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APEX Calculus for University of Lethbridge

Gregory Hartman, Ph.D., Sean Fitzpatrick, Ph.D. (Editor), Alex Jordan, Ph.D. (Editor), Carly Vollet, M.S. (Editor)

Appendix A Answers to Selected Exercises

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

Problems

1.1.3.7.
Answer.
\(-1\)
1.1.3.8.
Answer.
\(-2\)
1.1.3.9.
Answer.
\(\text{DNE}\)
1.1.3.10.
Answer.
\(-4\)
1.1.3.11.
Answer.
\(-{\frac{7}{3}}\)
1.1.3.12.
Answer.
\(\text{DNE}\hbox{ or }\infty \)
1.1.3.13.
Answer.
\(\text{DNE}\)
1.1.3.14.
Answer.
\(-4\)
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 1.
\(-0.1\)
Answer 2.
\(-7\)
Answer 3.
\(-0.01\)
Answer 4.
\(-7\)
Answer 5.
\(0.01\)
Answer 6.
\(-7\)
Answer 7.
\(0.1\)
Answer 8.
\(-7\)
Answer 9.
\(-7\)
1.1.3.22.
Answer 1.
\(-0.1\)
Answer 2.
\(9\)
Answer 3.
\(-0.01\)
Answer 4.
\(9\)
Answer 5.
\(0.01\)
Answer 6.
\(9\)
Answer 7.
\(0.1\)
Answer 8.
\(9\)
Answer 9.
\(9\)
1.1.3.23.
Answer 1.
\(-0.1\)
Answer 2.
\(4.9\)
Answer 3.
\(-0.01\)
Answer 4.
\(4.99\)
Answer 5.
\(0.01\)
Answer 6.
\(5.01\)
Answer 7.
\(0.1\)
Answer 8.
\(5.1\)
Answer 9.
\(5\)
1.1.3.24.
Answer 1.
\(-0.1\)
Answer 2.
\(-0.114943\)
Answer 3.
\(-0.01\)
Answer 4.
\(-0.111483\)
Answer 5.
\(0.01\)
Answer 6.
\(-0.110742\)
Answer 7.
\(0.1\)
Answer 8.
\(-0.107527\)
Answer 9.
\(-0.111111\)
1.1.3.25.
Answer 1.
\(-0.1\)
Answer 2.
\(29.4\)
Answer 3.
\(-0.01\)
Answer 4.
\(29.04\)
Answer 5.
\(0.01\)
Answer 6.
\(28.96\)
Answer 7.
\(0.1\)
Answer 8.
\(28.6\)
Answer 9.
\(29\)
1.1.3.26.
Answer 1.
\(-0.1\)
Answer 2.
\(0.202027\)
Answer 3.
\(-0.01\)
Answer 4.
\(0.2002\)
Answer 5.
\(0.01\)
Answer 6.
\(0.1998\)
Answer 7.
\(0.1\)
Answer 8.
\(0.198026\)
Answer 9.
\(0.2\)
1.1.3.27.
Answer 1.
\(-0.1\)
Answer 2.
\(-0.998334\)
Answer 3.
\(-0.01\)
Answer 4.
\(-0.999983\)
Answer 5.
\(0.01\)
Answer 6.
\(-0.999983\)
Answer 7.
\(0.1\)
Answer 8.
\(-0.998334\)
Answer 9.
\(-1\)
1.1.3.28.
Answer 1.
\(-0.1\)
Answer 2.
\(-0.0499583\)
Answer 3.
\(-0.01\)
Answer 4.
\(-0.00499996\)
Answer 5.
\(0.01\)
Answer 6.
\(0.00499996\)
Answer 7.
\(0.1\)
Answer 8.
\(0.0499583\)
Answer 9.
\(0\)

1.3 Finding Limits Analytically

Exercises

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.
\(101\)
1.3.20.
Answer.
\(\left(\frac{\pi -9}{\pi -2}\right)^{8}\)
1.3.21.
Answer.
\(\frac{\sqrt{3}}{4}\)
1.3.22.
Answer.
\({\frac{2}{3}}\)
1.3.23.
Answer.
\(\text{DNE}\)
1.3.24.
Answer.
\(8\)
1.3.25.
Answer.
\(\sqrt{2}\)
1.3.26.
Answer.
\(\ln\mathopen{}\left(6\right)\)
1.3.27.
Answer.
\(\frac{\pi ^{2}+3\pi -4}{4\pi ^{2}+3\pi +2}\)
1.3.28.
Answer.
\(\frac{4\pi +3}{-\left(4+5\pi \right)}\)
1.3.29.
Answer.
\({\frac{11}{8}}\)
1.3.30.
Answer.
\({\frac{4}{3}}\)
1.3.31.
Answer.
\(4\)
1.3.32.
Answer.
\({\frac{3}{10}}\)
1.3.33.
Answer.
\(2\)
1.3.34.
Answer.
\(4\)
1.3.35.
Answer.
\(0\)
1.3.36.
Answer.
\(0\)
1.3.37.
Answer.
\(1\)
1.3.38.
Answer.
\(9\)
1.3.39.
Answer.
\(2\)
1.3.40.
Answer.
\({\frac{3}{7}}\)
1.3.41.
Answer.
\(1\)
1.3.42.
Answer.
\(\frac{\pi }{180}\)

1.4 One-Sided Limits

Exercises

Problems
1.4.5.
1.4.5.a
Answer.
\(4\)
1.4.5.b
Answer.
\(4\)
1.4.5.c
Answer.
\(4\)
1.4.5.d
Answer.
\(1\)
1.4.5.e
Answer.
\(0\)
1.4.5.f
Answer.
\(\text{DNE}\)
1.4.6.
1.4.6.a
Answer.
\(2\)
1.4.6.b
Answer.
\(4\)
1.4.6.c
Answer.
\(\text{DNE}\)
1.4.6.d
Answer.
\(4\)
1.4.6.e
Answer.
\(0\)
1.4.6.f
Answer.
\(\text{DNE}\)
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.
\(\text{DNE}\)
1.4.7.f
Answer.
\(\text{DNE}\)
1.4.8.
1.4.8.a
Answer.
\(5\)
1.4.8.b
Answer.
\(4\)
1.4.8.c
Answer.
\(\text{DNE}\)
1.4.8.d
Answer.
\(3\)
1.4.9.
1.4.9.a
Answer.
\(3\)
1.4.9.b
Answer.
\(3\)
1.4.9.c
Answer.
\(3\)
1.4.9.d
Answer.
\(3\)
1.4.10.
1.4.10.a
Answer.
\(-1\)
1.4.10.b
Answer.
\(3\)
1.4.10.c
Answer.
\(\text{DNE}\)
1.4.10.d
Answer.
\(-6\)
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.
\(3\)
1.4.13.b
Answer.
\(2\)
1.4.13.c
Answer.
\(\text{DNE}\)
1.4.13.d
Answer.
\(3\)
1.4.14.
1.4.14.a
Answer.
\(-9\)
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.
\(-63\)
1.4.15.b
Answer.
\(-63\)
1.4.15.c
Answer.
\(-63\)
1.4.15.d
Answer.
\(-63\)
1.4.15.e
Answer.
\(9\)
1.4.15.f
Answer.
\(9\)
1.4.15.g
Answer.
\(9\)
1.4.15.h
Answer.
\(9\)
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.
\(1\)
1.4.18.b
Answer.
\(2\)
1.4.18.c
Answer.
\(\text{DNE}\)
1.4.18.d
Answer.
\(5\)
1.4.19.
1.4.19.a
Answer.
\(-2\)
1.4.19.b
Answer.
\(-2\)
1.4.19.c
Answer.
\(-2\)
1.4.19.d
Answer.
\(-5\)
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

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.
\(\text{No.}\)
Answer 2.
\(\text{Yes.}\)
Answer 3.
\(\text{No.}\)
1.5.18.
Answer.
\(\text{Yes.}\)
1.5.19.
1.5.19.a
Answer.
\(\text{Yes.}\)
1.5.19.b
Answer.
\(\text{Yes.}\)
1.5.20.
1.5.20.a
Answer.
\(\text{Yes.}\)
1.5.20.b
Answer.
\(\text{No.}\)
1.5.21.
1.5.21.a
Answer.
\(\text{Yes.}\)
1.5.21.b
Answer.
\(\text{Yes.}\)
1.5.22.
1.5.22.a
Answer.
\(\text{Yes.}\)
1.5.22.b
Answer.
\(\text{No.}\)
1.5.23.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.24.
Answer.
\(\left(-\infty ,-5\right], \left[5,\infty \right)\)
1.5.25.
Answer.
\(\left[-5,5\right]\)
1.5.26.
Answer.
\(\left[-3,1\right]\)
1.5.27.
Answer.
\(\left(-\infty ,-1.73205\right], \left[1.73205,\infty \right)\)
1.5.28.
Answer.
\(\left(-3,3\right)\)
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.
\(\left(-\infty ,2.07944\right]\)
1.5.34.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.39.
Answer 1.
\(\text{-}\)
Answer 2.
\(\left[1,1.25\right]\)
Answer 3.
\(\text{+}\)
Answer 4.
\(\left[1.125,1.25\right]\)
Answer 5.
\(\text{+}\)
Answer 6.
\(\left[1.1875,1.25\right]\)
Answer 7.
\(\text{+}\)
Answer 8.
\(\left[1.21875,1.25\right]\)
Answer 9.
\(\text{+}\)
Answer 10.
\(\left[1.23438,1.25\right]\)
Answer 11.
\(\text{-}\)
Answer 12.
\(\left[1.23438,1.24219\right]\)
Answer 13.
\(\text{-}\)
Answer 14.
\(\left[1.23438,1.23828\right]\)
Answer 15.
\(1.23633\)
1.5.40.
Answer 1.
\(\text{-}\)
Answer 2.
\(\left[0.5,0.525\right]\)
Answer 3.
\(\text{+}\)
Answer 4.
\(\left[0.5125,0.525\right]\)
Answer 5.
\(\text{+}\)
Answer 6.
\(\left[0.51875,0.525\right]\)
Answer 7.
\(\text{+}\)
Answer 8.
\(\left[0.521875,0.525\right]\)
Answer 9.
\(\text{+}\)
Answer 10.
\(\left[0.523438,0.525\right]\)
Answer 11.
\(\text{-}\)
Answer 12.
\(\left[0.523438,0.524219\right]\)
Answer 13.
\(\text{-}\)
Answer 14.
\(\left[0.523438,0.523828\right]\)
Answer 15.
\(0.523633\)
1.5.41.
Answer 1.
\(\text{+}\)
Answer 2.
\(\left[0.675,0.7\right]\)
Answer 3.
\(\text{+}\)
Answer 4.
\(\left[0.6875,0.7\right]\)
Answer 5.
\(\text{-}\)
Answer 6.
\(\left[0.6875,0.69375\right]\)
Answer 7.
\(\text{+}\)
Answer 8.
\(\left[0.690625,0.69375\right]\)
Answer 9.
\(\text{+}\)
Answer 10.
\(\left[0.692187,0.69375\right]\)
Answer 11.
\(\text{+}\)
Answer 12.
\(\left[0.692969,0.69375\right]\)
Answer 13.
\(\text{-}\)
Answer 14.
\(\left[0.692969,0.693359\right]\)
Answer 15.
\(0.693164\)
1.5.42.
Answer 1.
\(\text{-}\)
Answer 2.
\(\left[0.75,0.8\right]\)
Answer 3.
\(\text{-}\)
Answer 4.
\(\left[0.775,0.8\right]\)
Answer 5.
\(\text{+}\)
Answer 6.
\(\left[0.775,0.7875\right]\)
Answer 7.
\(\text{-}\)
Answer 8.
\(\left[0.78125,0.7875\right]\)
Answer 9.
\(\text{-}\)
Answer 10.
\(\left[0.784375,0.7875\right]\)
Answer 11.
\(\text{+}\)
Answer 12.
\(\left[0.784375,0.785937\right]\)
Answer 13.
\(\text{-}\)
Answer 14.
\(\left[0.785156,0.785937\right]\)
Answer 15.
\(0.785547\)

