Skip to main content
Logo image

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.
8.1.4.14.
Answer.
8.1.4.15.
Answer.
8.1.4.16.
Answer.
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.
8.1.4.22.
Answer.
8.1.4.23.
Answer.
8.1.4.24.
Answer.
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.
The solution will increase and begin to follow the line \(y=x-1\text{.}\)
\(y = x-1 + e^{-x}\)
8.3.2.22.
Answer.
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.)

9.2 Parametric Equations
9.2.4 Exercises

Terms and Concepts

9.2.4.1.
Answer.
\(\text{True}\)
9.2.4.2.
Answer.
\({\text{orientation}}\)
9.2.4.3.
Answer.
\({\text{rectangular}}\)

Problems

9.2.4.5.
Answer.
9.2.4.6.
Answer.
9.2.4.7.
Answer.
9.2.4.8.
Answer.
9.2.4.9.
Answer.
9.2.4.10.
Answer.
9.2.4.11.
Answer.
9.2.4.12.
Answer.
9.2.4.13.
Answer.
9.2.4.14.
Answer.
9.2.4.15.
Answer.
9.2.4.16.
Answer.
9.2.4.17.
Answer.
9.2.4.18.
Answer.
9.2.4.19.
9.2.4.19.a
Answer.
Traces the parabola \(y=x^2\text{,}\) moves from left to right.
9.2.4.19.b
Answer.
Traces the parabola \(y=x^2\text{,}\) but only from \(-1\leq x\leq 1\text{;}\) traces this portion back and forth infinitely.
9.2.4.19.c
Answer.
Traces the parabola \(y=x^2\text{,}\) but only for \(0\lt x\text{.}\) Moves left to right.
9.2.4.19.d
Answer.
Traces the parabola \(y=x^2\text{,}\) moves from right to left.
9.2.4.20.
9.2.4.20.a
Answer.
Traces a circle of radius 1 counterclockwise once.
9.2.4.20.b
Answer.
Traces a circle of radius 1 counterclockwise over 6 times.
9.2.4.20.c
Answer.
Traces a circle of radius 1 clockwise infinite times.
9.2.4.20.d
Answer.
Traces an arc of a circle of radius 1, from an angle of -1 radians to 1 radian, twice.
9.2.4.21.
Answer.
\(3x+2y = 17\)
9.2.4.25.
Answer.
\(y-2x = 3\)
9.2.4.30.
Answer.
\(x = 1-2y^{2}\)
9.2.4.35.
Answer 1.
\(\frac{t+11}{6}\)
Answer 2.
\(\frac{t^{2}-97}{12}\)
Answer 3.
\(\left(2,-8\right)\)
Answer 4.
\(6x-11\)
Answer 5.
\(1\)
9.2.4.36.
Answer 1.
\(\ln\mathopen{}\left(t\right)\)
Answer 2.
\(t\)
Answer 3.
\(\left(0,1\right)\)
Answer 4.
\(e^{x}\)
Answer 5.
\(1\)
9.2.4.37.
Answer 1.
\(\cos^{-1}\mathopen{}\left(t\right)\)
Answer 2.
\(\sqrt{1-t^{2}}\)
Answer 3.
\(\left(0,0\right)\)
Answer 4.
\(\cos\mathopen{}\left(x\right)\)
Answer 5.
\(1\)
9.2.4.39.
Answer 1.
\(-1, 1\)
Answer 2.
\(\left(3,-2\right)\)
9.2.4.44.
Answer 1.
\(2\)
Answer 2.
\(\left(-4,-8\right)\)
9.2.4.46.
Answer 1.
\(0\)
Answer 2.
\(\left(1,0\right)\)
9.2.4.50.
Answer.
\(2\cos\mathopen{}\left(t\right);\,-2\sin\mathopen{}\left(t\right)\)
9.2.4.51.
Answer.
\(3\cos\mathopen{}\left(2\pi t\right)+1;\,3\sin\mathopen{}\left(2\pi t\right)+1\)
9.2.4.52.
Answer.
\(3\cos\mathopen{}\left(2\pi t\right)+1;\,3\sin\mathopen{}\left(2\pi t\right)+1\)

