
5.5.6.22.a
Answer.
\(130\)
5.5.6.22.b
Answer.
\(18\)
5.5.6.23.
5.5.6.23.a
Answer.
\(994\)
5.5.6.23.b
Answer.
\(62\)
5.5.6.24.
5.5.6.24.a
Answer.
\(5591\)
5.5.6.24.b
Answer.
\(46\)
5.5.6.25.
Answer 1.
\(30.8667\ {\rm cm^{2}}\)
Answer 2.
\(308667\ {\rm ft^{2}}\)
5.5.6.26.
Answer 1.
\(25.0667\ {\rm cm^{2}}\)
Answer 2.
\(250667\ {\rm ft^{2}}\)

II Math 2560: Calculus II
6 Techniques of Antidifferentiation
6.1 Substitution
6.1.5 Exercises

Terms and Concepts

6.1.5.1.
Answer.
\(\text{the Chain Rule}\)
6.1.5.2.
Answer.
\(\text{True}\)

Problems

6.1.5.3.
Answer.
\({\frac{1}{6}}\mathopen{}\left(x^{4}+3\right)^{6}+C\)
6.1.5.4.
Answer.
\({\frac{1}{7}}\mathopen{}\left(x^{2}-9x-3\right)^{7}+C\)
6.1.5.5.
Answer.
\({\frac{1}{20}}\mathopen{}\left(x^{2}-7\right)^{10}+C\)
6.1.5.6.
Answer.
\(\left({\frac{2}{9}}\right)\mathopen{}\left(3x-5x^{2}-4\right)^{9}+C\)
6.1.5.7.
Answer.
\({\frac{1}{4}}\ln\mathopen{}\left(\left|4x+5\right|\right)+C\)
6.1.5.8.
Answer.
\(\left({\frac{2}{5}}\right)\sqrt{5x+9}+C\)
6.1.5.9.
Answer.
\({\frac{2}{3}}\mathopen{}\left(x-2\right)\sqrt{x+1}+C\)
6.1.5.10.
Answer.
\(x^{\left({\frac{3}{2}}\right)}\mathopen{}\left({\frac{2}{7}}x^{2}+2\right)+C\)
6.1.5.11.
Answer.
\(2e^{\sqrt{x}}+C\)
6.1.5.12.
Answer.
\(\left({\frac{1}{3}}\right)\sqrt{x^{6}+8}+C\)
6.1.5.13.
Answer.
\(C-{\frac{1}{2}}\mathopen{}\left(\frac{1}{x}-9\right)^{2}\)
6.1.5.14.
Answer.
\(\frac{\ln^{2}\mathopen{}\left(x\right)}{2}+C\)
6.1.5.15.
Answer.
\(\frac{\left(\sin\mathopen{}\left(x\right)\right)^{4}}{4}+C\)
6.1.5.16.
Answer.
\(C-\frac{\left(\cos\mathopen{}\left(x\right)\right)^{5}}{5}\)
6.1.5.17.
Answer.
\(C-\frac{\sin\mathopen{}\left(8-5x\right)}{5}\)
6.1.5.18.
Answer.
\(C-\frac{\tan\mathopen{}\left(5-4x\right)}{4}\)
6.1.5.19.
Answer.
\({\frac{1}{7}}\ln\mathopen{}\left(\left|\sec\mathopen{}\left(7x\right)+\tan\mathopen{}\left(7x\right)\right|\right)+C\)
6.1.5.20.
Answer.
\({\frac{1}{9}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{9}+C\)
6.1.5.21.
Answer.
\(C-{\frac{1}{9}}\cos\mathopen{}\left(x^{9}\right)\)
6.1.5.22.
Answer.
\(\tan\mathopen{}\left(x\right)-x+C\)
6.1.5.23.
Answer.
\(\ln\mathopen{}\left(\left|\sin\mathopen{}\left(x\right)\right|\right)+C\)
6.1.5.24.
Answer.
\(-\ln\mathopen{}\left(\left|\csc\mathopen{}\left(x\right)+\cot\mathopen{}\left(x\right)\right|\right)+C\)
6.1.5.25.
Answer.
\({\frac{1}{4}}e^{4x-9}+C\)
6.1.5.26.
Answer.
\({\frac{1}{5}}e^{x^{5}}+C\)
6.1.5.27.
Answer.
\({\frac{1}{2}}e^{\left(x+1\right)^{2}}+C\)
6.1.5.28.
Answer.
\(x-3e^{-x}+C\)
6.1.5.29.
Answer.
\(\ln\mathopen{}\left(e^{x}+8\right)+C\)
6.1.5.30.
Answer.
\(C-\left({\frac{1}{2}}e^{-2x}+{\frac{1}{4}}e^{-4x}\right)\)
6.1.5.31.
Answer.
\(\frac{2^{2x}}{1.38629}+C\)
6.1.5.32.
Answer.
\(\frac{2^{7x}}{4.85203}+C\)
6.1.5.33.
Answer.
\(\frac{\ln^{2}\mathopen{}\left(x\right)}{2}+C\)
6.1.5.34.
Answer.
\(\frac{\left(\ln\mathopen{}\left(x\right)\right)^{5}}{5}+C\)
6.1.5.35.
Answer.
\(\left({\frac{5}{2}}\right)\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)^{2}+C\)
6.1.5.36.
Answer.
\({\frac{1}{6}}\ln\mathopen{}\left(\left|\ln\mathopen{}\left(x^{6}\right)\right|\right)+C\)
6.1.5.37.
Answer.
\(\frac{x^{2}}{2}+4x+7\ln\mathopen{}\left(\left|x\right|\right)+C\)
6.1.5.38.
Answer.
\(\frac{x^{3}}{3}+\frac{x^{2}}{2}+x+\ln\mathopen{}\left(\left|x\right|\right)+C\)
6.1.5.39.
Answer.
\({\frac{1}{3}}\mathopen{}\left(x+1\right)^{3}+\left({\frac{3}{2}}\right)\mathopen{}\left(x+1\right)^{2}+3\mathopen{}\left(x+1\right)-5\ln\mathopen{}\left(\left|x+1\right|\right)+C\)
6.1.5.40.
Answer.
\(\frac{\left(x-3\right)^{2}}{2}+10\mathopen{}\left(x-3\right)+12\ln\mathopen{}\left(\left|x-3\right|\right)+C\)
6.1.5.41.
Answer.
\(C-\left(\left({\frac{7}{2}}\right)\mathopen{}\left(x-6\right)^{2}+85\mathopen{}\left(x-6\right)+250\ln\mathopen{}\left(\left|x-6\right|\right)\right)\)
6.1.5.42.
Answer.
\({\frac{1}{3}}\ln\mathopen{}\left(\left|x^{3}-6x^{2}-9x\right|\right)+C\)
6.1.5.43.
Answer.
\(2.44949\tan^{-1}\mathopen{}\left(\frac{x}{2.44949}\right)+C\)
6.1.5.44.
Answer.
\(5\sin^{-1}\mathopen{}\left(\frac{x}{5}\right)+C\)
6.1.5.45.
Answer.
\(3\sin^{-1}\mathopen{}\left(\frac{x}{3.16228}\right)+C\)
6.1.5.46.
Answer.
\(\left({\frac{8}{7}}\right)\sec^{-1}\mathopen{}\left(\frac{\left|x\right|}{7}\right)+C\)
6.1.5.47.
Answer.
\(\left({\frac{1}{2}}\right)\sec^{-1}\mathopen{}\left(\frac{\left|x\right|}{8}\right)+C\)
6.1.5.48.
Answer.
\(0.5\sin^{-1}\mathopen{}\left(x^{2}\right)+C\)
6.1.5.49.
Answer.
\(0.301511\tan^{-1}\mathopen{}\left(\frac{x+9}{11}\right)+C\)
6.1.5.50.
Answer.
\(7\sin^{-1}\mathopen{}\left(\frac{x-7}{4}\right)+C\)
6.1.5.51.
Answer.
\(2\sin^{-1}\mathopen{}\left(\frac{x-5}{9}\right)+C\)
6.1.5.52.
Answer.
\(\tan^{-1}\mathopen{}\left(\frac{x-3}{7}\right)+C\)
6.1.5.53.
Answer.
\(C-\frac{1}{6\mathopen{}\left(x^{6}-4\right)}\)
6.1.5.54.
Answer.
\({\frac{1}{7}}\mathopen{}\left(5x^{5}+9x^{4}-4\right)^{7}+C\)
6.1.5.55.
Answer.
\(\left({\frac{1}{2}}\right)\sqrt{6+2x^{2}}+C\)
6.1.5.56.
Answer.
\(\tan\mathopen{}\left(x^{8}-5\right)+C\)
6.1.5.57.
Answer.
\(C-{\frac{2}{3}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{\left({\frac{3}{2}}\right)}\)
6.1.5.58.
Answer.
\({\frac{1}{9}}\sin\mathopen{}\left(9x+1\right)+C\)
6.1.5.59.
Answer.
\(\ln\mathopen{}\left(\left|x-7\right|\right)+C\)
6.1.5.60.
Answer.
\(\left({\frac{1}{4}}\right)\ln\mathopen{}\left(\left|8x+7\right|\right)+C\)
6.1.5.61.
Answer.
\(x^{2}+2x+\ln\mathopen{}\left(\left|x^{2}-4x+1\right|\right)+C\)
6.1.5.62.
Answer.
\(\ln\mathopen{}\left(\left|x^{2}-2x-7\right|\right)+C\)
6.1.5.63.
Answer.
\(2\ln\mathopen{}\left(\left|x^{2}+6x-9\right|\right)+C\)
6.1.5.64.
Answer.
\(-\left({\frac{1}{2}}\right)x^{2}-x+\ln\mathopen{}\left(\left|x^{2}+3x-1\right|\right)+C\)
6.1.5.65.
Answer.
\({\frac{1}{16}}\tan^{-1}\mathopen{}\left(\frac{x^{2}}{8}\right)+C\)
6.1.5.66.
Answer.
\(\tan^{-1}\mathopen{}\left(9x\right)+C\)
6.1.5.67.
Answer.
\(\sec^{-1}\mathopen{}\left(\left|9x\right|\right)+C\)
6.1.5.68.
Answer.
\({\frac{1}{3}}\sin^{-1}\mathopen{}\left(3\frac{x}{2}\right)+C\)
6.1.5.69.
Answer.
\(\left({\frac{5}{2}}\right)\ln\mathopen{}\left(\left|x^{2}-10x+74\right|\right)+\left({\frac{1}{7}}\right)\tan^{-1}\mathopen{}\left(\frac{x-5}{7}\right)+C\)
6.1.5.70.
Answer.
\(\left({\frac{19}{5}}\right)\tan^{-1}\mathopen{}\left(\frac{x-3}{5}\right)+\ln\mathopen{}\left(\left|x^{2}-6x+34\right|\right)+C\)
6.1.5.71.
Answer.
\(x+14.1421\tan^{-1}\mathopen{}\left(\frac{x-1}{1.41421}\right)+\left({\frac{17}{2}}\right)\ln\mathopen{}\left(\left|x^{2}-2x+3\right|\right)+C\)
6.1.5.72.
Answer.
\(\frac{x^{2}}{2}-18\ln\mathopen{}\left(\left|x^{2}+36\right|\right)+C\)
6.1.5.73.
Answer.
\({\frac{1}{2}}x^{2}-6x+\left({\frac{7}{2}}\right)\ln\mathopen{}\left(\left|x^{2}+6x+15\right|\right)+4.49073\tan^{-1}\mathopen{}\left(\frac{x+3}{2.44949}\right)+C\)
6.1.5.74.
Answer.
\(-\tan^{-1}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)+C\)
6.1.5.75.
Answer.
\(\tan^{-1}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)+C\)
6.1.5.76.
Answer.
\(C-\ln\mathopen{}\left(\left|\csc\mathopen{}\left(x\right)+\cot\mathopen{}\left(x\right)\right|\right)\)
6.1.5.77.
Answer.
\(9\sqrt{x^{2}+16x+63}+C\)
6.1.5.78.
Answer.
\(\sqrt{x^{2}+12x+32}+C\)
6.1.5.79.
Answer.
\(\ln\mathopen{}\left(\left({\frac{3}{7}}\right)\right)\)
6.1.5.80.
Answer.
\({\frac{361568}{15}}\)
6.1.5.81.
Answer.
\(0\)
6.1.5.82.
Answer.
\({\frac{1}{8}}\)
6.1.5.83.
Answer.
\({\frac{1}{2}}\mathopen{}\left(e^{4}-e\right)\)
6.1.5.84.
Answer.
\(\frac{\pi }{2}\)
6.1.5.85.
Answer.
\(\frac{\pi }{2}\)
6.1.5.86.
Answer.
\(\left({\frac{5}{6}}\right)\pi \)