1.6 Limits Involving Infinity
1.6.4 Exercises

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.
\(4\)
1.6.4.11.c
Answer.
\(2\)
1.6.4.11.d
Answer.
\(2\)
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.
\(\infty \)
1.6.4.14.b
Answer.
\(-7\)
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.
\(0.857143\)
1.6.4.18.b
Answer.
\(0.857143\)
1.6.4.18.c
Answer.
\(0.857143\)
1.6.4.19.
Answer.
\(y = 4, x = -1, x = -5\)
1.6.4.20.
Answer.
\(y = \frac{-3}{4}, x = -7\)
1.6.4.21.
Answer.
\(y = 0, x = 0, x = 5\)
1.6.4.22.
Answer.
\(x = -2\)
1.6.4.23.
Answer.
\(\text{NONE}\)
1.6.4.24.
Answer.
\(y = \frac{2}{-5}\)
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

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.
\(2.9x+y = -1.9\)
2.1.3.24.
Answer.
\(0.00894855x+y = -0.195973\)
2.1.3.25.
Answer.
\(y-7.77114x = -8.15322\)
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.
\(-2, 0, 4\)
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.
\(-1, -0.25\)
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.
\(\left\{-1,0,1\right\}\)

2.2 Interpretations of the Derivative
2.2.5 Exercises

Problems

2.2.5.4.
Answer.
\(14\)
2.2.5.5.
Answer.
\(67\)
2.2.5.6.
Answer.
\(276\)
2.2.5.7.
Answer.
\(\text{f(10.1)}\)
2.2.5.8.
Answer.
\(-8\)
2.2.5.9.
Answer.
\(1\)
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.
\(\text{f is the derivative of g.}\)
2.2.5.16.
Answer.
\(\text{g is the derivative of f.}\)
2.2.5.17.
Answer.
\(\text{g is the derivative of f.}\)
2.2.5.18.
Answer.
\(\text{g is the derivative of f.}\)

2.3 Basic Differentiation Rules
2.3.3 Exercises

Terms and Concepts

2.3.3.2.
Answer.
\(\frac{1}{x}\)
2.3.3.3.
Answer.
\(e^{x}\)
2.3.3.4.
Answer.
\(10\)
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.
\(-\left(4x+9\right)\)
2.3.3.12.
Answer.
\(21x^{2}+26x+18\)
2.3.3.13.
Answer.
\(25t^{4}-{\frac{3}{4}}t^{2}-6\)
2.3.3.14.
Answer.
\(13\cos\mathopen{}\left(\theta\right)+19\sin\mathopen{}\left(\theta\right)\)
2.3.3.15.
Answer.
\(8e^{r}\)
2.3.3.16.
Answer.
\(\sin\mathopen{}\left(t\right)-54t^{2}+3\cos\mathopen{}\left(t\right)\)
2.3.3.17.
Answer.
\(9-\frac{7}{x}\)
2.3.3.18.
Answer.
\(s^{3}+s^{2}+s+1\)
2.3.3.19.
Answer.
\(-\left(e^{t}+\cos\mathopen{}\left(t\right)+\sin\mathopen{}\left(t\right)\right)\)
2.3.3.20.
Answer.
\(\frac{8}{x}\)
2.3.3.21.
Answer.
\(0\)
2.3.3.22.
Answer.
\(18t+42\)
2.3.3.23.
Answer.
\(192x^{2}+288x+108\)
2.3.3.24.
Answer.
\(81x^{2}+216x+144\)
2.3.3.25.
Answer.
\(8x+4\)
2.3.3.27.
Answer 1.
\(5x^{4}\)
Answer 2.
\(20x^{3}\)
Answer 3.
\(60x^{2}\)
Answer 4.
\(120x\)
2.3.3.28.
Answer 1.
\(-\left(5\cos\mathopen{}\left(x\right)\right)\)
Answer 2.
\(5\sin\mathopen{}\left(x\right)\)
Answer 3.
\(5\cos\mathopen{}\left(x\right)\)
Answer 4.
\(-5\sin\mathopen{}\left(x\right)\)
2.3.3.29.
Answer 1.
\(-\left(2t+3+e^{t}\right)\)
Answer 2.
\(-\left(2+e^{t}\right)\)
Answer 3.
\(-e^{t}\)
Answer 4.
\(-e^{t}\)
2.3.3.30.
Answer 1.
\(5\theta^{4}+8\theta^{7}\)
Answer 2.
\(20\theta^{3}+56\theta^{6}\)
Answer 3.
\(60\theta^{2}+336\theta^{5}\)
Answer 4.
\(120\theta+1680\theta^{4}\)
2.3.3.31.
Answer 1.
\(\cos\mathopen{}\left(\theta\right)+\sin\mathopen{}\left(\theta\right)\)
Answer 2.
\(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\)
Answer 3.
\(-\left(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\right)\)
Answer 4.
\(-\left(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\right)\)
2.3.3.32.
Answer 1.
\(0\)
Answer 2.
\(0\)
Answer 3.
\(0\)
Answer 4.
\(0\)
2.3.3.33.
Answer 1.
\(y = 17\mathopen{}\left(x-2\right)+18\)
Answer 2.
\(y = -{\frac{1}{17}}\mathopen{}\left(x-2\right)+18\)
2.3.3.34.
Answer 1.
\(y = e^{3}\ln\mathopen{}\left(e\right)\mathopen{}\left(t-3\right)+e^{3}-2\)
Answer 2.
\(y = \frac{-1}{e^{3}\ln\mathopen{}\left(e\right)}\mathopen{}\left(t-3\right)+e^{3}-2\)
2.3.3.35.
Answer 1.
\(y = x-1\)
Answer 2.
\(y = -\left(x-1\right)\)
2.3.3.36.
Answer 1.
\(y = 4\cdot 1\)
Answer 2.
\(x = \frac{\pi }{2}\)
2.3.3.37.
Answer 1.
\(y = \frac{3\sqrt{3}}{2}\mathopen{}\left(x-\frac{\pi }{3}\right)+\frac{-3\cdot 1}{2}\)
Answer 2.
\(y = -\left({\frac{1}{3}}\frac{2\sqrt{3}}{3}\right)\mathopen{}\left(x-\frac{\pi }{3}\right)+\frac{-3\cdot 1}{2}\)
2.3.3.38.
Answer 1.
\(y = -\left(9x+5\right)\)
Answer 2.
\(y = \frac{-1}{-9}\mathopen{}\left(x-\left(-2\right)\right)+13\)

2.4 The Product and Quotient Rules

Exercises

Problems
2.4.15.
Answer.
\(\cos\mathopen{}\left(z\right)-z\sin\mathopen{}\left(z\right)\)
2.4.16.
Answer.
\(9x^{8}\sin\mathopen{}\left(x\right)+x^{9}\cos\mathopen{}\left(x\right)\)
2.4.17.
Answer.
\(e^{r}\ln\mathopen{}\left(r\right)+e^{r}\frac{1}{r}\)
2.4.18.
Answer.
\(\frac{1}{z^{4}}\sec\mathopen{}\left(z\right)\tan\mathopen{}\left(z\right)-\frac{4z^{3}}{\left(z^{4}\right)^{2}}\mathopen{}\left(\sec\mathopen{}\left(z\right)-5\right)\)
2.4.19.
Answer.
\(\frac{t-8-\left(t+3\right)}{\left(t-8\right)^{2}}\)
2.4.20.
Answer.
\(\frac{3q^{2}\mathopen{}\left(\sin\mathopen{}\left(q\right)+6q^{8}\right)-q^{3}\mathopen{}\left(\cos\mathopen{}\left(q\right)+48q^{7}\right)}{\left(\sin\mathopen{}\left(q\right)+6q^{8}\right)^{2}}\)
2.4.21.
Answer.
\(\sec\mathopen{}\left(y\right)\tan\mathopen{}\left(y\right)-e^{y}\)
2.4.22.
Answer.
\(\frac{1}{t}\csc\mathopen{}\left(t\right)-\csc\mathopen{}\left(t\right)\cot\mathopen{}\left(t\right)\ln\mathopen{}\left(t\right)\)
2.4.23.
Answer.
\(10q+9\)
2.4.24.
Answer.
\(4y^{3}\)
2.4.25.
Answer.
\(\left(4t^{2}+14t+9\right)e^{t}\)
2.4.26.
Answer.
\(\frac{7q^{6}-q^{7}-q^{3}+3q^{2}}{e^{q}}\)
2.4.27.
Answer.
\(26\)
2.4.28.
Answer.
\(5t^{4}\mathopen{}\left(\csc\mathopen{}\left(t\right)+e^{t}\right)+t^{5}\mathopen{}\left(e^{t}-\csc\mathopen{}\left(t\right)\cot\mathopen{}\left(t\right)\right)\)
2.4.29.
Answer.
\(\frac{\left(\cos\mathopen{}\left(q\right)+8\right)\cos\mathopen{}\left(q\right)+\sin\mathopen{}\left(q\right)\sin\mathopen{}\left(q\right)}{\left(\cos\mathopen{}\left(q\right)+8\right)^{2}}\)
2.4.30.
Answer.
\(5\theta^{4}\sin\mathopen{}\left(\theta\right)+\theta^{5}\cos\mathopen{}\left(\theta\right)+\frac{\theta^{5}\cos\mathopen{}\left(\theta\right)-5\theta^{4}\sin\mathopen{}\left(\theta\right)}{\left(\theta^{5}\right)^{2}}\)
2.4.31.
Answer.
\(\frac{\tan\mathopen{}\left(r\right)-r\sec^{2}\mathopen{}\left(r\right)}{\tan^{2}\mathopen{}\left(r\right)}-\frac{\csc\mathopen{}\left(r\right)\cot\mathopen{}\left(r\right)r+\csc\mathopen{}\left(r\right)}{r^{2}}\)
2.4.32.
Answer.
\(0\)
2.4.33.
Answer.
\(15x^{2}e^{x}+5x^{3}e^{x}+\cos\mathopen{}\left(x\right)\cos\mathopen{}\left(x\right)-\sin\mathopen{}\left(x\right)\sin\mathopen{}\left(x\right)\)
2.4.34.
Answer.
\(\frac{\left(3r^{2}\cos\mathopen{}\left(r\right)-r^{3}\sin\mathopen{}\left(r\right)\right)\mathopen{}\left(r^{3}\sin\mathopen{}\left(r\right)-9\right)-\left(r^{3}\cos\mathopen{}\left(r\right)+8\right)\mathopen{}\left(3r^{2}\sin\mathopen{}\left(r\right)+r^{3}\cos\mathopen{}\left(r\right)\right)}{\left(r^{3}\sin\mathopen{}\left(r\right)-9\right)^{2}}\)
2.4.35.
Answer.
\(\left(3z^{2}\ln\mathopen{}\left(z\right)+z^{3}\frac{1}{z}\right)\cos\mathopen{}\left(z\right)-z^{3}\ln\mathopen{}\left(z\right)\sin\mathopen{}\left(z\right)\)
2.4.36.
Answer.
\(\left(7\cos\mathopen{}\left(x\right)-7x\sin\mathopen{}\left(x\right)\right)\csc\mathopen{}\left(x\right)-7x\cos\mathopen{}\left(x\right)\csc\mathopen{}\left(x\right)\cot\mathopen{}\left(x\right)\)
2.4.37.
Answer 1.
\(y = -\left(5x+5\right)\)
Answer 2.
\(y = \left({\frac{1}{5}}\right)x-5\)
2.4.38.
Answer 1.
\(y = -\left(9.18926\mathopen{}\left(x-\frac{7\pi }{4}\right)+\frac{7\pi }{2.82843}\right)\)
Answer 2.
\(y = \left({\frac{10455070}{96074343}}\right)\mathopen{}\left(x-\frac{7\pi }{4}\right)-\frac{7\pi }{2.82843}\)
2.4.39.
Answer 1.
\(y = 64-80\mathopen{}\left(x+8\right)\)
Answer 2.
\(y = \left({\frac{1}{80}}\right)\mathopen{}\left(x+8\right)+64\)
2.4.40.
Answer 1.
\(y = \left({\frac{34}{25}}\right)x-\left({\frac{1}{5}}\right)\)
Answer 2.
\(y = -\left(\left({\frac{25}{34}}\right)x+\left({\frac{1}{5}}\right)\right)\)
2.4.41.
Answer.
\(-{\frac{3}{2}}\)
2.4.42.
Answer.
\(0\)
2.4.43.
Answer.
\(\text{NONE}\)
2.4.44.
Answer.
\(0, 2\)
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.
\(2\sec\mathopen{}\left(x\right)\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)\)
2.4.48.
Answer.
\(0\)