9.3 Calculus and Parametric Equations
9.3.4 Exercises

Terms and Concepts

9.3.4.1.
Answer.
\(\text{False}\)
9.3.4.3.
Answer.
\(\text{False}\)
9.3.4.4.
Answer.
\(\text{True}\)

Problems

9.3.4.15.
Answer 1.
\(-0.5\)
Answer 2.
\(\left(0.75,-0.25\right)\)
9.3.4.18.
Answer 1.
\(\frac{\pi }{4}, \frac{3\pi }{4}, \frac{5\pi }{4}, \frac{7\pi }{4}\)
Answer 2.
\(\left(\frac{\sqrt{2}}{2},1\right), \left(\frac{-\sqrt{2}}{2},-1\right), \left(\frac{-\sqrt{2}}{2},1\right), \left(\frac{\sqrt{2}}{2},-1\right)\)
9.3.4.21.
Answer 1.
\(0\)
Answer 2.
\(0\)
9.3.4.22.
Answer 1.
\(2\)
Answer 2.
\(1\)
9.3.4.27.
Answer 1.
\(-\frac{4}{\left(2t-1\right)^{3}}\)
Answer 2.
\(\left(-\infty ,0.5\right]\)
Answer 3.
\(\left[0.5,\infty \right)\)
9.3.4.30.
Answer 1.
\(\frac{2\mathopen{}\left(\sin\mathopen{}\left(t\right)\mathopen{}\left(-2\right)\sin\mathopen{}\left(2t\right)-\cos\mathopen{}\left(2t\right)\cos\mathopen{}\left(t\right)\right)}{\sin^{3}\mathopen{}\left(t\right)}\)
Answer 2.
\(\left[\frac{\pi }{2}, \pi \right], \left[\frac{3\pi }{2}, 2\pi \right]\)
Answer 3.
\(\left[0, \frac{\pi }{2}\right], \left[\pi , \frac{3\pi }{2}\right]\)
9.3.4.33.
Answer.
\(6\pi \)
9.3.4.34.
Answer 1.
\(\sqrt{101}\mathopen{}\left(e^{\frac{\pi }{5}}-1\right)\)
Answer 2.
\(\sqrt{101}\mathopen{}\left(e^{\frac{2\pi }{5}}-e^{\frac{\pi }{5}}\right)\)
9.3.4.35.
Answer.
\(2\sqrt{34}\)

9.4 Introduction to Polar Coordinates
9.4.4 Exercises

Terms and Concepts

9.4.4.1.
Answer.
Answers will vary.
9.4.4.2.
Answer.
\(\text{False}\)
9.4.4.3.
Answer.
\(\text{True}\)
9.4.4.4.
Answer.
\(\text{False}\)