6.2 Integration by Parts

Exercises

Terms and Concepts
6.2.1.
Answer.
\(\text{True}\)
6.2.2.
Answer.
\(\text{False}\)
6.2.4.
Answer.
\(\text{False}\)
Problems
6.2.5.
Answer.
\(\sin\mathopen{}\left(x\right)-x\cos\mathopen{}\left(x\right)+C\)
6.2.6.
Answer.
\(-e^{-x}\mathopen{}\left(x+1\right)+C\)
6.2.7.
Answer.
\(-x^{2}\cos\mathopen{}\left(x\right)+2x\sin\mathopen{}\left(x\right)+2\cos\mathopen{}\left(x\right)+C\)
6.2.8.
Answer.
\(-x^{3}\cos\mathopen{}\left(x\right)+3x^{2}\sin\mathopen{}\left(x\right)+6x\cos\mathopen{}\left(x\right)-6\sin\mathopen{}\left(x\right)+C\)
6.2.9.
Answer.
\({\frac{1}{2}}e^{x^{2}}+C\)
6.2.10.
Answer.
\(e^{x}\mathopen{}\left(x^{3}-3x^{2}+6x-6\right)+C\)
6.2.11.
Answer.
\(-{\frac{1}{2}}xe^{-2x}-\frac{e^{-2x}}{4}+C\)
6.2.12.
Answer.
\({\frac{1}{2}}e^{x}\mathopen{}\left(\sin\mathopen{}\left(x\right)-\cos\mathopen{}\left(x\right)\right)+C\)
6.2.13.
Answer.
\({\frac{1}{5}}e^{2x}\mathopen{}\left(\sin\mathopen{}\left(x\right)+2\cos\mathopen{}\left(x\right)\right)+C\)
6.2.14.
Answer.
\(\left({\frac{1}{130}}\right)e^{7x}\mathopen{}\left(7\sin\mathopen{}\left(9x\right)-9\cos\mathopen{}\left(9x\right)\right)+C\)
6.2.15.
Answer.
\(\left({\frac{1}{16}}\right)e^{8x}\mathopen{}\left(\sin\mathopen{}\left(8x\right)+\cos\mathopen{}\left(8x\right)\right)+C\)
6.2.16.
Answer.
\(0.5\sin^{2}\mathopen{}\left(x\right)+C\)
6.2.17.
Answer.
\(\sqrt{1-x^{2}}+x\sin^{-1}\mathopen{}\left(x\right)+C\)
6.2.18.
Answer.
\(x\tan^{-1}\mathopen{}\left(2x\right)-0.25\ln\mathopen{}\left(4x^{2}+1\right)+C\)
6.2.19.
Answer.
\(0.5x^{2}\tan^{-1}\mathopen{}\left(x\right)-\frac{x}{2}+0.5\tan^{-1}\mathopen{}\left(x\right)+C\)
6.2.20.
Answer.
\(-\sqrt{1-x^{2}}+x\cos^{-1}\mathopen{}\left(x\right)+C\)
6.2.21.
Answer.
\(0.5x^{2}\ln\mathopen{}\left(x\right)-\frac{x^{2}}{4}+C\)
6.2.22.
Answer.
\({\frac{1}{2}}x^{2}\ln\mathopen{}\left(x\right)-\frac{x^{2}}{4}+x\ln\mathopen{}\left(x\right)-x+C\)
6.2.23.
Answer.
\({\frac{1}{2}}x^{2}\ln\mathopen{}\left(x-3\right)-{\frac{1}{4}}\mathopen{}\left(x-3\right)^{2}-3x-\left({\frac{9}{2}}\right)\ln\mathopen{}\left(x-3\right)+C\)
6.2.24.
Answer.
\(0.5x^{2}\ln\mathopen{}\left(x^{2}\right)-\frac{x^{2}}{2}+C\)
6.2.25.
Answer.
\(0.333333x^{3}\ln\mathopen{}\left(x\right)-\frac{x^{3}}{9}+C\)
6.2.26.
Answer.
\(2x+x\ln^{2}\mathopen{}\left(x\right)-2x\ln\mathopen{}\left(x\right)+C\)
6.2.27.
Answer.
\(2\mathopen{}\left(x-8\right)+\left(x-8\right)\mathopen{}\left(\ln\mathopen{}\left(x-8\right)\right)^{2}-2\mathopen{}\left(x-8\right)\ln\mathopen{}\left(x-8\right)+C\)
6.2.28.
Answer.
\(x\tan\mathopen{}\left(x\right)+\ln\mathopen{}\left(\left|\cos\mathopen{}\left(x\right)\right|\right)+C\)
6.2.29.
Answer.
\(\ln\mathopen{}\left(\left|\sin\mathopen{}\left(x\right)\right|\right)-x\cot\mathopen{}\left(x\right)+C\)
6.2.30.
Answer.
\(\left({\frac{2}{5}}\mathopen{}\left(x-2\right)^{2}+\left({\frac{4}{3}}\right)\mathopen{}\left(x-2\right)\right)\sqrt{x-2}+C\)
6.2.31.
Answer.
\({\frac{1}{3}}\mathopen{}\left(x^{2}-6\right)^{\left({\frac{3}{2}}\right)}+C\)
6.2.32.
Answer.
\(\sec\mathopen{}\left(x\right)+C\)
6.2.33.
Answer.
\(x\sec\mathopen{}\left(x\right)-\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)+C\)
6.2.34.
Answer.
\(-x\csc\mathopen{}\left(x\right)-\ln\mathopen{}\left(\left|\csc\mathopen{}\left(x\right)+\cot\mathopen{}\left(x\right)\right|\right)+C\)
6.2.35.
Answer.
\(\frac{x}{2}\mathopen{}\left(\sin\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)+\cos\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)\right)+C\)
6.2.36.
Answer.
\(\sin\mathopen{}\left(e^{x}\right)-e^{x}\cos\mathopen{}\left(e^{x}\right)+C\)
6.2.37.
Answer.
\(2\sin\mathopen{}\left(\sqrt{x}\right)-2\sqrt{x}\cos\mathopen{}\left(\sqrt{x}\right)+C\)
6.2.38.
Answer.
\(x\ln\mathopen{}\left(\sqrt{x}\right)-\frac{x}{2}+C\)
6.2.39.
Answer.
\(2\sqrt{x}e^{\sqrt{x}}-2e^{\sqrt{x}}+C\)
6.2.40.
Answer.
\(\frac{x^{2}}{2}+C\)
6.2.41.
Answer.
\(-1\)
6.2.42.
Answer.
\(-\left(2\frac{1}{e}+e^{2}\right)\)
6.2.43.
Answer.
\(0\)
6.2.44.
Answer.
\(\frac{3\pi ^{2}}{2}-12\)
6.2.45.
Answer.
\({\frac{1}{2}}\)
6.2.46.
Answer.
\(0.563436\)
6.2.47.
Answer.
\(\left(-{\frac{7}{4}}\right)e^{-6}-\left(-{\frac{5}{4}}\right)e^{-4}\)
6.2.48.
Answer.
\(0.5e^{\pi }+0.5\)
6.2.49.
Answer.
\(0.2\mathopen{}\left(-e^{3\pi }-e^{-3\pi }\right)\)