2.5 The Chain Rule

Exercises

Problems
2.5.7.
Answer.
\(10\mathopen{}\left(4x^{3}-x\right)^{9}\mathopen{}\left(12x^{2}-1\right)\)
2.5.8.
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.
\(7\mathopen{}\left(\ln\mathopen{}\left(y\right)-y^{5}\right)^{6}\mathopen{}\left(\frac{1}{y}-5y^{4}\right)\)
2.5.12.
Answer.
\(1.09861\cdot 3^{r^{3}+5r}\mathopen{}\left(3r^{2}+5\right)\)
2.5.13.
Answer.
\(3\mathopen{}\left(z-\frac{1}{z}\right)^{2}\mathopen{}\left(1+\frac{1}{z^{2}}\right)\)
2.5.14.
Answer.
\(8\cos\mathopen{}\left(8x\right)\)
2.5.15.
Answer.
\(6\sec^{2}\mathopen{}\left(6r\right)\)
2.5.16.
Answer.
\(-\csc^{2}\mathopen{}\left(\theta^{2}+8\right)\cdot 2\theta\)
2.5.17.
Answer.
\(-\left(2x-\frac{3x^{2}}{\left(x^{3}\right)^{2}}\right)\sin\mathopen{}\left(x^{2}+\frac{1}{x^{3}}\right)\)
2.5.18.
Answer.
\(-8\cos^{7}\mathopen{}\left(7r\right)\cdot 7\sin\mathopen{}\left(7r\right)\)
2.5.19.
Answer.
\(6\sin^{5}\mathopen{}\left(z^{2}+4z-7\right)\mathopen{}\left(2z+4\right)\cos\mathopen{}\left(z^{2}+4z-7\right)\)
2.5.20.
Answer.
\(\frac{1}{\sec\mathopen{}\left(x\right)}\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)\)
2.5.21.
Answer.
\(\frac{1}{r^{9}}\cdot 9r^{8}\)
2.5.22.
Answer.
\(5\frac{1}{z}\)
2.5.23.
Answer.
\(2.07944\cdot 8^{x}\)
2.5.24.
Answer.
\(1.09861\cdot 3^{\sin\mathopen{}\left(q\right)}\cos\mathopen{}\left(q\right)\)
2.5.25.
Answer.
\(0\)
2.5.26.
Answer.
\(\frac{1.79176\cdot 6^{x}\cdot 7^{x}-6^{x}\cdot 1.94591\cdot 7^{x}}{\left(7^{x}\right)^{2}}\)
2.5.27.
Answer.
\(\frac{1.60944\cdot 5^{q}\mathopen{}\left(3^{q}+5\right)-\left(5^{q}+3\right)\cdot 1.09861\cdot 3^{q}}{\left(3^{q}+5\right)^{2}}\)
2.5.28.
Answer.
\(\frac{2.19722\cdot 9^{z}\cdot 8^{z}-\left(9^{z}+1\right)\cdot 2.07944\cdot 8^{z}}{\left(8^{z}\right)^{2}}\)
2.5.29.
Answer.
\(\frac{\left(1.38629\cdot 4^{t^{2}}\cdot 2t-1\right)\cdot 7^{t^{2}}-\left(4^{t^{2}}-t\right)\cdot 1.94591\cdot 7^{t^{2}}\cdot 2t}{\left(7^{t^{2}}\right)^{2}}\)
2.5.30.
Answer.
\(3q^{2}\cos\mathopen{}\left(6q\right)-q^{3}\cdot 6\sin\mathopen{}\left(6q\right)\)
2.5.31.
Answer.
\(2\mathopen{}\left(z^{2}+2z\right)\mathopen{}\left(2z+2\right)\mathopen{}\left(6z^{3}+z\right)^{5}+\left(z^{2}+2z\right)^{2}\cdot 5\mathopen{}\left(6z^{3}+z\right)^{4}\mathopen{}\left(18z^{2}+1\right)\)
2.5.32.
Answer.
\(\cos\mathopen{}\left(6t^{2}-5t\right)\cos\mathopen{}\left(t-4\right)-\left(12t-5\right)\sin\mathopen{}\left(6t^{2}-5t\right)\sin\mathopen{}\left(t-4\right)\)
2.5.33.
Answer.
\(9\cos\mathopen{}\left(7+9q\right)\cos\mathopen{}\left(5q-2\right)-5\sin\mathopen{}\left(5q-2\right)\sin\mathopen{}\left(7+9q\right)\)
2.5.34.
Answer.
\(e^{4y^{2}}\cdot 8y\sin\mathopen{}\left(\frac{1}{y}\right)-e^{4y^{2}}\frac{1}{y^{2}}\cos\mathopen{}\left(\frac{1}{y}\right)\)
2.5.35.
Answer.
\(-\frac{\left(3t-9\right)^{4}\sin\mathopen{}\left(t+5\right)+12\mathopen{}\left(3t-9\right)^{3}\cos\mathopen{}\left(t+5\right)}{\left(\left(3t-9\right)^{4}\right)^{2}}\)
2.5.36.
Answer.
\(\frac{28\mathopen{}\left(7w+4\right)^{3}\sin\mathopen{}\left(6w\right)-\left(7w+4\right)^{4}\cdot 6\cos\mathopen{}\left(6w\right)}{\sin^{2}\mathopen{}\left(6w\right)}\)
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

Problems

2.6.4.5.
Answer.
\(\frac{1}{2\sqrt{x}}-\frac{\frac{1}{2\sqrt{x}}}{\left(\sqrt{x}\right)^{2}}\)
2.6.4.6.
Answer.
\({\frac{1}{5}}\frac{1}{\left(\sqrt[5]{r}\right)^{4}}-\left({\frac{4}{5}}\right)\frac{1}{r^{0.2}}\)
2.6.4.7.
Answer.
\(-\frac{1}{2\sqrt{4-z^{2}}}\cdot 2z\)
2.6.4.8.
Answer.
\(\frac{1}{2\sqrt{x}}\cos\mathopen{}\left(x\right)-\sqrt{x}\sin\mathopen{}\left(x\right)\)
2.6.4.9.
Answer.
\(3.7r^{2.7}\)
2.6.4.10.
Answer.
\(\pi z^{\pi -1}+1.4z^{0.4}\)
2.6.4.11.
Answer.
\(\frac{\sqrt{x}-\left(x+3\right)\frac{1}{2\sqrt{x}}}{\left(\sqrt{x}\right)^{2}}\)
2.6.4.12.
Answer.
\({\frac{1}{3}}\frac{1}{\left(\sqrt[3]{q}\right)^{2}}\mathopen{}\left(\csc\mathopen{}\left(q\right)+e^{q}\right)+\left(e^{q}-\csc\mathopen{}\left(q\right)\cot\mathopen{}\left(q\right)\right)\sqrt[3]{q}\)
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

Problems
2.7.9.
Answer.
\({\frac{1}{4}}\)
2.7.10.
Answer.
\(-{\frac{1}{2}}\)
2.7.11.
Answer.
\(-0.471405\)
2.7.12.
Answer.
\({\frac{1}{33}}\)
2.7.13.
Answer.
\(-{\frac{1089}{80}}\)
2.7.14.
Answer.
\({\frac{1}{54}}\)
2.7.15.
Answer.
\(-\frac{1}{\sqrt{1-\left(6r\right)^{2}}}\cdot 6\)
2.7.16.
Answer.
\(\frac{1}{\left|9w\right|\sqrt{\left(9w\right)^{2}-1}}\cdot 9\)
2.7.17.
Answer.
\(\frac{1}{1+\left(4x\right)^{2}}\cdot 4\)
2.7.18.
Answer.
\(\cos^{-1}\mathopen{}\left(r\right)-r\frac{1}{\sqrt{1-r^{2}}}\)
2.7.19.
Answer.
\(\cos\mathopen{}\left(w\right)\tan^{-1}\mathopen{}\left(w\right)+\frac{1}{1+w^{2}}\sin\mathopen{}\left(w\right)\)
2.7.20.
Answer.
\(\frac{e^{t}}{t}+\ln\mathopen{}\left(t\right)e^{t}\)
2.7.21.
Answer.
\(\frac{\frac{1}{\sqrt{1-r^{2}}}\tan^{-1}\mathopen{}\left(r\right)-\frac{1}{1+r^{2}}\sin^{-1}\mathopen{}\left(r\right)}{\left(\tan^{-1}\mathopen{}\left(r\right)\right)^{2}}\)
2.7.22.
Answer.
\(\left(\sec\mathopen{}\left(\sqrt[4]{z}\right)\right)^{2}{\frac{1}{4}}\frac{1}{\left(\sqrt[4]{z}\right)^{3}}\)
2.7.23.
Answer.
\(\csc\mathopen{}\left(\frac{1}{x^{2}}\right)\cot\mathopen{}\left(\frac{1}{x^{2}}\right)\frac{2x}{\left(x^{2}\right)^{2}}\)
2.7.24.
Answer.
\(1\)
2.7.29.
Answer.
\(y = \frac{2\sqrt{3}}{3}\mathopen{}\left(x-{\frac{1}{2}}\right)+\frac{\pi }{6}\)
2.7.30.
Answer.
\(y = \frac{-4\sqrt{3}}{3}\mathopen{}\left(x--{\frac{1}{4}}\right)+\frac{2\pi }{3}\)

3 The Graphical Behavior of Functions
3.1 Extreme Values

Exercises

Terms and Concepts
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.
\(\text{A}, \text{E}\)
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.
\(1\)
Answer 2.
\(-{\frac{21}{4}}\)
3.1.18.
Answer 1.
\(9\)
Answer 2.
\(-{\frac{1683}{2}}\)
3.1.19.
Answer 1.
\(-2.82843\)
Answer 2.
\(-4\)
3.1.20.
Answer 1.
\(0.238018\)
Answer 2.
\(0\)
3.1.21.
Answer 1.
\(8\)
Answer 2.
\(5.2915\)
3.1.22.
Answer 1.
\({\frac{64}{69}}\)
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.
\(0.811425\)
Answer 2.
\(-1022.59\)