Problems

9.4.4.5.
Answer.
9.4.4.6.
Answer.
9.4.4.7.
Answer.
\(A=P(2.5,\pi/4)\) and \(P(-2.5,5\pi/4)\text{;}\)
\(B=P(-1,5\pi/6)\) and \(P(1,11\pi/6)\text{;}\)
\(C=P(3,4\pi/3)\) and \(P(-3,\pi/3)\text{;}\)
\(D=P(1.5,2\pi/3)\) and \(P(-1.5,5\pi/3)\text{;}\)
9.4.4.8.
Answer 1.
\(\left(2, 0.523599\right), \left(-2, -2.61799\right)\)
Answer 2.
\(\left(1, -1.0472\right), \left(-1, 2.0944\right)\)
Answer 3.
\(\left(2, 2.35619\right), \left(-2, -0.785398\right)\)
Answer 4.
\(\left(2.5, 3.14159\right), \left(2.5, -3.14159\right)\)
9.4.4.9.
Answer 1.
\(\left(\sqrt{2},\sqrt{2}\right)\)
Answer 2.
\(\left(\sqrt{2},-\sqrt{2}\right)\)
Answer 3.
\(\left(\sqrt{5},\tan^{-1}\mathopen{}\left(\frac{-1}{2}\right)\right)\)
Answer 4.
\(\left(\sqrt{5},\pi +\tan^{-1}\mathopen{}\left(\frac{-1}{2}\right)\right)\)
9.4.4.10.
Answer 1.
\(\left(-3,0\right)\)
Answer 2.
\(\left(\frac{-1}{2},\frac{\sqrt{3}}{2}\right)\)
Answer 3.
\(\left(4,\frac{\pi }{2}\right)\)
Answer 4.
\(\left(2,\frac{-\pi }{3}\right)\)
9.4.4.11.
Answer.
9.4.4.12.
Answer.
9.4.4.13.
Answer.
9.4.4.14.
Answer.
9.4.4.15.
Answer.
9.4.4.16.
Answer.
9.4.4.17.
Answer.
9.4.4.18.
Answer.
9.4.4.19.
Answer.
9.4.4.20.
Answer.
9.4.4.21.
Answer.
9.4.4.22.
Answer.
9.4.4.23.
Answer.
9.4.4.24.
Answer.
9.4.4.25.
Answer.
9.4.4.26.
Answer.
9.4.4.27.
Answer.
9.4.4.28.
Answer.
9.4.4.29.
Answer.
9.4.4.30.
Answer.
9.4.4.31.
Answer.
\(\left(x-3\right)^{2}+y^{2} = 9\)
9.4.4.32.
Answer.
\(x^{2}+\left(y+2\right)^{2} = 4\)
9.4.4.33.
Answer.
\(\left(x-0.5\right)^{2}+\left(y-0.5\right)^{2} = 0.5\)
9.4.4.34.
Answer.
\(y = 0.4x+1.4\)
9.4.4.35.
Answer.
\(x = 3\)
9.4.4.36.
Answer.
\(y = 4\)
9.4.4.38.
Answer.
\(y^4+x^2y^2-x^2=0\)
9.4.4.39.
Answer.
\(x^{2}+y^{2} = 4\)
9.4.4.40.
Answer.
\(y = \frac{x}{1.73205}\)
9.4.4.41.
Answer.
\(\theta = \frac{\pi }{4}\)
9.4.4.42.
Answer.
\(r = \frac{7}{\sin\mathopen{}\left(\theta\right)-4\cos\mathopen{}\left(\theta\right)}\)
9.4.4.43.
Answer.
\(r = 5\sec\mathopen{}\left(\theta\right)\)
9.4.4.44.
Answer.
\(r = 5\csc\mathopen{}\left(\theta\right)\)
9.4.4.45.
Answer.
\(r = \frac{\cos\mathopen{}\left(\theta\right)}{\sin^{2}\mathopen{}\left(\theta\right)}\)
9.4.4.47.
Answer.
\(r = \sqrt{7}\)
9.4.4.49.
Answer.
\(P\left(\frac{\sqrt{3}}{2},\frac{\pi }{6}\right), P\left(0,\frac{\pi }{2}\right), P\left(\frac{-\sqrt{3}}{2},\frac{5\pi }{6}\right)\)
9.4.4.51.
Answer.
\(P\left(0,0\right), P\left(\sqrt{2},\frac{\pi }{4}\right)\)
9.4.4.54.
Answer.
\(P\left(\frac{3}{2},\frac{\pi }{3}\right), P\left(\frac{3}{2},\frac{-\pi }{3}\right), P\left(0,\pi \right)\)