6.3 Trigonometric Integrals
6.3.4 Exercises

Terms and Concepts

6.3.4.1.
Answer.
\(\text{False}\)
6.3.4.2.
Answer.
\(\text{False}\)
6.3.4.3.
Answer.
\(\text{False}\)
6.3.4.4.
Answer.
\(\text{False}\)

Problems

6.3.4.5.
Answer.
\(-0.2\cos^{5}\mathopen{}\left(x\right)+C\)
6.3.4.6.
Answer.
\(0.25\sin^{4}\mathopen{}\left(x\right)+C\)
6.3.4.7.
Answer.
\({\frac{1}{7}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{7}-{\frac{1}{5}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{5}+C\)
6.3.4.8.
Answer.
\({\frac{1}{8}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{8}-{\frac{1}{6}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{6}+C\)
6.3.4.9.
Answer.
\({\frac{1}{11}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{11}-{\frac{2}{9}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{9}+{\frac{1}{7}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{7}+C\)
6.3.4.10.
Answer.
\(-0.111111\sin^{9}\mathopen{}\left(x\right)+0.428571\sin^{7}\mathopen{}\left(x\right)-0.6\sin^{5}\mathopen{}\left(x\right)+0.333333\sin^{3}\mathopen{}\left(x\right)+C\)
6.3.4.11.
Answer.
\(\frac{x}{8}-0.03125\sin\mathopen{}\left(4x\right)+C\)
6.3.4.12.
Answer.
\(0.5\mathopen{}\left(-0.125\cos\mathopen{}\left(8x\right)-0.5\cos\mathopen{}\left(2x\right)\right)+C\)
6.3.4.13.
Answer.
\(C-\left(\left({\frac{1}{4}}\right)\cos\mathopen{}\left(2x\right)+\left({\frac{1}{8}}\right)\cos\mathopen{}\left(4x\right)\right)\)
6.3.4.14.
Answer.
\(\left({\frac{1}{14}}\right)\sin\mathopen{}\left(7x\right)-\left({\frac{1}{22}}\right)\sin\mathopen{}\left(11x\right)+C\)
6.3.4.15.
Answer.
\(\frac{1}{12\pi }\sin\mathopen{}\left(6\pi x\right)-\frac{1}{16\pi }\sin\mathopen{}\left(8\pi x\right)+C\)
6.3.4.16.
Answer.
\(0.5\mathopen{}\left(\sin\mathopen{}\left(x\right)+0.333333\sin\mathopen{}\left(3x\right)\right)+C\)
6.3.4.17.
Answer.
\(\frac{3}{4\pi }\cos\mathopen{}\left(\frac{2\pi }{3}\pi x\right)+\frac{3}{8\pi }\cos\mathopen{}\left(\frac{4\pi }{3}\pi x\right)+C\)
6.3.4.18.
Answer.
\(\frac{\tan^{5}\mathopen{}\left(x\right)}{5}+C\)
6.3.4.19.
Answer.
\(\frac{\tan^{5}\mathopen{}\left(x\right)}{5}+\frac{\tan^{3}\mathopen{}\left(x\right)}{3}+C\)
6.3.4.20.
Answer.
\({\frac{1}{10}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{10}+{\frac{1}{8}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{8}+C\)
6.3.4.21.
Answer.
\({\frac{1}{9}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{9}+C\)
6.3.4.22.
Answer.
\({\frac{1}{11}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{11}-{\frac{1}{9}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{9}+C\)
6.3.4.23.
Answer.
\({\frac{1}{6}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{6}-{\frac{1}{2}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{4}+{\frac{1}{2}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{2}+C\)
6.3.4.24.
Answer.
\(\frac{\tan^{3}\mathopen{}\left(x\right)}{3}-\tan\mathopen{}\left(x\right)+x+C\)
6.3.4.25.
Answer.
\(0.25\tan\mathopen{}\left(x\right)\sec^{3}\mathopen{}\left(x\right)+0.375\mathopen{}\left(\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)+\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)\right)+C\)
6.3.4.26.
Answer.
\(0.5\mathopen{}\left(\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)-\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)\right)+C\)
6.3.4.27.
Answer.
\(0.25\tan\mathopen{}\left(x\right)\sec^{3}\mathopen{}\left(x\right)-0.125\mathopen{}\left(\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)+\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)\right)+C\)
6.3.4.28.
Answer.
\({\frac{1}{5}}\)
6.3.4.29.
Answer.
\(0\)
6.3.4.30.
Answer.
\(0\)
6.3.4.31.
Answer.
\(0\)
6.3.4.32.
Answer.
\({\frac{2}{3}}\)
6.3.4.33.
Answer.
\({\frac{1}{5}}\)
6.3.4.34.
Answer.
\({\frac{8}{15}}\)