3.2 The Mean Value Theorem

Exercises

Problems
3.2.3.
Answer.
\(\left(-1,1\right)\)
3.2.4.
Answer.
\(\text{does not apply}\)
3.2.5.
Answer.
\(-{\frac{1}{2}}\)
3.2.6.
Answer.
\(-{\frac{1}{2}}\)
3.2.7.
Answer.
\(\text{does not apply}\)
3.2.8.
Answer.
\(\frac{\pi }{2}\)
3.2.9.
Answer.
\(\text{does not apply}\)
3.2.10.
Answer.
\(\text{does not apply}\)
3.2.11.
Answer.
\(0\)
3.2.12.
Answer.
\({\frac{5}{2}}\)
3.2.13.
Answer.
\(3\frac{\sqrt{2}}{2}\)
3.2.14.
Answer.
\({\frac{19}{4}}\)
3.2.15.
Answer.
\(\text{does not apply}\)
3.2.16.
Answer.
\(\frac{4}{\ln\mathopen{}\left(5\right)}\)
3.2.17.
Answer.
\(-\sec^{-1}\mathopen{}\left(\frac{2}{\sqrt{\pi }}\right), \sec^{-1}\mathopen{}\left(\frac{2}{\sqrt{\pi }}\right)\)
3.2.18.
Answer.
\(-{\frac{2}{3}}\)
3.2.19.
Answer.
\(5+7\frac{\sqrt{7}}{6}, 5-7\frac{\sqrt{7}}{6}\)
3.2.20.
Answer.
\(\frac{\sqrt{\pi ^{2}-4}}{\pi }, \frac{-\sqrt{\pi ^{2}-4}}{\pi }\)

3.3 Increasing and Decreasing Functions

Exercises

Terms and Concepts
3.3.3.
Answer.
Answers will vary; graphs should be steeper near \(x=0\) than near \(x=2\text{.}\)
Problems
3.3.15.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(1.5\)
Answer 3.
\(\left[1.5,\infty \right)\)
Answer 4.
\(\left(-\infty ,1.5\right]\)
Answer 5.
\(\text{NONE}\)
Answer 6.
\(1.5\)
3.3.16.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(-{\frac{10}{3}}, 0\)
Answer 3.
\(\left(-\infty ,-3.33333\right], \left[0,\infty \right)\)
Answer 4.
\(\left[-3.33333,0\right]\)
Answer 5.
\(-3.33333\)
Answer 6.
\(0\)
3.3.17.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\({\frac{4}{3}}, 8\)
Answer 3.
\(\left(-\infty ,1.33333\right], \left[8,\infty \right)\)
Answer 4.
\(\left[1.33333,8\right]\)
Answer 5.
\({\frac{4}{3}}\)
Answer 6.
\(8\)
3.3.18.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(5\)
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.
\(\left(-\infty ,-8\right)\cup \left(-8,8\right)\cup \left(8,\infty \right)\)
Answer 2.
\(0\)
Answer 3.
\(\left(-\infty ,-8\right), \left(-8,0\right]\)
Answer 4.
\(\left[0,8\right), \left(8,\infty \right)\)
Answer 5.
\(0\)
Answer 6.
\(\text{NONE}\)
3.3.21.
Answer 1.
\(\left(-\infty ,-4\right)\cup \left(-4,-1\right)\cup \left(-1,\infty \right)\)
Answer 2.
\(-2, 2\)
Answer 3.
\(\left[-2,-1\right), \left(-1,2\right]\)
Answer 4.
\(\left(-\infty ,-4\right), \left(-4,-2\right], \left[2,\infty \right)\)
Answer 5.
\(2\)
Answer 6.
\(-2\)
3.3.22.
Answer 1.
\(\left(-\infty ,0\right)\cup \left(0,\infty \right)\)
Answer 2.
\(-2, -6\)
Answer 3.
\(\left[-6,-2\right]\)
Answer 4.
\(\left(-\infty ,-6\right], \left[-2,0\right), \left(0,\infty \right)\)
Answer 5.
\(-2\)
Answer 6.
\(-6\)
3.3.23.
Answer 1.
\(\left(-\pi ,\pi \right)\)
Answer 2.
\(-2.35619, -0.785398, 0.785398, 2.35619\)
Answer 3.
\(\left(-3.14159,-2.35619\right), \left(-0.785398,0.785398\right), \left(2.35619,3.14159\right)\)
Answer 4.
\(\left(-2.35619,-0.785398\right), \left(0.785398,2.35619\right)\)
Answer 5.
\(-2.35619, 0.785398\)
Answer 6.
\(-0.785398, 2.35619\)
3.3.24.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(-1, 1\)
Answer 3.
\(\left(-\infty ,-1\right], \left[1,\infty \right)\)
Answer 4.
\(\left[-1,1\right]\)
Answer 5.
\(-1\)
Answer 6.
\(1\)

3.4 Concavity and the Second Derivative
3.4.3 Exercises

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.
\({\frac{3}{8}}\)
Answer 2.
\(\left(-\infty ,0.375\right]\)
Answer 3.
\(\left[0.375,\infty \right)\)
3.4.3.19.
Answer 1.
\(-8, 0\)
Answer 2.
\(\left(-\infty ,-8\right], \left[0,\infty \right)\)
Answer 3.
\(\left[-8,0\right]\)
3.4.3.20.
Answer 1.
\(-4.51188, 2.51188\)
Answer 2.
\(\left[-4.51188,2.51188\right]\)
Answer 3.
\(\left(-\infty ,-4.51188\right], \left[2.51188,\infty \right)\)
3.4.3.21.
Answer 1.
\(-1\)
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.
\(1, 3\)
Answer 2.
\(\left(-\infty ,1\right], \left[3,\infty \right)\)
Answer 3.
\(\left[1,3\right]\)
3.4.3.24.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\left(-\infty ,5\right), \left(8,\infty \right)\)
Answer 3.
\(\left(5,8\right)\)
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.
\(-4\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(-4\)
3.4.3.30.
Answer 1.
\(-1\)
Answer 2.
\(-1\)
Answer 3.
\(\text{NONE}\)
3.4.3.31.
Answer 1.
\(-1.29099, 1.29099\)
Answer 2.
\(-1.29099\)
Answer 3.
\(1.29099\)
3.4.3.32.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\text{NONE}\)
3.4.3.33.
Answer 1.
\(-6\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(-6\)
3.4.3.34.
Answer 1.
\(-5, -2, 4\)
Answer 2.
\(-5, 4\)
Answer 3.
\(-2\)
3.4.3.35.
Answer 1.
\(4\)
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.
\(-6\)
Answer 2.
\(-6\)
Answer 3.
\(\text{NONE}\)
3.4.3.38.
Answer 1.
\(-{\frac{13}{2}}\)
Answer 2.
\(-{\frac{13}{2}}\)
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.
\(-{\frac{2}{9}}\)
Answer 2.
\(\text{NONE}\)
3.4.3.47.
Answer 1.
\(-{\frac{20}{3}}\)
Answer 2.
\(0\)
3.4.3.48.
Answer 1.
\(-2.52753\)
Answer 2.
\(0.527525\)
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.
\(2.70901\)
Answer 2.
\(5.29099\)
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.
\(0.707107\)

4 Applications of the Derivative
4.1 Newton’s Method

Exercises

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

Problems
4.2.3.
4.2.3.a
Answer.
\(0.568411\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.3.b
Answer.
\(0.0568411\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.3.c
Answer.
\(0.00568411\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.4.
4.2.4.a
Answer.
\(0.31831\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.4.b
Answer.
\(0.0031831\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.4.c
Answer.
\(3.1831\times 10^{-5}\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.5.
Answer.
\(43.9949\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut h}}\)
4.2.6.
4.2.6.a
Answer.
\(62.5\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut h}}\)
4.2.6.b
Answer.
\(66.6667\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut h}}\)
4.2.7.
4.2.7.a
Answer.
\(237.363\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
4.2.7.b
Answer.
\(371.511\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
4.2.7.c
Answer.
\(380.471\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
4.2.8.
4.2.8.a
Answer.
\(0.0202578\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut s}}\)
4.2.8.b
Answer.
\(1.01149\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut s}}\)
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.
\(16.3362\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.11.b
Answer.
\(0.0726052\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.11.c
Answer.
\(0.0452526\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.11.d
Answer.
\(401.624\ {\rm s}\)
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.
\(91\ {\rm ft}\)
4.2.14.b
Answer.
\(23.7365\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.15.
Answer.
\(0.00266081\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)

4.3 Optimization

Exercises

Problems
4.3.3.
Answer.
\(2916\)
4.3.4.
Answer.
\(2\sqrt{500}\)
4.3.5.
Answer.
\(\text{DNE}\)
4.3.6.
Answer.
\({\frac{811}{4}}\)
4.3.7.
Answer.
\(16\)
4.3.8.
Answer.
\(250\ {\rm ft};\,\left({\frac{375}{2}}\right)\ {\rm ft}\)
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.
\(8.66025\ {\rm in};\,12.2475\ {\rm in}\)
4.3.14.
Answer 1.
\(1.81782\ {\rm mi}\)
Answer 2.
\(\$463{,}693.18\)
4.3.15.
Answer 1.
\(0\ {\rm mi}\)
Answer 2.
\(\$400{,}000\)
4.3.16.
Answer.
\(21.4908\ {\rm ft}\)
4.3.17.
Answer.
\(11.5311\ {\rm ft}\)
4.3.18.
Answer.
\(\sqrt{2};\,\sqrt{2}\)

4.4 Differentials

Exercises

Problems
4.4.7.
Answer.
\(9.36\)
4.4.8.
Answer.
\(15.68\)
4.4.9.
Answer.
\(155\)
4.4.10.
Answer.
\(110\)
4.4.11.
Answer.
\(5.03\)
4.4.12.
Answer.
\(5.9\)
4.4.13.
Answer.
\(5.99352\)
4.4.14.
Answer.
\(6.00278\)
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.
\(\left(5x^{4}-8x^{7}\right)dx\)
4.4.19.
Answer.
\(-\frac{15x^{2}}{\left(5x^{3}\right)^{2}}dx\)
4.4.20.
Answer.
\(3\mathopen{}\left(6x+\sin\mathopen{}\left(x\right)\right)^{2}\mathopen{}\left(6+\cos\mathopen{}\left(x\right)\right)dx\)
4.4.21.
Answer.
\(\left(7x^{6}+6e^{6x}\right)dx\)
4.4.22.
Answer.
\(-\frac{32x^{3}}{\left(x^{4}\right)^{2}}dx\)
4.4.23.
Answer.
\(\frac{9\mathopen{}\left(\tan\mathopen{}\left(x\right)+9\right)-9x\sec^{2}\mathopen{}\left(x\right)}{\left(\tan\mathopen{}\left(x\right)+9\right)^{2}}dx\)
4.4.24.
Answer.
\(\frac{2}{2x}dx\)
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.
\(\csc^{2}\mathopen{}\left(x\right)\sin\mathopen{}\left(\cot\mathopen{}\left(x\right)\right)dx\)
4.4.27.
Answer.
\(\frac{x+2-\left(x-3\right)}{\left(x+2\right)^{2}}dx\)
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.
\(3.92699\ {\rm cm^{3}}\)
4.4.32.
4.4.32.a
Answer.
\(51.2\)
4.4.32.b
Answer.
\(102.4\)
4.4.33.
Answer.
\(4.71239\)
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.
\(\text{Isosceles triangle at 50 feet}\)
4.4.39.
Answer.
\(1\%\)