9.5 Calculus and Polar Functions
9.5.5 Exercises

Problems

9.5.5.3.
Answer 1.
\(-\cot\mathopen{}\left(\theta\right)\)
Answer 2.
\(y = -\left(x-\frac{\sqrt{2}}{2}\right)+\frac{\sqrt{2}}{2}\)
Answer 3.
\(y = x\)
9.5.5.4.
Answer 1.
\(0.5\mathopen{}\left(\tan\mathopen{}\left(\theta\right)-\cot\mathopen{}\left(\theta\right)\right)\)
Answer 2.
\(y = \frac{1}{2}\)
Answer 3.
\(x = \frac{1}{2}\)
9.5.5.7.
Answer 1.
\(\frac{\theta\cos\mathopen{}\left(\theta\right)+\sin\mathopen{}\left(\theta\right)}{\cos\mathopen{}\left(\theta\right)-\theta\sin\mathopen{}\left(\theta\right)}\)
Answer 2.
\(y = \frac{-2}{\pi }x+\frac{\pi }{2}\)
Answer 3.
\(y = \frac{\pi }{2}x+\frac{\pi }{2}\)
9.5.5.8.
Answer 1.
\(\frac{\cos\mathopen{}\left(\theta\right)\cos\mathopen{}\left(3\theta\right)-3\sin\mathopen{}\left(\theta\right)\sin\mathopen{}\left(3\theta\right)}{-\cos\mathopen{}\left(3\theta\right)\sin\mathopen{}\left(\theta\right)-3\cos\mathopen{}\left(\theta\right)\sin\mathopen{}\left(3\theta\right)}\)
Answer 2.
\(y = \frac{x}{\sqrt{3}}\)
Answer 3.
\(y = -\sqrt{3}x\)
9.5.5.9.
Answer 1.
\(\frac{4\sin\mathopen{}\left(\theta\right)\cos\mathopen{}\left(4\theta\right)+\sin\mathopen{}\left(4\theta\right)\cos\mathopen{}\left(\theta\right)}{4\cos\mathopen{}\left(\theta\right)\cos\mathopen{}\left(4\theta\right)-\sin\mathopen{}\left(\theta\right)\sin\mathopen{}\left(4\theta\right)}\)
Answer 2.
\(y = 5\sqrt{3}\mathopen{}\left(x+\frac{\sqrt{3}}{4}\right)-\frac{3}{4}\)
Answer 3.
\(y = \frac{-1}{5\sqrt{3}}\mathopen{}\left(x+\frac{\sqrt{3}}{4}\right)-\frac{3}{4}\)
9.5.5.14.
Answer 1.
\(\frac{\pi }{3}, \pi , \frac{5\pi }{3}\)
Answer 2.
\(0, \frac{2\pi }{3}, \frac{4\pi }{3}\)
9.5.5.19.
Answer.
\(\frac{\pi }{12}\)
9.5.5.20.
Answer.
area = \(\pi/(4n)\)
9.5.5.21.
Answer.
\(\frac{3\pi }{2}\)
9.5.5.23.
Answer.
\(2\pi +\frac{3\cdot 1.73205}{2}\)
9.5.5.24.
Answer.
\(\pi +3\cdot 1.73205\)
9.5.5.25.
Answer.
\(1\)
9.5.5.26.
Answer.
\(\frac{1}{32}\mathopen{}\left(4\pi -3\cdot 1.73205\right)\)
9.5.5.29.
Answer.
\(4\pi \)
9.5.5.30.
Answer.
\(4\pi \)
9.5.5.31.
Answer.
\(\sqrt{2}\pi\)
9.5.5.32.
Answer.
\(8\)
9.5.5.33.
Answer.
\(2.2592\hbox{ or }2.22748\)
9.5.5.40.
Answer.
\(SA = 9\pi\)

III Math 2570: Calculus III
10 Sequences and Series
10.1 Sequences

Exercises

Terms and Concepts
10.1.1.
Answer.
Answers will vary.
10.1.2.
Answer.
natural
10.1.3.
Answer.
Answers will vary.
10.1.4.
Answer.
Answers will vary.
Problems
10.1.5.
Answer.
\(2,\frac{8}{3},\frac{8}{3},\frac{32}{15},\frac{64}{45}\)
10.1.6.
Answer.
\(-\frac{3}{2},\frac{9}{4},-\frac{27}{8},\frac{81}{16}, -\frac{243}{32}\)
10.1.7.
Answer.
\(-\frac{1}{3},-2,-\frac{81}{5},-\frac{512}{3},-\frac{15625}{7}\)
10.1.8.
Answer.
\(1, 1, 2, 3, 5\)
10.1.9.
Answer.
\(a_n = 3n+1\)
10.1.10.
Answer.
\(a_n = (-1)^{n+1}\frac{3}{2^{n-1}}\)
10.1.11.
Answer.
\(a_n = 10\cdot 2^{n-1}\)
10.1.12.
Answer.
\(a_n = 1/(n-1)!\)
10.1.13.
Answer.
\(1/7\)
10.1.14.
Answer.
\(3e^2-1\)
10.1.15.
Answer.
\(0\)
10.1.16.
Answer.
\(e^4\)
10.1.17.
Answer.
diverges
10.1.18.
Answer.
converges to \(4/3\)
10.1.19.
Answer.
converges to \(0\)
10.1.20.
Answer.
converges to \(0\)
10.1.21.
Answer.
diverges
10.1.22.
Answer.
converges to 3
10.1.23.
Answer.
converges to \(e\)
10.1.24.
Answer.
converges to 5
10.1.25.
Answer.
converges to 0
10.1.26.
Answer.
diverges
10.1.27.
Answer.
converges to 2
10.1.28.
Answer.
converges to 0
10.1.29.
Answer.
bounded
10.1.30.
Answer.
neither bounded above or below
10.1.31.
Answer.
bounded
10.1.32.
Answer.
bounded below
10.1.33.
Answer.
neither bounded above or below
10.1.34.
Answer.
bounded above
10.1.35.
Answer.
monotonically increasing
10.1.36.
Answer.
monotonically increasing for \(n\geq 3\)
10.1.37.
Answer.
never monotonic
10.1.38.
Answer.
monotonically decreasing for \(n\geq 3\)
10.1.40.
10.1.40.b
Answer.
\(a_n = 1/3^n\) and \(b_n = 1/2^n\)