6.4 Trigonometric Substitution

Exercises

Terms and Concepts
6.4.1.
Answer.
\(\text{backward}\)
6.4.2.
Answer.
\(6\sin\mathopen{}\left(\theta\right)\hbox{ or }6\cos\mathopen{}\left(\theta\right)\)
6.4.3.
Answer 1.
\(\tan^{2}\mathopen{}\left(\theta\right)+1 = \sec^{2}\mathopen{}\left(\theta\right)\)
Answer 2.
\(6\sec^{2}\mathopen{}\left(\theta\right)\)
Problems
6.4.5.
Answer.
\({\frac{1}{2}}\mathopen{}\left(x\sqrt{x^{2}+1}+\ln\mathopen{}\left(\sqrt{x^{2}+1}+x\right)\right)+C\)
6.4.6.
Answer.
\(\frac{x}{2}\sqrt{x^{2}+4}+2\ln\mathopen{}\left(\frac{\sqrt{x^{2}+4}}{2}+\frac{x}{2}\right)+C\)
6.4.7.
Answer.
\({\frac{1}{2}}\sin^{-1}\mathopen{}\left(x\right)+\frac{x}{2}\sqrt{1-x^{2}}+C\)
6.4.8.
Answer.
\({\frac{9}{2}}\sin^{-1}\mathopen{}\left(\frac{x}{3}\right)+\frac{x}{2}\sqrt{9-x^{2}}+C\)
6.4.9.
Answer.
\({\frac{1}{2}}x\sqrt{x^{2}-1}-{\frac{1}{2}}\ln\mathopen{}\left(\left|x+\sqrt{x^{2}-1}\right|\right)+C\)
6.4.10.
Answer.
\({\frac{1}{2}}x\sqrt{x^{2}-16}-8\ln\mathopen{}\left(\left|\frac{x}{4}+\frac{\sqrt{x^{2}-16}}{4}\right|\right)+C\)
6.4.11.
Answer.
\(\frac{x}{2}\sqrt{36x^{2}+1}+{\frac{1}{12}}\ln\mathopen{}\left(6x+\sqrt{36x^{2}+1}\right)+C\)
6.4.12.
Answer.
\(\frac{x}{2}\sqrt{1-36x^{2}}+{\frac{1}{12}}\sin^{-1}\mathopen{}\left(6x\right)+C\)
6.4.13.
Answer.
\(\frac{x}{2}\sqrt{49x^{2}-1}-{\frac{1}{14}}\ln\mathopen{}\left(\left|7x+\sqrt{49x^{2}-1}\right|\right)+C\)
6.4.14.
Answer.
\(8\ln\mathopen{}\left(\frac{x}{1.73205}+\sqrt{\frac{x^{2}}{3}+1}\right)+C\)
6.4.15.
Answer.
\(9\sin^{-1}\mathopen{}\left(\frac{x}{3.60555}\right)+C\)
6.4.16.
Answer.
\(2\ln\mathopen{}\left(\left|\frac{x}{2.64575}+\sqrt{\frac{x^{2}}{7}-1}\right|\right)+C\)
6.4.17.
Answer.
\(\sqrt{x^{2}-3}-1.73205\sec^{-1}\mathopen{}\left(\frac{x}{1.73205}\right)+C\)
6.4.18.
Answer.
\({\frac{1}{2}}\tan^{-1}\mathopen{}\left(x\right)+\frac{x}{2\mathopen{}\left(x^{2}+1\right)}+C\)
6.4.19.
Answer.
\(\sqrt{x^{2}-6}+C\)
6.4.20.
Answer.
\({\frac{1}{8}}\sin^{-1}\mathopen{}\left(x\right)+\frac{x}{8}\sqrt{1-x^{2}}\mathopen{}\left(2x^{2}-1\right)+C\)
6.4.21.
Answer.
\(C-\frac{1}{\sqrt{x^{2}+36}}\)
6.4.22.
Answer.
\(\frac{7x}{2}\sqrt{x^{2}-6}+21\ln\mathopen{}\left(\left|\frac{x}{2.44949}+\sqrt{\frac{x^{2}}{6}-1}\right|\right)+C\)
6.4.23.
Answer.
\(\left({\frac{1}{162}}\right)\frac{x-6}{x^{2}-12x+117}+\left({\frac{1}{1458}}\right)\tan^{-1}\mathopen{}\left(\frac{x-6}{9}\right)+C\)
6.4.24.
Answer.
\(\frac{x}{\sqrt{1-x^{2}}}-\sin^{-1}\mathopen{}\left(x\right)+C\)
6.4.25.
Answer.
\(C-\left(\frac{\sqrt{5-x^{2}}}{2x}+{\frac{1}{2}}\sin^{-1}\mathopen{}\left(\frac{x}{2.23607}\right)\right)\)
6.4.26.
Answer.
\(\frac{x}{2}\sqrt{x^{2}+3}-\left({\frac{3}{2}}\right)\ln\mathopen{}\left(\frac{x}{1.73205}+\sqrt{\frac{x^{2}}{3}+1}\right)+C\)
6.4.27.
Answer.
\(\frac{\pi }{2}\)
6.4.28.
Answer.
\(\left({\frac{7}{2}}\right)\sqrt{33}-8\ln\mathopen{}\left(\left|\left({\frac{7}{4}}\right)+\left({\frac{1}{4}}\right)\sqrt{33}\right|\right)\)
6.4.29.
Answer.
\(\left({\frac{5}{2}}\right)\sqrt{29}+2\ln\mathopen{}\left(\left({\frac{5}{2}}\right)+\left({\frac{1}{2}}\right)\sqrt{29}\right)\)
6.4.30.
Answer.
\(\tan^{-1}\mathopen{}\left(7\right)+\left({\frac{7}{50}}\right)\)
6.4.31.
Answer.
\(9\sin^{-1}\mathopen{}\left(\left({\frac{2}{3}}\right)\right)+2\sqrt{5}\)
6.4.32.
Answer.
\(\frac{\pi }{8}\)

6.5 Partial Fraction Decomposition

Exercises

Terms and Concepts
6.5.1.
Answer.
\(\text{rational}\)
6.5.2.
Answer.
\(\text{True}\)
6.5.3.
Answer.
\(\frac{A}{x}+\frac{B}{x-6}\)
6.5.4.
Answer.
\(\frac{A}{x-3}+\frac{B}{x+3}\)
6.5.5.
Answer.
\(\frac{A}{x-\sqrt{6}}+\frac{B}{x+\sqrt{6}}\)
6.5.6.
Answer.
\(\frac{A}{x}+\frac{Bx+C}{x^{2}+5}\)
Problems
6.5.7.
Answer.
\(5\ln\mathopen{}\left(\left|x+3\right|\right)+9\ln\mathopen{}\left(\left|x-2\right|\right)+C\)
6.5.8.
Answer.
\(8\ln\mathopen{}\left(\left|x\right|\right)-8\ln\mathopen{}\left(\left|x-4\right|\right)+C\)
6.5.9.
Answer.
\(\left({\frac{3}{4}}\right)\ln\mathopen{}\left(\left|x-2\right|\right)-\left({\frac{3}{4}}\right)\ln\mathopen{}\left(\left|x+2\right|\right)+C\)
6.5.10.
Answer.
\(\ln\mathopen{}\left(\left|x-8\right|\right)+\ln\mathopen{}\left(\left|1-4x\right|\right)+C\)
6.5.11.
Answer.
\(\ln\mathopen{}\left(\left|x+9\right|\right)-\frac{3}{x+9}+C\)
6.5.12.
Answer.
\(7\ln\mathopen{}\left(\left|x+7\right|\right)-\frac{5}{x+7}+C\)
6.5.13.
Answer.
\(3\ln\mathopen{}\left(\left|x\right|\right)+\ln\mathopen{}\left(\left|x+4\right|\right)+\frac{4}{x+4}+C\)
6.5.14.
Answer.
\(C-\left(2\ln\mathopen{}\left(\left|9-3x\right|\right)+\ln\mathopen{}\left(\left|x+3\right|\right)+5\ln\mathopen{}\left(\left|x-9\right|\right)\right)\)
6.5.15.
Answer.
\(\left({\frac{1}{7}}\right)\ln\mathopen{}\left(\left|7x+1\right|\right)-\left({\frac{2}{5}}\right)\ln\mathopen{}\left(\left|5x+3\right|\right)+\frac{\left({\frac{1}{3}}\right)}{9x-9}+C\)
6.5.16.
Answer.
\(x-2\ln\mathopen{}\left(\left|x-2\right|\right)-\ln\mathopen{}\left(\left|x+5\right|\right)+C\)
6.5.17.
Answer.
\({\frac{1}{2}}x^{2}+12x-16\ln\mathopen{}\left(\left|x-4\right|\right)+128\ln\mathopen{}\left(\left|x-8\right|\right)+C\)
6.5.18.
Answer.
\(2x+C\)
6.5.19.
Answer.
\(\left({\frac{1}{18}}\right)\ln\mathopen{}\left(\left|x\right|\right)-\left({\frac{1}{36}}\right)\ln\mathopen{}\left(x^{2}-8x+18\right)+0.157135\tan^{-1}\mathopen{}\left(\frac{x-4}{1.41421}\right)+C\)
6.5.20.
Answer.
\(x+4\ln\mathopen{}\left(x^{2}+8x+22\right)-15.1052\tan^{-1}\mathopen{}\left(\frac{x+4}{2.44949}\right)+C\)
6.5.21.
Answer.
\(\ln\mathopen{}\left(\left|3x^{2}+x-4\right|\right)-2\ln\mathopen{}\left(\left|x-9\right|\right)+C\)
6.5.22.
Answer.
\(5\ln\mathopen{}\left(\left|x+6\right|\right)+4\ln\mathopen{}\left(x^{2}+4x+5\right)-2\tan^{-1}\mathopen{}\left(x+2\right)+C\)
6.5.23.
Answer.
\(\left({\frac{129}{58}}\right)\ln\mathopen{}\left(\left|x-7\right|\right)+\left({\frac{45}{116}}\right)\ln\mathopen{}\left(x^{2}+9\right)+\left({\frac{199}{174}}\right)\tan^{-1}\mathopen{}\left(\frac{x}{3}\right)+C\)
6.5.24.
Answer.
\(\ln\mathopen{}\left(x^{2}-2x+5\right)-\ln\mathopen{}\left(\left|x+4\right|\right)-2\tan^{-1}\mathopen{}\left(\frac{x-1}{2}\right)+C\)
6.5.25.
Answer.
\(4\ln\mathopen{}\left(\left|x+9\right|\right)-2\ln\mathopen{}\left(x^{2}-2x+4\right)+2.88675\tan^{-1}\mathopen{}\left(\frac{x-1}{1.73205}\right)+C\)
6.5.26.
Answer.
\(\ln\mathopen{}\left(\left|x+1\right|\right)-\left({\frac{3}{2}}\right)\ln\mathopen{}\left(x^{2}-8x+21\right)-0.894427\tan^{-1}\mathopen{}\left(\frac{x-4}{2.23607}\right)+C\)
6.5.27.
Answer.
\(\ln\mathopen{}\left(\left({\frac{48828125}{14155776}}\right)\right)\)
6.5.28.
Answer.
\(-4.35712\)
6.5.29.
Answer.
\(\ln\mathopen{}\left(\left({\frac{5}{7}}\right)\right)+\tan^{-1}\mathopen{}\left(5\right)-\tan^{-1}\mathopen{}\left(3\right)\)
6.5.30.
Answer.
\({\frac{1}{8}}\)