4.5 Taylor Polynomials

Exercises

Terms and Concepts
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
5.1.8.
Answer.
\(F\mathopen{}\left(x\right)+G\mathopen{}\left(x\right)\)
Problems
5.1.9.
Answer.
\(\left({\frac{3}{2}}\right)x^{6}+C\)
5.1.10.
Answer.
\({\frac{1}{3}}x^{3}+C\)
5.1.11.
Answer.
\(\left({\frac{3}{4}}\right)x^{8}+7x+C\)
5.1.12.
Answer.
\(t+C\)
5.1.13.
Answer.
\(s+C\)
5.1.14.
Answer.
\(C-\frac{1}{30t^{6}}\)
5.1.15.
Answer.
\(C-\frac{2}{t^{3}}\)
5.1.16.
Answer.
\(2\sqrt{x}+C\)
5.1.17.
Answer.
\(\csc\mathopen{}\left(\theta\right)+C\)
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.
\(3e^{\theta}+C\)
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.
\(\left({\frac{25}{3}}\right)t^{3}+5t^{2}+t+\left({\frac{1}{15}}\right)+C\)
5.1.24.
Answer.
\(\frac{t^{10}}{10}-3\frac{t^{6}}{2}+9t^{2}+C\)
5.1.25.
Answer.
\(\frac{x^{16}}{16}+C\)
5.1.26.
Answer.
\(1.41421^{e}x+C\)
5.1.27.
Answer.
\(rx+C\)
5.1.30.
Answer.
\(6-\sin\mathopen{}\left(x\right)\)
5.1.31.
Answer.
\(3e^{x}+5\)
5.1.32.
Answer.
\(3\frac{x^{4}}{4}-7\frac{x^{3}}{3}+\left({\frac{44}{3}}\right)\)
5.1.33.
Answer.
\(\sec\mathopen{}\left(x\right)+3\)
5.1.34.
Answer.
\(\frac{5^{x}}{\ln\mathopen{}\left(5\right)}-\frac{25}{\ln\mathopen{}\left(5\right)}+4\)
5.1.35.
Answer.
\(3x^{2}+2x+3\)
5.1.36.
Answer.
\(\left({\frac{5}{6}}\right)x^{3}+\left({\frac{7}{2}}\right)x+\left(-{\frac{22}{3}}\right)\)
5.1.37.
Answer.
\(8e^{x}-12x+1\)
5.1.38.
Answer.
\(4\theta-\cos\mathopen{}\left(\theta\right)+4\)
5.1.39.
Answer.
\(\frac{x^{6}}{3}+\frac{7^{x}}{3.78657}+\cos\mathopen{}\left(x\right)-3.5139x+0.735908\)
5.1.40.
Answer.
\(-\left(4x+23\right)\)

5.2 The Definite Integral

Exercises

Terms and Concepts
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

5.3.4.2.
Answer.
\(15\)
5.3.4.3.
Answer.
\(\text{rectangles}\)

Problems

5.3.4.5.
Answer 1.
\(25+36+49+64\)
Answer 2.
\(174\)
5.3.4.6.
Answer 1.
\(-13+\left(-8\right)+\left(-3\right)+2+7\)
Answer 2.
\(-15\)
5.3.4.7.
Answer 1.
\(1+0+\left(-1\right)+0+1+0+\left(-1\right)\)
Answer 2.
\(0\)
5.3.4.8.
Answer 1.
\(8+8+8+8+8+8\)
Answer 2.
\(48\)
5.3.4.9.
Answer 1.
\(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\)
Answer 2.
\({\frac{137}{60}}\)
5.3.4.10.
Answer 1.
\(-1+2+\left(-3\right)+4+\left(-5\right)+6\)
Answer 2.
\(3\)
5.3.4.11.
Answer 1.
\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}\)
Answer 2.
\({\frac{4}{5}}\)
5.3.4.12.
Answer 1.
\(1+1+1+1\)
Answer 2.
\(4\)
5.3.4.13.
Answer.
\(1;\,6;\,8i\)
5.3.4.14.
Answer.
\(0;\,9;\,i^{2}-1\)
5.3.4.15.
Answer.
\(1;\,6;\,\frac{i}{i+1}\)
5.3.4.16.
Answer.
\(0;\,4;\,\left(-e\right)^{i}\)
5.3.4.17.
Answer.
\(42\)
5.3.4.18.
Answer.
\(276\)
5.3.4.19.
Answer.
\(1870\)
5.3.4.20.
Answer.
\(74096\)
5.3.4.21.
Answer.
\(-51194\)
5.3.4.22.
Answer.
\(6996\)
5.3.4.23.
Answer.
\(6670\)
5.3.4.24.
Answer.
\(9455\)
5.3.4.25.
Answer.
\(150\)
5.3.4.26.
Answer.
\(78975\)
5.3.4.27.
Answer.
\(63\)
5.3.4.28.
Answer.
\(24168\)
5.3.4.35.
Answer 1.
\(\frac{\left(n+1\right)^{2}}{4n^{2}}\)
Answer 2.
\(0.3025\)
Answer 3.
\(0.255025\)
Answer 4.
\(0.2505\)
Answer 5.
\({\frac{1}{4}}\)
5.3.4.36.
Answer 1.
\(15+\frac{45}{2n}+\frac{45}{2n^{2}}\)
Answer 2.
\(17.475\)
Answer 3.
\(15.2272\)
Answer 4.
\(15.0225\)
Answer 5.
\(15\)
5.3.4.37.
Answer 1.
\(4\)
Answer 2.
\(4\)
Answer 3.
\(4\)
Answer 4.
\(4\)
Answer 5.
\(4\)
5.3.4.38.
Answer 1.
\(112+\frac{144}{1n}+\frac{32}{1n^{2}}\)
Answer 2.
\(126.72\)
Answer 3.
\(113.443\)
Answer 4.
\(112.144\)
Answer 5.
\(112\)
5.3.4.39.
Answer 1.
\(56+\frac{98}{n}\)
Answer 2.
\(65.8\)
Answer 3.
\(56.98\)
Answer 4.
\(56.098\)
Answer 5.
\(56\)
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.
\(0\)

Problems

5.4.6.5.
Answer.
\(11\)
5.4.6.6.
Answer.
\({\frac{28}{3}}\)
5.4.6.7.
Answer.
\(0\)
5.4.6.8.
Answer.
\(-1\)
5.4.6.9.
Answer.
\(1\)
5.4.6.10.
Answer.
\(8\)
5.4.6.11.
Answer.
\(\frac{\left({\frac{6560}{9}}\right)}{\ln\mathopen{}\left(9\right)}\)
5.4.6.12.
Answer.
\(-{\frac{292}{3}}\)
5.4.6.13.
Answer.
\(-16\)
5.4.6.14.
Answer.
\(e^{4}-e^{2}\)
5.4.6.15.
Answer.
\({\frac{14}{3}}\)
5.4.6.16.
Answer.
\(4\)
5.4.6.17.
Answer.
\({\frac{968}{5}}\)
5.4.6.18.
Answer.
\(\ln\mathopen{}\left(8\right)\)
5.4.6.19.
Answer.
\({\frac{7}{8}}\)
5.4.6.20.
Answer.
\({\frac{16806}{84035}}\)
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.
\({\frac{1}{98}}\)
5.4.6.25.
Answer.
\(10\)
5.4.6.26.
Answer.
\(3\)
5.4.6.27.
Answer.
\(0\)
5.4.6.28.
Answer.
\(\frac{2}{\sqrt{3}}-1\)
5.4.6.31.
Answer.
\(2.3094\)
5.4.6.32.
Answer.
\(-5.19615, 5.19615\)
5.4.6.33.
Answer.
\(0.541325\)
5.4.6.34.
Answer.
\(4\)
5.4.6.35.
Answer.
\(\frac{\frac{1}{\frac{\pi }{2}-0}\cdot 3.14159}{\pi }\)
5.4.6.36.
Answer.
\(\frac{\frac{-1}{\pi -\frac{\pi }{2}}\cdot 3.14159}{\pi }\)
5.4.6.37.
Answer.
\(3\)
5.4.6.38.
Answer.
\({\frac{49}{3}}\)
5.4.6.39.
Answer.
\(128\)
5.4.6.40.
Answer.
\(\frac{9}{e^{9}-1}\)
5.4.6.41.
Answer.
\(-216\ {\rm ft}\)
5.4.6.42.
Answer.
\(896\ {\rm ft}\)
5.4.6.43.
Answer.
\(40\ {\rm ft}\)
5.4.6.44.
Answer.
\(331.777\ {\rm mi}\)
5.4.6.45.
Answer.
\(-1\ {\rm ft}\)
5.4.6.46.
Answer.
\({\frac{128}{5}}\ {\rm ft}\)
5.4.6.47.
Answer.
\(-224\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.48.
Answer.
\(112\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.49.
Answer.
\({\frac{81}{2}}\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.50.
Answer.
\(-1\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.55.
Answer.
\(\frac{3x^{2}+1}{x^{3}+x}\)
5.4.6.56.
Answer.
\(-3x^{11}\)
5.4.6.57.
Answer.
\(4x^{3}\mathopen{}\left(x^{4}-2\right)-\left(x-2\right)\)
5.4.6.58.
Answer.
\(-\left(e^{\cos\mathopen{}\left(x\right)}\sin\mathopen{}\left(x\right)+e^{\sin\mathopen{}\left(x\right)}\cos\mathopen{}\left(x\right)\right)\)
5.4.6.59.
Answer.
\(2x\sin\mathopen{}\left(3x^{4}\right)\)
5.4.6.60.
Answer.
\(-\left(\sin\mathopen{}\left(x\right)\sqrt{\cos^{4}\mathopen{}\left(x\right)+5\cos^{2}\mathopen{}\left(x\right)}+\frac{1}{2\sqrt{x}}\sqrt{\left(\sqrt{x}\right)^{4}+5\mathopen{}\left(\sqrt{x}\right)^{2}}\right)\)