10.2 Infinite Series
10.2.4 Exercises

Terms and Concepts

10.2.4.1.
Answer.
Answers will vary.
10.2.4.2.
Answer.
Answers will vary.
10.2.4.4.
Answer.
Answers will vary.
10.2.4.5.
Answer.
F
10.2.4.6.
Answer.
F

10.3 Integral and Comparison Tests
10.3.4 Exercises

Terms and Concepts

10.3.4.1.
Answer.
continuous, positive and decreasing
10.3.4.2.
Answer.
F

Problems

10.3.4.5.
Answer.
Converges
10.3.4.6.
Answer.
Converges
10.3.4.7.
Answer.
Diverges
10.3.4.8.
Answer.
Diverges
10.3.4.9.
Answer.
Converges
10.3.4.10.
Answer.
Converges
10.3.4.11.
Answer.
Converges
10.3.4.12.
Answer.
Converges

10.4 Ratio and Root Tests
10.4.3 Exercises

Terms and Concepts

10.4.3.1.
Answer.
algebraic, or polynomial.
10.4.3.2.
Answer.
factorial and/or exponential
10.4.3.3.
Answer.
Integral Test, Limit Comparison Test, and Root Test
10.4.3.4.
Answer.
raised to a power

Problems

10.4.3.5.
Answer.
Converges
10.4.3.6.
Answer.
Diverges
10.4.3.7.
Answer.
Converges
10.4.3.8.
Answer.
Converges
10.4.3.9.
Answer.
The Ratio Test is inconclusive; the \(p\)-Series Test states it diverges.
10.4.3.10.
Answer.
The Ratio Test is inconclusive; the Direct Comparison Test with \(1/n^3\) shows it converges.
10.4.3.11.
Answer.
Converges
10.4.3.12.
Answer.
Converges
10.4.3.13.
Answer.
Converges; note the summation can be rewritten as \(\ds\infser \frac{2^nn!}{3^nn!}\text{,}\) from which the Ratio Test or Geometric Series Test can be applied.
10.4.3.14.
Answer.
Converges; rewrite the summation as \(\ds\infser \frac{n!}{5^nn!}\) then apply the Ratio Test or Geometric Series Test.
10.4.3.15.
Answer.
Converges
10.4.3.16.
Answer.
Converges
10.4.3.17.
Answer.
Converges
10.4.3.18.
Answer.
Converges
10.4.3.19.
Answer.
Diverges
10.4.3.20.
Answer.
Converges
10.4.3.21.
Answer.
Diverges. The Root Test is inconclusive, but the \(n\)th-Term Test shows divergence. (The terms of the sequence approach \(e^{-2}\text{,}\) not 0, as \(n\to\infty\text{.}\))
10.4.3.22.
Answer.
Converges
10.4.3.23.
Answer.
Converges
10.4.3.24.
Answer.
Converges

10.5 Alternating Series and Absolute Convergence

Exercises

Terms and Concepts
10.5.2.
Answer.
postive, decreasing, 0
10.5.3.
Answer.
Many examples exist; one common example is \(a_n = (-1)^n/n\text{.}\)
10.5.4.
Answer.
conditionally

11 Vectors
11.1 Introduction to Cartesian Coordinates in Space
11.1.7 Exercises

Terms and Concepts

11.1.7.2.
Answer 1.
\({\text{line}}\)
Answer 2.
\({\text{plane}}\)
11.1.7.4.
Answer.