6.6 Hyperbolic Functions
6.6.3 Exercises

Problems

6.6.3.11.
Answer.
\(2\cosh\mathopen{}\left(2x\right)\)
6.6.3.12.
Answer.
\(2\cosh\mathopen{}\left(x\right)\sinh\mathopen{}\left(x\right)\)
6.6.3.13.
Answer.
\(\mathop{\rm sech}\nolimits^{2}\mathopen{}\left(x^{2}\right)\cdot 2x\)
6.6.3.14.
Answer.
\(\frac{1}{\sinh\mathopen{}\left(x\right)}\cosh\mathopen{}\left(x\right)\)
6.6.3.15.
Answer.
\(\cosh\mathopen{}\left(x\right)\cosh\mathopen{}\left(x\right)+\sinh\mathopen{}\left(x\right)\sinh\mathopen{}\left(x\right)\)
6.6.3.16.
Answer.
\(\sinh\mathopen{}\left(x\right)+x\cosh\mathopen{}\left(x\right)-\sinh\mathopen{}\left(x\right)\)
6.6.3.17.
Answer.
\(-\frac{1}{x^{2}\sqrt{1-\left(x^{2}\right)^{2}}}\cdot 2x\)
6.6.3.18.
Answer.
\(3\frac{1}{\sqrt{1+\left(3x\right)^{2}}}\)
6.6.3.19.
Answer.
\(\frac{1}{\sqrt{\left(2x^{2}\right)^{2}-1}}\cdot 2\cdot 2x\)
6.6.3.20.
Answer.
\(\frac{1}{1-\left(x+5\right)^{2}}\)
6.6.3.21.
Answer.
\(-\frac{1}{1-\cos^{2}\mathopen{}\left(x\right)}\sin\mathopen{}\left(x\right)\)
6.6.3.22.
Answer.
\(\frac{1}{\sqrt{\sec^{2}\mathopen{}\left(x\right)-1}}\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)\)
6.6.3.23.
Answer.
\(1\mathopen{}\left(x-0\right)+0\)
6.6.3.24.
Answer.
\(0.75\mathopen{}\left(x-0.693147\right)+1.25\)
6.6.3.25.
Answer.
\(0.36\mathopen{}\left(x-\left(-1.09861\right)\right)+\left(-0.8\right)\)
6.6.3.26.
Answer.
\(-0.576\mathopen{}\left(x-1.09861\right)+0.36\)
6.6.3.27.
Answer.
\(1\mathopen{}\left(x-0\right)+0\)
6.6.3.28.
Answer.
\(1\mathopen{}\left(x-1.41421\right)+0.881374\)
6.6.3.29.
Answer.
\(0.5\ln\mathopen{}\left(\cosh\mathopen{}\left(2x\right)\right)+C\)
6.6.3.30.
Answer.
\(0.333333\sinh\mathopen{}\left(3x-7\right)+C\)
6.6.3.31.
Answer.
\(0.5\sinh^{2}\mathopen{}\left(x\right)+C\)
6.6.3.32.
Answer.
\(x\sinh\mathopen{}\left(x\right)-\cosh\mathopen{}\left(x\right)+C\)
6.6.3.33.
Answer.
\(x\cosh\mathopen{}\left(x\right)-\sinh\mathopen{}\left(x\right)+C\)
6.6.3.34.
Answer.
\(\sinh^{-1} x +C=\ln\big(x+\sqrt{x^2+1}\big)+C\)
6.6.3.35.
Answer.
\(\cosh^{-1} x/3 +C=\ln\big(x+\sqrt{x^2-9}\big)+C\)
6.6.3.36.
Answer.
\(0.5\ln\mathopen{}\left(\left|x+1\right|\right)-0.5\ln\mathopen{}\left(\left|x-1\right|\right)+C\)
6.6.3.37.
Answer.
\(\cosh^{-1}\mathopen{}\left(\frac{x^{2}}{2}\right)+C\)
6.6.3.38.
Answer.
\(0.666667\sinh^{-1}\mathopen{}\left(x^{1.5}\right)+C\)
6.6.3.39.
Answer.
\(-0.0625\tan^{-1}\mathopen{}\left(\frac{x}{2}\right)+0.03125\ln\mathopen{}\left(\left|x-2\right|\right)-0.03125\ln\mathopen{}\left(\left|x+2\right|\right)+C\)
6.6.3.40.
Answer.
\(\ln\mathopen{}\left(x\right)-\ln\mathopen{}\left(\left|x+1\right|\right)+C\)
6.6.3.41.
Answer.
\(\tan^{-1}\mathopen{}\left(e^{x}\right)+C\)
6.6.3.42.
Answer.
\(x\sinh^{-1}\mathopen{}\left(x\right)-\sqrt{x^{2}+1}+C\)
6.6.3.43.
Answer.
\(x\tanh^{-1}\mathopen{}\left(x\right)+0.5\ln\mathopen{}\left(\left|x^{2}-1\right|\right)+C\)
6.6.3.44.
Answer.
\(\tan^{-1}\mathopen{}\left(\sinh\mathopen{}\left(x\right)\right)+C\)
6.6.3.45.
Answer.
\(0\)
6.6.3.46.
Answer.
\(1.5\)
6.6.3.47.
Answer.
\(0.761594\)
6.6.3.48.
Answer.
\(1.44364\)

6.7 L’Hospital’s Rule
6.7.4 Exercises

Terms and Concepts

6.7.4.2.
Answer.
\(\text{False}\)
6.7.4.3.
Answer.
\(\text{False}\)

Problems

6.7.4.9.
Answer.
\(3\)
6.7.4.10.
Answer.
\(-1.66667\)
6.7.4.11.
Answer.
\(-1\)
6.7.4.12.
Answer.
\(-0.707107\)
6.7.4.13.
Answer.
\(5\)
6.7.4.14.
Answer.
\(0\)
6.7.4.15.
Answer.
\(0.666667\)
6.7.4.16.
Answer.
\(\frac{a\cos\mathopen{}\left(a\cdot 0\right)}{b\cos\mathopen{}\left(b\cdot 0\right)}\)
6.7.4.17.
Answer.
\(\infty \)
6.7.4.18.
Answer.
\(0.5\)
6.7.4.19.
Answer.
\(0\)
6.7.4.20.
Answer.
\(0\)
6.7.4.21.
Answer.
\(0\)
6.7.4.23.
Answer.
\(\infty \)
6.7.4.24.
Answer.
\(\infty \)
6.7.4.25.
Answer.
\(0\)
6.7.4.26.
Answer.
\(2\)
6.7.4.27.
Answer.
\(-2\)
6.7.4.28.
Answer.
\(0\)
6.7.4.29.
Answer.
\(0\)
6.7.4.30.
Answer.
\(0\)
6.7.4.31.
Answer.
\(0\)
6.7.4.32.
Answer.
\(0\)
6.7.4.33.
Answer.
\(\infty \)
6.7.4.34.
Answer.
\(\infty \)
6.7.4.35.
Answer.
\(\infty \)
6.7.4.36.
Answer.
\(0\)
6.7.4.37.
Answer.
\(0\)
6.7.4.38.
Answer.
\(e\)
6.7.4.39.
Answer.
\(1\)
6.7.4.40.
Answer.
\(1\)
6.7.4.41.
Answer.
\(1\)
6.7.4.42.
Answer.
\(1\)
6.7.4.43.
Answer.
\(1\)
6.7.4.44.
Answer.
\(0\)
6.7.4.45.
Answer.
\(1\)
6.7.4.46.
Answer.
\(1\)
6.7.4.47.
Answer.
\(1\)
6.7.4.48.
Answer.
\(1\)
6.7.4.49.
Answer.
\(2\)
6.7.4.50.
Answer.
\(\frac{1}{2}\)
6.7.4.51.
Answer.
\(-\infty \)
6.7.4.52.
Answer.
\(1\)
6.7.4.53.
Answer.
\(0\)
6.7.4.54.
Answer.
\(3\)

6.8 Improper Integration
6.8.4 Exercises

Terms and Concepts

6.8.4.4.
Answer.
\(p\gt 1\)
6.8.4.5.
Answer.
\(p\gt 1\)
6.8.4.6.
Answer.
\(p\lt 1\)

Problems

6.8.4.7.
Answer.
\(\frac{e^{5}}{2}\)
6.8.4.8.
Answer.
\(\frac{1}{2}\)
6.8.4.9.
Answer.
\(\frac{1}{3}\)
6.8.4.10.
Answer.
\(\frac{\pi }{3}\)
6.8.4.11.
Answer.
\(\frac{1}{\ln\mathopen{}\left(2\right)}\)
6.8.4.12.
Answer.
\(\infty \)
6.8.4.13.
Answer.
\(\infty \)
6.8.4.14.
Answer.
\(\infty \)
6.8.4.15.
Answer.
\(1\)
6.8.4.16.
Answer.
\(\infty \)
6.8.4.17.
Answer.
\(\infty \)
6.8.4.18.
Answer.
\(\infty \)
6.8.4.19.
Answer.
\(\infty \)
6.8.4.20.
Answer.
\(\infty \)
6.8.4.21.
Answer.
\(\infty \)
6.8.4.22.
Answer.
\(2+2\sqrt{2}\)
6.8.4.23.
Answer.
\(1\)
6.8.4.24.
Answer.
\(\frac{1}{2}\)
6.8.4.25.
Answer.
\(0\)
6.8.4.26.
Answer.
\(\frac{\pi }{2}\)
6.8.4.27.
Answer.
\(\frac{-1}{4}\)
6.8.4.28.
Answer.
\(\frac{-1}{9}\)
6.8.4.29.
Answer.
\(\infty \)
6.8.4.30.
Answer.
\(-1\)
6.8.4.31.
Answer.
\(1\)
6.8.4.32.
Answer.
\(\infty \)
6.8.4.33.
Answer.
\(\frac{1}{2}\)
6.8.4.34.
Answer.
\(\frac{1}{2}\)
6.8.4.35.
Answer 1.
\(\text{Limit Comparison Test}\)
Answer 2.
\(\text{diverges}\)
Answer 3.
\(\frac{1}{x}\)
6.8.4.36.
Answer 1.
\(\text{Limit Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(\frac{1}{x^{1.5}}\)
6.8.4.37.
Answer 1.
\(\text{Limit Comparison Test}\)
Answer 2.
\(\text{diverges}\)
Answer 3.
\(\frac{1}{x}\)
6.8.4.38.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(xe^{-x}\)
6.8.4.39.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(e^{-x}\)
6.8.4.40.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(xe^{-x}\)
6.8.4.41.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(\frac{1}{x^{2}-1}\)
6.8.4.42.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{diverges}\)
Answer 3.
\(\frac{x}{x^{2}+1}\)
6.8.4.43.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(\frac{1}{e^{x}}\)
6.8.4.44.
Answer 1.
\(\text{Limit Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(\frac{1}{e^{x}}\)