5.5 Numerical Integration
5.5.6 Exercises

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

Problems

6.1.5.3.
Answer.
\({\frac{1}{8}}\mathopen{}\left(x^{3}+6\right)^{8}+C\)
6.1.5.4.
Answer.
\({\frac{1}{10}}\mathopen{}\left(x^{2}-2x+1\right)^{10}+C\)
6.1.5.5.
Answer.
\({\frac{1}{12}}\mathopen{}\left(x^{2}-7\right)^{6}+C\)
6.1.5.6.
Answer.
\(\left({\frac{1}{3}}\right)\mathopen{}\left(3x^{2}+6x+5\right)^{6}+C\)
6.1.5.7.
Answer.
\({\frac{1}{7}}\ln\mathopen{}\left(\left|7x+5\right|\right)+C\)
6.1.5.8.
Answer.
\(\left({\frac{1}{4}}\right)\sqrt{8x+2}+C\)
6.1.5.9.
Answer.
\({\frac{2}{3}}\mathopen{}\left(x-18\right)\sqrt{x+9}+C\)
6.1.5.10.
Answer.
\(x^{\left({\frac{3}{2}}\right)}\mathopen{}\left({\frac{2}{9}}x^{3}-\left({\frac{16}{3}}\right)\right)+C\)
6.1.5.11.
Answer.
\(2e^{\sqrt{x}}+C\)
6.1.5.12.
Answer.
\(\left({\frac{2}{7}}\right)\sqrt{x^{7}-4}+C\)
6.1.5.13.
Answer.
\(C-{\frac{1}{4}}\mathopen{}\left(\frac{1}{x^{2}}-2\right)^{2}\)
6.1.5.14.
Answer.
\(\frac{\ln^{2}\mathopen{}\left(x\right)}{2}+C\)
6.1.5.15.
Answer.
\(\frac{\left(\sin\mathopen{}\left(x\right)\right)^{8}}{8}+C\)
6.1.5.16.
Answer.
\(C-\frac{\left(\cos\mathopen{}\left(x\right)\right)^{8}}{8}\)
6.1.5.17.
Answer.
\(\frac{\cos\mathopen{}\left(9-2x\right)}{2}+C\)
6.1.5.18.
Answer.
\(C-\tan\mathopen{}\left(6-x\right)\)
6.1.5.19.
Answer.
\({\frac{1}{2}}\ln\mathopen{}\left(\left|\sec\mathopen{}\left(2x\right)+\tan\mathopen{}\left(2x\right)\right|\right)+C\)
6.1.5.20.
Answer.
\({\frac{1}{4}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{4}+C\)
6.1.5.21.
Answer.
\({\frac{1}{4}}\sin\mathopen{}\left(x^{4}\right)+C\)
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.
\({\frac{1}{7}}e^{7x-6}+C\)
6.1.5.26.
Answer.
\({\frac{1}{8}}e^{x^{8}}+C\)
6.1.5.27.
Answer.
\({\frac{1}{2}}e^{\left(x+9\right)^{2}}+C\)
6.1.5.28.
Answer.
\(x+8e^{-x}+C\)
6.1.5.29.
Answer.
\(\ln\mathopen{}\left(e^{x}+2\right)+C\)
6.1.5.30.
Answer.
\({\frac{1}{4}}e^{-4x}-{\frac{1}{2}}e^{-2x}+C\)
6.1.5.31.
Answer.
\(\frac{5^{5x}}{8.04719}+C\)
6.1.5.32.
Answer.
\(\frac{6^{8x}}{14.3341}+C\)
6.1.5.33.
Answer.
\(\frac{\ln^{2}\mathopen{}\left(x\right)}{2}+C\)
6.1.5.34.
Answer.
\(\frac{\left(\ln\mathopen{}\left(x\right)\right)^{8}}{8}+C\)
6.1.5.35.
Answer.
\(\left({\frac{9}{2}}\right)\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)^{2}+C\)
6.1.5.36.
Answer.
\({\frac{1}{8}}\ln\mathopen{}\left(\left|\ln\mathopen{}\left(x^{8}\right)\right|\right)+C\)
6.1.5.37.
Answer.
\(\frac{x^{2}}{2}-8x-9\ln\mathopen{}\left(\left|x\right|\right)+C\)
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.
\(\frac{\left(x+7\right)^{2}}{2}-16\mathopen{}\left(x+7\right)+72\ln\mathopen{}\left(\left|x+7\right|\right)+C\)
6.1.5.41.
Answer.
\(\left({\frac{1}{2}}\right)\mathopen{}\left(x+5\right)^{2}-7\mathopen{}\left(x+5\right)+6\ln\mathopen{}\left(\left|x+5\right|\right)+C\)
6.1.5.42.
Answer.
\({\frac{1}{3}}\ln\mathopen{}\left(\left|x^{3}+3x^{2}-6x+6\right|\right)+C\)
6.1.5.43.
Answer.
\(3.74166\tan^{-1}\mathopen{}\left(\frac{x}{3.74166}\right)+C\)
6.1.5.44.
Answer.
\(8\sin^{-1}\mathopen{}\left(\frac{x}{8}\right)+C\)
6.1.5.45.
Answer.
\(6\sin^{-1}\mathopen{}\left(\frac{x}{3.87298}\right)+C\)
6.1.5.46.
Answer.
\(\left({\frac{9}{2}}\right)\sec^{-1}\mathopen{}\left(\frac{\left|x\right|}{2}\right)+C\)
6.1.5.47.
Answer.
\(2\sec^{-1}\mathopen{}\left(\frac{\left|x\right|}{3}\right)+C\)
6.1.5.48.
Answer.
\(0.5\sin^{-1}\mathopen{}\left(x^{2}\right)+C\)
6.1.5.49.
Answer.
\(0.267261\tan^{-1}\mathopen{}\left(\frac{x-2}{14}\right)+C\)
6.1.5.50.
Answer.
\(6\sin^{-1}\mathopen{}\left(\frac{x-8}{3}\right)+C\)
6.1.5.51.
Answer.
\(6\sin^{-1}\mathopen{}\left(\frac{x+2}{3}\right)+C\)
6.1.5.52.
Answer.
\(\tan^{-1}\mathopen{}\left(\frac{x+4}{8}\right)+C\)
6.1.5.53.
Answer.
\(C-\frac{1}{8\mathopen{}\left(x^{8}-1\right)}\)
6.1.5.54.
Answer.
\({\frac{1}{40}}\mathopen{}\left(9x^{4}-8x^{2}+5\right)^{10}+C\)
6.1.5.55.
Answer.
\(\left({\frac{1}{5}}\right)\sqrt{1+5x^{2}}+C\)
6.1.5.56.
Answer.
\(C-\cot\mathopen{}\left(x^{4}-2\right)\)
6.1.5.57.
Answer.
\({\frac{2}{3}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{\left({\frac{3}{2}}\right)}+C\)
6.1.5.58.
Answer.
\(C-{\frac{1}{5}}\cos\mathopen{}\left(5x+4\right)\)
6.1.5.59.
Answer.
\(\ln\mathopen{}\left(\left|x-3\right|\right)+C\)
6.1.5.60.
Answer.
\(6\ln\mathopen{}\left(\left|x+3\right|\right)+C\)
6.1.5.61.
Answer.
\(2x^{2}-6x+\ln\mathopen{}\left(\left|x^{2}+4x+3\right|\right)+C\)
6.1.5.62.
Answer.
\(\ln\mathopen{}\left(\left|x^{2}+6x-4\right|\right)+C\)
6.1.5.63.
Answer.
\(4\ln\mathopen{}\left(\left|8x^{2}+9x-1\right|\right)+C\)
6.1.5.64.
Answer.
\(-\left({\frac{1}{2}}\right)x^{2}+8x+\ln\mathopen{}\left(\left|x^{2}-9x+3\right|\right)+C\)
6.1.5.65.
Answer.
\({\frac{1}{6}}\tan^{-1}\mathopen{}\left(\frac{x^{2}}{3}\right)+C\)
6.1.5.66.
Answer.
\(\tan^{-1}\mathopen{}\left(4x\right)+C\)
6.1.5.67.
Answer.
\(\sec^{-1}\mathopen{}\left(\left|5x\right|\right)+C\)
6.1.5.68.
Answer.
\({\frac{1}{3}}\sin^{-1}\mathopen{}\left(3\frac{x}{5}\right)+C\)
6.1.5.69.
Answer.
\(\left({\frac{9}{2}}\right)\ln\mathopen{}\left(\left|x^{2}+4x+85\right|\right)-\tan^{-1}\mathopen{}\left(\frac{x+2}{9}\right)+C\)
6.1.5.70.
Answer.
\(\left({\frac{3}{2}}\right)\tan^{-1}\mathopen{}\left(\frac{x+4}{6}\right)+\left({\frac{7}{2}}\right)\ln\mathopen{}\left(\left|x^{2}+8x+52\right|\right)+C\)
6.1.5.71.
Answer.
\(x+24.8261\tan^{-1}\mathopen{}\left(\frac{x+6}{1.73205}\right)+\left({\frac{9}{2}}\right)\ln\mathopen{}\left(\left|x^{2}+12x+39\right|\right)+C\)
6.1.5.72.
Answer.
\(\frac{x^{2}}{2}-\left({\frac{81}{2}}\right)\ln\mathopen{}\left(\left|x^{2}+81\right|\right)+C\)
6.1.5.73.
Answer.
\({\frac{1}{2}}x^{2}+18x+9\ln\mathopen{}\left(\left|x^{2}-18x+88\right|\right)+13.2288\tan^{-1}\mathopen{}\left(\frac{x-9}{2.64575}\right)+C\)
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.
\(\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)+C\)
6.1.5.77.
Answer.
\(8\sqrt{x^{2}+16x+58}+C\)
6.1.5.78.
Answer.
\(\sqrt{x^{2}-4x+1}+C\)
6.1.5.79.
Answer.
\(\ln\mathopen{}\left(\left({\frac{3}{4}}\right)\right)\)
6.1.5.80.
Answer.
\({\frac{11232}{5}}\)
6.1.5.81.
Answer.
\({\frac{1}{5}}\)
6.1.5.82.
Answer.
\({\frac{1}{4}}\)
6.1.5.83.
Answer.
\({\frac{1}{2}}\mathopen{}\left(e^{4}-e^{4}\right)\)
6.1.5.84.
Answer.
\(\frac{\pi }{2}\)
6.1.5.85.
Answer.
\(\frac{\pi }{2}\)
6.1.5.86.
Answer.
\(\left({\frac{1}{2}}\right)\pi \)

6.2 Integration by Parts

Exercises

Problems
6.2.5.
Answer.
\(\sin\mathopen{}\left(x\right)-x\cos\mathopen{}\left(x\right)+C\)
6.2.6.
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.
\(\left({\frac{1}{61}}\right)e^{6x}\mathopen{}\left(6\sin\mathopen{}\left(5x\right)-5\cos\mathopen{}\left(5x\right)\right)+C\)
6.2.15.
Answer.
\(\left({\frac{1}{14}}\right)e^{7x}\mathopen{}\left(\sin\mathopen{}\left(7x\right)+\cos\mathopen{}\left(7x\right)\right)+C\)
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.
\({\frac{1}{2}}x^{2}\ln\mathopen{}\left(x\right)-\frac{x^{2}}{4}+4x\ln\mathopen{}\left(x\right)-4x+C\)
6.2.23.
Answer.
\({\frac{1}{2}}x^{2}\ln\mathopen{}\left(x-2\right)-{\frac{1}{4}}\mathopen{}\left(x-2\right)^{2}-2x-2\ln\mathopen{}\left(x-2\right)+C\)
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.
\(2\mathopen{}\left(x-9\right)+\left(x-9\right)\mathopen{}\left(\ln\mathopen{}\left(x-9\right)\right)^{2}-2\mathopen{}\left(x-9\right)\ln\mathopen{}\left(x-9\right)+C\)
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.
\(\left({\frac{2}{5}}\mathopen{}\left(x-3\right)^{2}+2\mathopen{}\left(x-3\right)\right)\sqrt{x-3}+C\)
6.2.31.
Answer.
\({\frac{1}{3}}\mathopen{}\left(x^{2}-5\right)^{\left({\frac{3}{2}}\right)}+C\)
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.
\(2\frac{1}{e}-3\frac{1}{e^{2}}\)
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.
\(\left(-{\frac{5}{4}}\right)e^{-4}-\left(-{\frac{3}{4}}\right)e^{-2}\)
6.2.48.
Answer.
\(0.5e^{\pi }+0.5\)
6.2.49.
Answer.
\(0.4\mathopen{}\left(-e^{2\pi }+e^{-2\pi }\right)\)