7 Applications of Integration
7.1 Area Between Curves

Exercises

Terms and Concepts
7.1.1.
Answer.
\(\text{True}\)
7.1.2.
Answer.
\(\text{True}\)
Problems
7.1.5.
Answer.
\(22.436\)
7.1.6.
Answer.
\(5.33333\)
7.1.7.
Answer.
\(3.14159\)
7.1.8.
Answer.
\(3.14159\)
7.1.9.
Answer.
\(0.5\)
7.1.10.
Answer.
\(2.82843\)
7.1.11.
Answer.
\(0.721354\)
7.1.12.
Answer.
\(4/3\)
7.1.13.
Answer.
\(4.5\)
7.1.14.
Answer.
\(1.33333\)
7.1.15.
Answer.
\(0.429204\)
7.1.16.
Answer.
\(8\)
7.1.17.
Answer.
\(0.166667\)
7.1.18.
Answer.
\(3.08333\)
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 \(2\sqrt{2}\text{.}\)
7.1.20.
Answer.
\(3.89711\)
7.1.21.
Answer.
\(1\)
7.1.22.
Answer.
\(1.66667\)
7.1.23.
Answer.
\(4.5\)
7.1.24.
Answer.
\(2.25\)
7.1.25.
Answer.
\(0.514298\)
7.1.26.
Answer.
\(4/3\)
7.1.27.
Answer.
\(1\)
7.1.28.
Answer.
\(5\)
7.1.29.
Answer.
\(4\)
7.1.30.
Answer.
\(10.5\)
7.1.31.
Answer.
\(262800\ {\rm ft^{2}}\)
7.1.32.
Answer.
\(623333\ {\rm ft^{2}}\)

7.2 Volume by Cross-Sectional Area; Disk and Washer Methods

Exercises

Terms and Concepts
7.2.1.
Answer.
T
7.2.2.
Answer.
Answers will vary.
Problems
7.2.4.
Answer.
\(48\pi\sqrt{3}/5\) units\(^3\)
7.2.5.
Answer.
\(175\pi/3\) units\(^3\)
7.2.6.
Answer.
\(\pi^2/4\) units\(^3\)
7.2.7.
Answer.
\(\pi/6\) units\(^3\)
7.2.8.
Answer.
\(9\pi/2\) units\(^3\)
7.2.9.
Answer.
\(35\pi/3\) units\(^3\)
7.2.10.
Answer.
\(\pi^2-2\pi\) units\(^3\)
7.2.11.
Answer.
\(2\pi/15\) units\(^3\)
7.2.12.
7.2.12.a
Answer.
\(\pi/2\)
7.2.12.b
Answer.
\(5\pi/6\)
7.2.12.c
Answer.
\(4\pi/5\)
7.2.12.d
Answer.
\(8\pi/15\)
7.2.13.
7.2.13.a
Answer.
\(512\pi/15\)
7.2.13.b
Answer.
\(256\pi/5\)
7.2.13.c
Answer.
\(832\pi/15\)
7.2.13.d
Answer.
\(128\pi/3\)
7.2.14.
7.2.14.a
Answer.
\(4\pi/3\)
7.2.14.b
Answer.
\(2\pi/3\)
7.2.14.c
Answer.
\(4\pi/3\)
7.2.14.d
Answer.
\(\pi/3\)
7.2.15.
7.2.15.a
Answer.
\(104\pi/15\)
7.2.15.b
Answer.
\(64\pi/15\)
7.2.15.c
Answer.
\(32\pi/5\)
7.2.16.
7.2.16.a
Answer.
\(\pi^2/2\)
7.2.16.b
Answer.
\(\pi^2/2-4\pi\sinh^{-1}(1)\)
7.2.16.c
Answer.
\(\pi^2/2+4\pi\sinh^{-1}(1)\)
7.2.17.
7.2.17.a
Answer.
\(8\pi\)
7.2.17.b
Answer.
\(8\pi\)
7.2.17.c
Answer.
\(16\pi/3\)
7.2.17.d
Answer.
\(8\pi/3\)
7.2.18.
Answer.
\(250\pi/3\)
7.2.19.
Answer.
\(250\pi/3\)
7.2.20.
Answer.
\(80/3\)
7.2.21.
Answer.
\(187.5\)

7.3 The Shell Method

Exercises

Terms and Concepts
7.3.1.
Answer.
T
7.3.2.
Answer.
F
7.3.3.
Answer.
F
7.3.4.
Answer.
T
Problems
7.3.5.
Answer.
\(9\pi/2\) units\(^3\)
7.3.6.
Answer.
\(70\pi/3\) units\(^3\)
7.3.7.
Answer.
\(\pi^2-2\pi\) units\(^3\)
7.3.8.
Answer.
\(2\pi/15\) units\(^3\)
7.3.9.
Answer.
\(48\pi\sqrt{3}/5\) units\(^3\)
7.3.10.
Answer.
\(350\pi/3\) units\(^3\)
7.3.11.
Answer.
\(\pi^2/4\) units\(^3\)
7.3.12.
Answer.
\(\pi/6\) units\(^3\)
7.3.13.
7.3.13.a
Answer.
\(4\pi/5\)
7.3.13.b
Answer.
\(8\pi/15\)
7.3.13.c
Answer.
\(\pi/2\)
7.3.13.d
Answer.
\(5\pi/6\)
7.3.14.
7.3.14.a
Answer.
\(128\pi/3\)
7.3.14.b
Answer.
\(128\pi/3\)
7.3.14.c
Answer.
\(512\pi/15\)
7.3.14.d
Answer.
\(256\pi/5\)
7.3.15.
7.3.15.a
Answer.
\(4\pi/3\)
7.3.15.b
Answer.
\(\pi/3\)
7.3.15.c
Answer.
\(4\pi/3\)
7.3.15.d
Answer.
\(2\pi/3\)
7.3.16.
7.3.16.a
Answer.
\(16\pi/3\)
7.3.16.b
Answer.
\(8\pi/3\)
7.3.16.c
Answer.
\(8\pi\)
7.3.17.
7.3.17.a
Answer.
\(2\pi(\sqrt{2}-1)\)
7.3.17.b
Answer.
\(2\pi(1-\sqrt{2}+\sinh^{-1}(1))\)
7.3.18.
7.3.18.a
Answer.
\(16\pi/3\)
7.3.18.b
Answer.
\(8\pi/3\)
7.3.18.c
Answer.
\(8\pi\)
7.3.18.d
Answer.
\(8\pi\)

7.4 Arc Length and Surface Area
7.4.3 Exercises

Problems

7.4.3.3.
Answer.
\(\sqrt{2}\)
7.4.3.4.
Answer.
\(6\)
7.4.3.5.
Answer.
\(\frac{10}{3}\)
7.4.3.6.
Answer.
\(6\)
7.4.3.7.
Answer.
\(\frac{157}{3}\)
7.4.3.8.
Answer.
\(\frac{3}{2}\)
7.4.3.9.
Answer.
\(\frac{12}{5}\)
7.4.3.10.
Answer.
\(\frac{7.99533\times 10^{7}}{400000}\)
7.4.3.11.
Answer.
\(-\ln(2-\sqrt{3}) \approx 1.31696\)
7.4.3.12.
Answer.
\(\sinh^{-1}(1)\)
7.4.3.13.
Answer.
\(\int_0^1 \sqrt{1+4x^2}\, dx\)
7.4.3.14.
Answer.
\(\int_0^1 \sqrt{1+100x^{18}}\, dx\)
7.4.3.15.
Answer.
\(\int_1^e \sqrt{1+\frac1{x^2}}\, dx\)
7.4.3.16.
Answer.
\(\int_{1}^2 \sqrt{1+\frac1{x^4}}\, dx\)
7.4.3.17.
Answer.
\(\int_0^{\pi/2}\sqrt{1+\sin^2(x)}\,dx\)
7.4.3.18.
Answer.
\(\int_{-\pi/4}^{\pi/4} \sqrt{1+\sec^2(x) \tan^2(x) }\, dx\)
7.4.3.19.
Answer.
\(1.4790\)
7.4.3.20.
Answer.
\(1.8377\)
7.4.3.21.
Answer.
\(2.1300\)
7.4.3.22.
Answer.
\(1.3254\)
7.4.3.23.
Answer.
\(1.00013\)
7.4.3.24.
Answer.
\(1.7625\)
7.4.3.25.
Answer.
\(2\pi\int_0^1 2x\sqrt{5}\, dx = 2\pi\sqrt{5}\)
7.4.3.26.
Answer.
\(2\pi\int_0^1 x\sqrt{5}\, dx = \pi\sqrt{5}\)
7.4.3.27.
Answer.
\(2\pi\int_0^1 x\sqrt{1+4x^2}\, dx = \pi/6(5\sqrt{5}-1)\)
7.4.3.28.
Answer.
\(2\pi\int_0^1 x^3\sqrt{1+9x^4}\, dx = \pi/27(10\sqrt{10}-1)\)
7.4.3.29.
Answer.
\(\int_0^1 \sqrt{1+\frac{1}{4x}}\, dx\)
7.4.3.30.
Answer.
\(\int_{-1}^1 \sqrt{1+\frac{x^2}{1-x^2}}\, dx\)
7.4.3.31.
Answer.
\(\int_{-3}^3 \sqrt{1+\frac{x^2}{81-9x^2}}\, dx\)
7.4.3.32.
Answer.
\(2\pi\int_0^1 \sqrt{x}\sqrt{1+1/(4x)}\, dx = \pi/6(5\sqrt{5}-1)\)
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·ms2m In Imperial Units, it is ft–lb.
7.5.4.2.
Answer.
The same.
7.5.4.3.
Answer.
Smaller.
7.5.4.4.
Answer.
force; distance