6.3 Trigonometric Integrals
6.3.4 Exercises

Problems

6.3.4.5.
Answer.
\(-0.2\cos^{5}\mathopen{}\left(x\right)+C\)
6.3.4.6.
Answer.
\(0.25\sin^{4}\mathopen{}\left(x\right)+C\)
6.3.4.7.
Answer.
\({\frac{1}{11}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{11}-{\frac{1}{9}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{9}+C\)
6.3.4.8.
Answer.
\({\frac{1}{12}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{12}-{\frac{1}{10}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{10}+C\)
6.3.4.9.
Answer.
\({\frac{1}{7}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{7}-{\frac{2}{5}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{5}+{\frac{1}{3}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{3}+C\)
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.
\(\left({\frac{1}{4}}\right)\sin\mathopen{}\left(2x\right)-\left({\frac{1}{28}}\right)\sin\mathopen{}\left(14x\right)+C\)
6.3.4.15.
Answer.
\(\frac{1}{8\pi }\sin\mathopen{}\left(4\pi x\right)-\frac{1}{12\pi }\sin\mathopen{}\left(6\pi x\right)+C\)
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.
\(\frac{1}{\pi }\cos\mathopen{}\left(\frac{\pi }{2}\pi x\right)+\frac{1}{3\pi }\cos\mathopen{}\left(\frac{3\pi }{2}\pi x\right)+C\)
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.
\({\frac{1}{6}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{6}+{\frac{1}{4}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{4}+C\)
6.3.4.21.
Answer.
\({\frac{1}{5}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{5}+C\)
6.3.4.22.
Answer.
\({\frac{1}{7}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{7}-{\frac{1}{5}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{5}+C\)
6.3.4.23.
Answer.
\({\frac{1}{10}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{10}-{\frac{1}{4}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{8}+{\frac{1}{6}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{6}+C\)
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.
\({\frac{1}{4}}\)
6.3.4.32.
Answer.
\(-{\frac{2}{3}}\)
6.3.4.33.
Answer.
\({\frac{2}{5}}\)
6.3.4.34.
Answer.
\({\frac{16}{15}}\)

6.4 Trigonometric Substitution

Exercises

Terms and Concepts
6.4.2.
Answer.
\(9\sin\mathopen{}\left(\theta\right)\hbox{ or }9\cos\mathopen{}\left(\theta\right)\)
6.4.3.
Answer 1.
\(\tan^{2}\mathopen{}\left(\theta\right)+1 = \sec^{2}\mathopen{}\left(\theta\right)\)
Answer 2.
\(2\sec^{2}\mathopen{}\left(\theta\right)\)
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.
\(\frac{x}{2}\sqrt{64x^{2}+1}+{\frac{1}{16}}\ln\mathopen{}\left(8x+\sqrt{64x^{2}+1}\right)+C\)
6.4.12.
Answer.
\(\frac{x}{2}\sqrt{1-81x^{2}}+{\frac{1}{18}}\sin^{-1}\mathopen{}\left(9x\right)+C\)
6.4.13.
Answer.
\(\frac{x}{2}\sqrt{4x^{2}-1}-{\frac{1}{4}}\ln\mathopen{}\left(\left|2x+\sqrt{4x^{2}-1}\right|\right)+C\)
6.4.14.
Answer.
\(2\ln\mathopen{}\left(\frac{x}{3.60555}+\sqrt{\frac{x^{2}}{13}+1}\right)+C\)
6.4.15.
Answer.
\(3\sin^{-1}\mathopen{}\left(\frac{x}{2.44949}\right)+C\)
6.4.16.
Answer.
\(4\ln\mathopen{}\left(\left|\frac{x}{1.41421}+\sqrt{\frac{x^{2}}{2}-1}\right|\right)+C\)
6.4.17.
Answer.
\(\sqrt{x^{2}-7}-2.64575\sec^{-1}\mathopen{}\left(\frac{x}{2.64575}\right)+C\)
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.
\(\sqrt{x^{2}-11}+C\)
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.
\(C-\frac{1}{\sqrt{x^{2}+81}}\)
6.4.22.
Answer.
\(x\sqrt{x^{2}-15}+15\ln\mathopen{}\left(\left|\frac{x}{3.87298}+\sqrt{\frac{x^{2}}{15}-1}\right|\right)+C\)
6.4.23.
Answer.
\(\left({\frac{1}{72}}\right)\frac{x+7}{x^{2}+14x+85}+\left({\frac{1}{432}}\right)\tan^{-1}\mathopen{}\left(\frac{x+7}{6}\right)+C\)
6.4.24.
Answer.
\(\frac{x}{\sqrt{1-x^{2}}}-\sin^{-1}\mathopen{}\left(x\right)+C\)
6.4.25.
Answer.
\(C-\left(\frac{\sqrt{15-x^{2}}}{4x}+{\frac{1}{4}}\sin^{-1}\mathopen{}\left(\frac{x}{3.87298}\right)\right)\)
6.4.26.
Answer.
\(\frac{x}{2}\sqrt{x^{2}+7}-\left({\frac{7}{2}}\right)\ln\mathopen{}\left(\frac{x}{1.73205}+\sqrt{\frac{x^{2}}{7}+1}\right)+C\)
6.4.27.
Answer.
\(\frac{\pi }{2}\)
6.4.28.
Answer.
\(\left({\frac{9}{2}}\right)\sqrt{65}-8\ln\mathopen{}\left(\left|\left({\frac{9}{4}}\right)+\left({\frac{1}{4}}\right)\sqrt{65}\right|\right)\)
6.4.29.
Answer.
\(8\sqrt{17}+2\ln\mathopen{}\left(4+1\sqrt{17}\right)\)
6.4.30.
Answer.
\(\tan^{-1}\mathopen{}\left(9\right)+\left({\frac{9}{82}}\right)\)
6.4.31.
Answer.
\(9\sin^{-1}\mathopen{}\left(\left({\frac{1}{3}}\right)\right)+2\sqrt{2}\)
6.4.32.
Answer.
\(\frac{\pi }{8}\)

6.5 Partial Fraction Decomposition

Exercises

Terms and Concepts
6.5.3.
Answer.
\(\frac{A}{x}+\frac{B}{x-5}\)
6.5.4.
Answer.
\(\frac{A}{x-4}+\frac{B}{x+4}\)
6.5.5.
Answer.
\(\frac{A}{x-\sqrt{7}}+\frac{B}{x+\sqrt{7}}\)
6.5.6.
Answer.
\(\frac{A}{x}+\frac{Bx+C}{x^{2}+6}\)
Problems
6.5.7.
Answer.
\(9\ln\mathopen{}\left(\left|x-3\right|\right)+7\ln\mathopen{}\left(\left|x-5\right|\right)+C\)
6.5.8.
Answer.
\(C-\left(3\ln\mathopen{}\left(\left|x-5\right|\right)+2\ln\mathopen{}\left(\left|x\right|\right)\right)\)
6.5.9.
Answer.
\(3\ln\mathopen{}\left(\left|x+2\right|\right)-3\ln\mathopen{}\left(\left|x-2\right|\right)+C\)
6.5.10.
Answer.
\(\ln\mathopen{}\left(\left|x-9\right|\right)+\ln\mathopen{}\left(\left|5x+1\right|\right)+C\)
6.5.11.
Answer.
\(4\ln\mathopen{}\left(\left|x+7\right|\right)-\frac{7}{x+7}+C\)
6.5.12.
Answer.
\(9\ln\mathopen{}\left(\left|x+5\right|\right)-\frac{1}{x+5}+C\)
6.5.13.
Answer.
\(6\ln\mathopen{}\left(\left|x\right|\right)+4\ln\mathopen{}\left(\left|x+3\right|\right)+\frac{1}{x+3}+C\)
6.5.14.
Answer.
\(\ln\mathopen{}\left(\left|8x+6\right|\right)+8\ln\mathopen{}\left(\left|x+2\right|\right)+6\ln\mathopen{}\left(\left|x+4\right|\right)+C\)
6.5.15.
Answer.
\(\left({\frac{1}{3}}\right)\ln\mathopen{}\left(\left|6x-1\right|\right)+\left({\frac{1}{3}}\right)\ln\mathopen{}\left(\left|9x+1\right|\right)+\frac{\left({\frac{1}{3}}\right)}{3x-1}+C\)
6.5.16.
Answer.
\(x+2\ln\mathopen{}\left(\left|x-3\right|\right)-2\ln\mathopen{}\left(\left|x-2\right|\right)+C\)
6.5.17.
Answer.
\({\frac{1}{2}}x^{2}+\left({\frac{25}{2}}\right)\ln\mathopen{}\left(\left|x-5\right|\right)+\left({\frac{25}{2}}\right)\ln\mathopen{}\left(\left|x+5\right|\right)+C\)
6.5.18.
Answer.
\(3x+C\)
6.5.19.
Answer.
\(\left({\frac{1}{19}}\right)\ln\mathopen{}\left(\left|x\right|\right)-\left({\frac{1}{38}}\right)\ln\mathopen{}\left(x^{2}-8x+19\right)+0.121547\tan^{-1}\mathopen{}\left(\frac{x-4}{1.73205}\right)+C\)
6.5.20.
Answer.
\(x-\left({\frac{3}{2}}\right)\ln\mathopen{}\left(x^{2}+8x+18\right)+3.53553\tan^{-1}\mathopen{}\left(\frac{x+4}{1.41421}\right)+C\)
6.5.21.
Answer.
\(\ln\mathopen{}\left(\left|3x^{2}+7x+5\right|\right)-\ln\mathopen{}\left(\left|x-7\right|\right)+C\)
6.5.22.
Answer.
\(4\ln\mathopen{}\left(\left|x+5\right|\right)+3\ln\mathopen{}\left(x^{2}+4x+5\right)+4\tan^{-1}\mathopen{}\left(x+2\right)+C\)
6.5.23.
Answer.
\(\left({\frac{55}{9}}\right)\ln\mathopen{}\left(\left|x+6\right|\right)+\left({\frac{4}{9}}\right)\ln\mathopen{}\left(x^{2}+9\right)-\left({\frac{28}{9}}\right)\tan^{-1}\mathopen{}\left(\frac{x}{3}\right)+C\)
6.5.24.
Answer.
\(\ln\mathopen{}\left(x^{2}-6x+13\right)-\ln\mathopen{}\left(\left|x+8\right|\right)+2\tan^{-1}\mathopen{}\left(\frac{x-3}{2}\right)+C\)
6.5.25.
Answer.
\(3\ln\mathopen{}\left(\left|x+1\right|\right)-\ln\mathopen{}\left(x^{2}-2x+8\right)-0.377964\tan^{-1}\mathopen{}\left(\frac{x-1}{2.64575}\right)+C\)
6.5.26.
Answer.
\(\left({\frac{5}{2}}\right)\ln\mathopen{}\left(x^{2}-10x+27\right)-\ln\mathopen{}\left(\left|x-7\right|\right)+9.19239\tan^{-1}\mathopen{}\left(\frac{x-5}{1.41421}\right)+C\)
6.5.27.
Answer.
\(\ln\mathopen{}\left(\left({\frac{243}{784}}\right)\right)\)
6.5.28.
Answer.
\(1.4594\)
6.5.29.
Answer.
\(\ln\mathopen{}\left(\left({\frac{5}{4}}\right)\right)+\tan^{-1}\mathopen{}\left(5\right)-\tan^{-1}\mathopen{}\left(3\right)\)
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.
\(\frac{1}{\sqrt{\left(2x^{2}\right)^{2}-1}}\cdot 4x\)
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

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

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

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

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

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·m⁄s2m In Imperial Units, it is ft–lb.