Problems

7.5.4.5.
7.5.4.5.a
Answer.
500 ft–lb
7.5.4.5.b
Answer.
\(100-50\sqrt{2} \approx 29.29\) ft
7.5.4.6.
7.5.4.6.a
Answer.
2450 J
7.5.4.6.b
Answer.
1568 J
7.5.4.7.
7.5.4.7.a
Answer.
\(\frac12\cdot d\cdot l^2\) ft–lb
7.5.4.7.b
Answer.
75 %
7.5.4.7.c
Answer.
\(\ell(1-\sqrt{2}/2) \approx 0.2929\ell\)
7.5.4.8.
Answer.
735 J
7.5.4.9.
7.5.4.9.a
Answer.
756 ft–lb
7.5.4.9.b
Answer.
60,000 ft–lb
7.5.4.9.c
Answer.
Yes, for the cable accounts for about 1% of the total work.
7.5.4.10.
Answer.
11,100 ft–lb
7.5.4.11.
Answer.
575 ft–lb
7.5.4.12.
Answer.
125 ft–lb
7.5.4.13.
Answer.
0.05 J
7.5.4.14.
Answer.
12.5 ft–lb
7.5.4.15.
Answer.
5/3 ft–lb
7.5.4.16.
Answer.
0.2625 = 21/80 J
7.5.4.17.
Answer.
\(f\cdot d/2\) J
7.5.4.18.
Answer.
45 ft–lb
7.5.4.19.
Answer.
5 ft–lb
7.5.4.20.
Answer.
\(953,284\) J
7.5.4.21.
7.5.4.21.a
Answer.
52,929.6 ft–lb
7.5.4.21.b
Answer.
18,525.3 ft–lb
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.
7.5.4.22.
Answer.
192,767 ft–lb. Note that the tank is oriented horizontally. Let the origin be the center of one of the circular ends of the tank. Since the radius is 3.75 ft, the fluid is being pumped to \(y=4.75\text{;}\) thus the distance the gas travels is \(h(y)=4.75-y\text{.}\) A differential element of water is a rectangle, with length 20 and width \(2\sqrt{3.75^2-y^2}\text{.}\) Thus the force required to move that slab of gas is \(F(y) = 40\cdot45.93\cdot\sqrt{3.75^2-y^2}dy\text{.}\) Total work is \(\int_{-3.75}^{3.75} 40\cdot45.93\cdot(4.75-y)\sqrt{3.75^2-y^2}\, dy\text{.}\) This can be evaluated without actual integration; split the integral into \(\int_{-3.75}^{3.75} 40\cdot45.93\cdot(4.75)\sqrt{3.75^2-y^2}\, dy + \int_{-3.75}^{3.75} 40\cdot45.93\cdot(-y)\sqrt{3.75^2-y^2}\, dy\text{.}\) The first integral can be evaluated as measuring half the area of a circle; the latter integral can be shown to be 0 without much difficulty. (Use substitution and realize the bounds are both 0.)
7.5.4.23.
Answer.
212,135 ft–lb
7.5.4.24.
7.5.4.24.a
Answer.
approx. 577,000 J
7.5.4.24.b
Answer.
approx. 399,000 J
7.5.4.24.c
Answer.
approx 110,000 J (By volume, half of the water is between the base of the cone and a height of 3.9685 m. If one rounds this to 4 m, the work is approx 104,000 J.)
7.5.4.25.
Answer.
187,214 ft–lb
7.5.4.26.
Answer.
617,400 J
7.5.4.27.
Answer.
4,917,150 J

7.6 Fluid Forces

Exercises

Terms and Concepts
7.6.1.
Answer.
Answers will vary.
7.6.2.
Answer.
Answers will vary.
Problems
7.6.3.
Answer.
499.2 lb
7.6.4.
Answer.
249.6 lb
7.6.5.
Answer.
6739.2 lb
7.6.6.
Answer.
5241.6 lb
7.6.7.
Answer.
3920.7 lb
7.6.8.
Answer.
15682.8 lb
7.6.9.
Answer.
2496 lb
7.6.10.
Answer.
2496 lb
7.6.11.
Answer.
602.59 lb
7.6.12.
Answer.
291.2 lb
7.6.13.
Answer.
  1. 2340 lb
  2. 5625 lb
7.6.14.
Answer.
  1. 1064.96 lb
  2. 2560 lb
7.6.15.
Answer.
  1. 1597.44 lb
  2. 3840 lb
7.6.16.
Answer.
  1. 41.6 lb
  2. 100 lb
7.6.17.
Answer.
  1. 56.42 lb
  2. 135.62 lb
7.6.18.
Answer.
  1. 1123.2 lb
  2. 2700 lb
7.6.19.
Answer.
5.1 ft
7.6.20.
Answer.
4.1 ft

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.
8.1.4.2.
Answer.
Answers will vary.
8.1.4.3.
Answer.
Substitute the proposed function into the differential equation, and show the the statement is satisfied.
8.1.4.4.
Answer.
A particular solution is one specifica member of a family of solutions, and has no arbitrary constants. A general solution is a family of solutions, includes all possible solutions to the differential equation, and typically includes one or more arbitrary constants.
8.1.4.5.
Answer.
Many differential equations are impossible to solve analytically.
8.1.4.6.
Answer.
A smaller \(h\) value leads to a numerical solution that is closer to the true solution, but decreasing the \(h\) value leads to more computational effort.

Problems

8.1.4.7.
Answer.
Answers will vary.
8.1.4.8.
Answer.
Answers will vary.
8.1.4.9.
Answer.
Answers will vary.
8.1.4.10.
Answer.
Answers will vary.
8.1.4.11.
Answer.
\(C = 2\)
8.1.4.12.
Answer.
\(C = 6\)
8.1.4.13.
Answer.
Graph showing slope field for the given differential equation.
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.
8.1.4.14.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated. The field lines form concentric ovals facing away from the origin on both positive and negative \(x\) and \(y\) axes. The concentric shorter arcs are on either end of the \(x\) axis. On the two ends of the \(y\) axis concentric wider arcs are drawn. The field lines intermix to form an ’X’ with centre at the origin.
8.1.4.15.
Answer.
Graph showing slope field for the given differential equation.
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.
8.1.4.16.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated. The field lines run almost parallel to the \(x\) axis. Above the axis the field lines are slightly facing north east. Below the \(x\) axis the lines are directed facing southeast.
8.1.4.17.
Answer.
b
8.1.4.18.
Answer.
c
8.1.4.19.
Answer.
d
8.1.4.20.
Answer.
a
8.1.4.21.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated, the field lines in the first quadrant are shown. The field lines very close to the \(y\) axis are almost north facing for higher values of \(y\) and almost east facing for lower values of \(y\text{.}\) With smaller values of \(x\text{,}\) the field lines, from left to right the lines first face northeast then east and southeast after for greater values of \(x\text{.}\)
A curve is drawn that starts at a point for some small value of \(x\) and a high value of \(y\text{.}\) The curve has a positive slope at first after reaching a peak it declines almost close to the \(x\) axis.
8.1.4.22.
Answer.
Graph showing slope field for the given differential equation.
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.
8.1.4.23.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated, the field lines in the first quadrant are shown. There are two instances where the field lines are parallel to the \(x\) axis. From under the \(x\) axis to the first such line the field lines transition from almost north facing to northeast facing. Between the horizontal field line for a small \(y\) value and a greater \(y\) value the field lines are facing southeast. Above the line with a higher \(y\) value the field lines transition from northeast facing to north facing.
8.1.4.24.
Answer.
Graph showing slope field for the given differential equation.
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.
8.1.4.25.
Answer.
\begin{align*} x_i \amp \quad \amp \quad \amp y_i\\ 0.00 \amp \quad \amp \quad \amp 1.0000 \\ 0.25 \amp \quad \amp \quad \amp 1.5000 \\ 0.50 \amp \quad \amp \quad \amp 2.3125 \\ 0.75 \amp \quad \amp \quad \amp 3.5938\\ 1.00 \amp \quad \amp \quad \amp 5.5781 \end{align*}
8.1.4.26.
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*}
8.1.4.27.
Answer.
\begin{align*} x_i \amp \quad \amp \quad \amp y_i \\ 0.0 \amp \quad \amp \quad \amp 2.0000 \\ 0.2 \amp \quad \amp \quad \amp 2.4000 \\ 0.4 \amp \quad \amp \quad \amp 2.9197 \\ 0.6 \amp \quad \amp \quad \amp 3.5816 \\ 0.8 \amp \quad \amp \quad \amp 4.4108 \\ 1.0 \amp \quad \amp \quad \amp 5.4364 \end{align*}
8.1.4.28.
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*}
8.1.4.29.
Answer.
\(x\) \(0.0\) \(0.2\) \(0.4\) \(0.6\) \(0.8\) \(1.0\)
\(y(x)\) 1.0000 1.0204 1.0870 1.2195 1.4706 2.0000
\(h = 0.2\) 1.0000 1.0000 1.0400 1.1265 1.2788 1.5405
\(h = 0.1\) 1.0000 1.0100 1.0623 1.1687 1.3601 1.7129
8.1.4.30.
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. \(\displaystyle \frac{1}{y^2-y}\,dy = dx\)
8.2.2.2.
Answer.
Not separable.
8.2.2.3.
Answer.
Not separable.
8.2.2.4.
Answer.
Separable. \(\displaystyle \frac{1}{\cos y - y}\,dy = (x^2+1)\,dx\)
8.2.2.5.
Answer.
\(\left \{ \displaystyle y = \frac{1 + Ce^{2x}}{1 - Ce^{2x}}, y = -1\right \}\)
8.2.2.6.
Answer.
\(y = 2 + Ce^x\)
8.2.2.7.
Answer.
\(y = Cx^4\)
8.2.2.8.
Answer.
\(y^2 - 4x^2 = C\)
8.2.2.9.
Answer.
\(\displaystyle (y-1)e^y = -e^{-x} - \frac{1}{3}e^{-3x} + C\)
8.2.2.10.
Answer.
\(\displaystyle (y-1)^2 = \ln(x^2+1) + C\)
8.2.2.11.
Answer.
\(\left \{ \arcsin{2y} - \arctan(x^2+1) = C, y = \pm \displaystyle \frac{1}{2} \right \}\)
8.2.2.12.
Answer.
\(\left \{ \displaystyle y = \frac{1}{C - \arctan x}, y = 0 \right \}\)
8.2.2.13.
Answer.
\(\sin y + \cos(x) = 2\)
8.2.2.14.
Answer.
\(-x^3 + 3y - y^3 = 2\)
8.2.2.15.
Answer.
\(\frac{1}{2}y^2 - \ln(1+x^2) = 8\)
8.2.2.16.
Answer.
\(y^2+2xe^x - 2e^x = 2\)
8.2.2.17.
Answer.
\(\displaystyle \frac{1}{2}y^2 - y = \frac{1}{2}\big ( (x^2+1)\ln(x^2+1) - (x^2 + 1)\big) + \frac{1}{2}\)
8.2.2.18.
Answer.
\(\sin(y^2)-(\arcsin x)^2 = -\frac{1}{2}\)
8.2.2.19.
Answer.
\(2\tan 2y = 2x + \sin 2x\)
8.2.2.20.
Answer.
\(x = exp \displaystyle \left ( -\frac{\sqrt{1-y^2}}{y}\right )\)

8.3 First Order Linear Differential Equations
8.3.2 Exercises

Problems

8.3.2.1.
Answer.
\(y = \displaystyle \frac{3}{2} + Ce^{2x}\)
8.3.2.2.
Answer.
\(y = \displaystyle \frac{\ln \abs{x} + C}{x}\)
8.3.2.3.
Answer.
\(y = \displaystyle -\frac{1}{2x} + Cx\)
8.3.2.4.
Answer.
\(y = \displaystyle \frac{x^3}{7} - \frac{x}{5} + \frac{C}{x^4}\)
8.3.2.5.
Answer.
\(y = \sec x + C(\csc x)\)
8.3.2.6.
Answer.
\(y = \displaystyle \frac{1}{2} + Ce^{-x^2}\)
8.3.2.7.
Answer.
\(y = \displaystyle Ce^{3x}-(x+1)e^{2x}\)
8.3.2.8.
Answer.
\(y = sin(2x) - 2\cos(2x) + Ce^{-x}\)
8.3.2.9.
Answer.
\(y = (x^2+2)e^x\)
8.3.2.10.
Answer.
\(y = \displaystyle \frac{1}{4}x^2-\frac{1}{3}x+\frac{1}{2}+\frac{7}{12x^2}\)
8.3.2.11.
Answer.
\(y = \displaystyle 1 - \frac{2}{x} + \frac{2-e^{1-x}}{x^2}\)
8.3.2.12.
Answer.
\(y = \displaystyle 3e^{-2x}\)
8.3.2.13.
Answer.
\(y = \displaystyle \frac{x^2+1}{x+1}e^{-x}\)
8.3.2.14.
Answer.
\(y = \sin(x) - 3\cos(x)\)
8.3.2.15.
Answer.
\(y = \displaystyle \frac{(x-2)(x+1)}{x-1}\)
8.3.2.16.
Answer.
\(y = \displaystyle x^2\left (\arctan x - \frac{\pi}{4}\right )\)
8.3.2.17.
Answer.
Both; \(\displaystyle y = -5e^{x + \frac{1}{3}x^3}\)
8.3.2.18.
Answer.
separable; \(\displaystyle e^y = \sin(x) - x\cos(x) + 1\)
8.3.2.19.
Answer.
linear; \(\displaystyle y = \frac{x^3-3x-6}{3(x-1)}\)
8.3.2.20.
Answer.
separable; \(\displaystyle y = 1\)
8.3.2.21.
Answer.
Graph showing slope field for the given differential equation.
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 \(y=x-1\text{.}\)
\(y = x-1 + e^{-x}\)
8.3.2.22.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated, the field lines in the first quadrant are shown. The lines in the top are southeast facing, for lower values of \(y\) from left to right the field lines are northeast facing then they transition to east facing. A downward sloping curve is shown on the field lines.
The solution will decrease and approach \(y=0\text{.}\)
\(\displaystyle y = \frac{2 + \ln(x+1)}{x+1}\)

8.4 Modeling with Differential Equations
8.4.3 Exercises

Problems

8.4.3.1.
Answer.
\(y = 10 + Ce^{-kx}\)
8.4.3.2.
Answer.
13.66 days
8.4.3.3.
Answer.
4.43 days
8.4.3.4.
Answer.
13,304.65 years old
8.4.3.5.
Answer.
\(x = \begin{cases}\displaystyle\frac{ab(1 - e^{(a-b)kt})}{b-ae^{(a-b)kt}} \amp \text{ if } a \neq b\\ \displaystyle \frac{a^2kt}{1+akt} \amp \text{ if } a = b \end{cases}\)
8.4.3.6.
Answer.
24.57 minutes
8.4.3.7.
Answer.
\(\displaystyle y = 60 - 3.69858e^{-\frac{1}{4}t} + 43.69858e^{-0.0390169 t}\)
8.4.3.8.
Answer.
0.06767 g/gal
8.4.3.9.
Answer.
\(y = 8(1-e^{-\frac{1}{2}t})\) g/cm\(^2\)
8.4.3.10.
Answer.
\(y = \displaystyle 20 - \frac{10}{17}\left (4\cos(2t)- \sin(2t)\right) - \frac{300}{17}e^{-\frac{1}{2}t}\) g
8.4.3.11.
Answer.
11.00075 g
8.4.3.12.
Answer.
pond 1: 50.4853 grams per million gallons
pond 2: 32.8649 grams per million gallons

9 Curves in the Plane
9.1 Conic Sections
9.1.4 Exercises

Terms and Concepts

9.1.4.6.
Answer.
line

Problems

9.1.4.19.
Answer.
\(\frac{(x+1)^2}{9}+\frac{(y-2)^2}{4}=1\text{;}\) foci at \((-1\pm\sqrt{5},2)\text{;}\) \(e=\sqrt{5}/3\)
9.1.4.20.
Answer.
\(\frac{(x-1)^2}{1/4}+\frac{y^2}{9}=1\text{;}\) foci at \((1,\pm \sqrt{8.75})\text{;}\) \(e=\sqrt{8.75}/3\approx 0.99\)
9.1.4.29.
Answer.
\(x^2-\frac{y^2}{3}=1\)
9.1.4.30.
Answer.
\(y^2-\frac{x^2}{24}=1\)
9.1.4.31.
Answer.
\(\frac{(y-3)^2}{4}-\frac{(x-1)^2}{9}=1\)
9.1.4.32.
Answer.
\(\frac{(x-1)^2}{9}-\frac{(y-3)^2}{4}=1\)
9.1.4.45.
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.)