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

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.
8.1.4.3.
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

8.1.4.13.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated.In the first quadrant in the top left, the field lines are north-east facing and in the bottom right they are southeast facing. In the second quadrant the field lines are all north-east facing. In the third quadrant like in the first quadrant in the top left the field lines are northeast facing and in the bottom right they are southeast facing. In the fourth quadrant all lines are southeast facing.
8.1.4.14.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated. The field lines form concentric ovals facing away from the origin on both positive and negative \(x\) and \(y\) axes. The concentric shorter arcs are on either end of the \(x\) axis. On the two ends of the \(y\) axis concentric wider arcs are drawn. The field lines intermix to form an ’X’ with centre at the origin.
8.1.4.15.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated. There are five instances where the field lines run parallel to the \(x\) axis. One of them is on the \(x\) axis itself, other two pairs of such field lines are above and below the \(x\) axis. In between the \(x\) axis and the first horizontal field line for some positive \(y\) value, the field lines are all northeast facing. Above the horizontal field line for some \(y\) value until another with a higher \(y\) value, the field lines in between are southeast facing.
Similarly below the \(x\) axis till the first horizontal line with some negative \(y\) value, the field lines in between are southeast facing. In between this horizontal line and another horizontal line with a higher negative \(y\) value, the field lines are northeast facing.
8.1.4.16.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated. The field lines run almost parallel to the \(x\) axis. Above the axis the field lines are slightly facing north east. Below the \(x\) axis the lines are directed facing southeast.
8.1.4.18.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated, the field lines in the first quadrant are shown. The field lines very close to the \(y\) axis are almost north facing for higher values of \(y\) and almost east facing for lower values of \(y\text{.}\) With smaller values of \(x\text{,}\) the field lines, from left to right the lines first face northeast then east and southeast after for greater values of \(x\text{.}\)
A curve is drawn that starts at a point for some small value of \(x\) and a high value of \(y\text{.}\) The curve has a positive slope at first after reaching a peak it declines almost close to the \(x\) axis.
8.1.4.19.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated, the field lines in the first quadrant are shown. Front left to right, a little away from the x axis the field lines are northeast facing that transition to north facing. Moving further right then again become northeast facing then transition to southeast facing, further right they become south facing then east facing. The pattern then repeats. Very close to the \(x\) axis the field lines are almost parallel to it.
A wave is drawn that starts at some y intercept above the origin. It has a high positive slope, it reaches peak when the field lines change from northeast facing to southeast facing, then it declines until the point the field lines are parallel to the \(x\) axis. The curve continues to form a second wave.
8.1.4.20.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated, the field lines in the first quadrant are shown. There are two instances where the field lines are parallel to the \(x\) axis. From under the \(x\) axis to the first such line the field lines transition from almost north facing to northeast facing. Between the horizontal field line for a small \(y\) value and a greater \(y\) value the field lines are facing southeast. Above the line with a higher \(y\) value the field lines transition from northeast facing to north facing.
8.1.4.21.
Answer.
Graph showing slope field for the given differential equation.
The \(x\) and \(y\) axes are uncalibrated, the field lines in the first quadrant are shown. In the top right and the centre the field lines are southeast facing, very close to the \(x\) and \(y\) axis the field lines are almost parallel to the \(x\) axis. A curve is drawn that starts from a \(y\) intercept and decreases along the slope lines coming close to the \(x\) axis.
8.1.4.22.
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*}
8.1.4.23.
Answer.
\begin{align*} x_i \amp \quad \amp \quad \amp y_i \\ 0.0 \amp \quad \amp \quad \amp 1.0000 \\ 0.1 \amp \quad \amp \quad \amp 1.0000 \\ 0.2 \amp \quad \amp \quad \amp 1.0037 \\ 0.3 \amp \quad \amp \quad \amp 1.0110 \\ 0.4 \amp \quad \amp \quad \amp 1.0219 \\ 0.5 \amp \quad \amp \quad \amp 1.0363 \end{align*}
8.1.4.24.
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*}
8.1.4.25.
Answer.
\begin{align*} x_i \amp \quad \amp \quad \amp y_i \\ 0.0 \amp \quad \amp \quad \amp 0.0000 \\ 0.5 \amp \quad \amp \quad \amp 0.5000 \\ 1.0 \amp \quad \amp \quad \amp 1.8591 \\ 1.5 \amp \quad \amp \quad \amp 10.5824 \\ 2.0 \amp \quad \amp \quad \amp 88378.1190 \end{align*}
8.1.4.26.
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
8.1.4.27.
Answer.
\(x\) \(0.0\) \(0.2\) \(0.4\) \(0.6\) \(0.8\) \(1.0\)
\(y(x)\) 0.5000 0.5412 0.6806 0.9747 1.5551 2.7183
\(h = 0.2\) 0.5000 0.5000 0.5816 0.7686 1.1250 1.7885
\(h = 0.1\) 0.5000 0.5201 0.6282 0.8622 1.3132 2.1788

8.2 Separable Differential Equations
8.2.2 Exercises

Problems

8.2.2.1.
Answer.
Separable. \(\displaystyle \frac{1}{y^2-y}\,dy = dx\)
8.2.2.2.
Answer.
Not separable.
8.2.2.3.
Answer.
Not separable.
8.2.2.4.
Answer.
Separable. \(\displaystyle \frac{1}{\cos(y) - y}\,dy = (x^2+1)\,dx\)
8.2.2.5.
Answer.
\(\left\{ \displaystyle y = \frac{1 + Ce^{2x}}{1 - Ce^{2x}}, y = -1\right\}\)
8.2.2.6.
Answer.
\(y = 2 + Ce^x\)
8.2.2.7.
Answer.
\(y = Cx^4\)
8.2.2.8.
Answer.
\(y^2 - 4x^2 = C\)
8.2.2.9.
Answer.
\(\displaystyle (y-1)e^y = -e^{-x} - \frac{1}{3}e^{-3x} + C\)
8.2.2.10.
Answer.
\(\displaystyle (y-1)^2 = \ln(x^2+1) + C\)
8.2.2.11.
Answer.
\(\left\{ \arcsin{2y} - \arctan(x^2+1) = C, y = \pm \displaystyle \frac{1}{2} \right\}\)
8.2.2.12.
Answer.
\(\left\{ \displaystyle y = \frac{1}{C - \arctan x}, y = 0 \right\}\)
8.2.2.13.
Answer.
\(\sin(y) + \cos(x) = 2\)
8.2.2.14.
Answer.
\(-x^3 + 3y - y^3 = 2\)
8.2.2.15.
Answer.
\(\frac{1}{2}y^2 - \ln(1+x^2) = 8\)
8.2.2.16.
Answer.
\(y^2+2xe^x - 2e^x = 2\)
8.2.2.17.
Answer.
\(\displaystyle \frac{1}{2}y^2 - y = \frac{1}{2}\big ( (x^2+1)\ln(x^2+1) - (x^2 + 1)\big) + \frac{1}{2}\)
8.2.2.18.
Answer.
\(\sin(y^2)-(\arcsin x)^2 = -\frac{1}{2}\)
8.2.2.19.
Answer.
\(2\tan(2y) = 2x + \sin(2x)\)
8.2.2.20.
Answer.
\(x = exp \displaystyle \left( -\frac{\sqrt{1-y^2}}{y}\right)\)

8.3 First Order Linear Differential Equations
8.3.2 Exercises

Problems

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

8.4 Modeling with Differential Equations
8.4.3 Exercises

Problems

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

9 Curves in the Plane
9.1 Conic Sections
9.1.4 Exercises

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

Problems

9.2.4.5.
Answer.
Sketch of the parametric curve in this exercise.
The sketch for this exercise is a curve that lies mostly in the fourth quadrant. It resembles part of a slingshot orbit for a comet passing around the sun: the curve passes through the origin from below, turns quickly in the second quadrant, crossing the \(y\) axis at \((0,1)\text{,}\) and then the \(x\) axis at \((2,0)\text{,}\) where it returns to the fourth quadrant.
9.2.4.6.
Answer.
The vertical line x=1.
The curve for this exercise is the vertical line \(x=1\text{.}\) An arrow on the line indicates that the direction of travel is up.
9.2.4.7.
Answer.
The horizontal line y=2, marked with two arrows.
The horizontal line \(y=2\text{.}\) On the line there are two arrows pointing in opposite directions. These indicate that the direction of travel is to the left when \(t\lt 0\text{,}\) and to the right when \(t\gt 0\text{.}\)
9.2.4.8.
Answer.
The solution curve for this exercise.
The curve begins to the left of the \(y\) axis, and crosses near \((0,4)\text{.}\) It passes through the first quadrant to the point \((3,2)\text{;}\) it then bends downward and makes a teardrop-shaped loop before passing through \((3,2)\) a second time, and then continuing up and to the right, through the first quadrant.
9.2.4.9.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
A curve resembling a check mark, with a cusp at the origin. Direction of travel is from the second quadrant toward the cusp, and then up from the cusp to a \(y\) intercept at \((0,4)\text{,}\) and then into the first quadrant.
9.2.4.10.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
A curve resembling a sine wave with a period that gets longer for large values of \(x\text{.}\) The direction of travel is that of decreasing \(x\) value, with the \(y\) value oscillating.
9.2.4.11.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
The curve is an ellipse, centered at the origin, with counter-clockwise direction of travel.
9.2.4.12.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
An ellipse with center \((2,3)\) and counter-clockwise direction of travel.
9.2.4.13.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
The curve resembles a parabola, with vertex at \((0,-1)\text{.}\) The direction of travel is from right to left.
9.2.4.14.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
A figure-eight curve, centered at the origin. The orientation is counter-clockwise in the fourth and first quadrants; once the curve passes through the origin (a point of self-intersection) this direction becomes clockwise it the second and third quadrants.
9.2.4.15.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
The curve resembles one branch of a hyperbola, opening to the right, with a vertex at \((2,0)\text{.}\) The direction of travel is that of increasing \(y\) value.
9.2.4.16.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
The curve resembles one branch of a hyperbola, opening to the right, with a vertex at \((1,0)\text{.}\) The direction of travel is that of increasing \(y\) value.
9.2.4.17.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
A flower-shaped curve, with 7 “petals”. Each petal is an arc that loops around and intersects itself before continuing to the next arc.
9.2.4.18.
Answer.
Computer-generated sketch of the parametric curve in this exercise.
A curve that spirals around the origin, with several self-intersecting loops. This curve has 9 arcs and 9 loops. It is similar to the curve in the previous exercise, except that this time, the arcs bend inward toward the origin, rather than outward.
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

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

Problems

9.4.4.5.
Answer.
The four points plotted in this exercise.
On a polar grid, four points are plotted. The point \(A\) is at the intersection of the initial ray and the circle of radius 2. Points \(B\) and \(D\) are both on the circle of radius 1. The point \(B\) is on the same line as the initial ray, but in the opposite direction. The point \(D\) lies above the initial ray, making an angle of \(\pi/4\text{.}\) Finally, the point \(C\) is at the bottom of the circle of radius \(2\text{.}\)
9.4.4.6.
Answer.
The four points plotted in this exercise.
On a polar grid, four points are plotted. Points \(A\) and \(B\) are both on the same line as the initial ray, but to the left of the origin \(O\text{,}\) with \(A\) on the circle of radius 2, and \(B\) on the circle of radius 1. The point \(C\) is on the circle of radius 1, and makes an angle slightly greater than a right angle with the initial ray, placing it above and just to the left of the origin \(O\text{.}\) The point \(D\) lies almost immediately below the point \(C\text{;}\) the circle of radius \(1/2\) is not marked as part of the grid.
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.
Portion of the circle of radius 2, centered at the origin, in the first quadrant.
An arc of the circle \(r=2\) is shown, for \(0\leq \theta\leq \pi/2\text{.}\) This is the quarter of a circle of radius 2, centered at the origin, that lies in the first quadrant.
9.4.4.12.
Answer.
A line segment through the origin with positive slope.
A line segment through the origin is shown. The segment makes an angle of \(\pi/6\) with the positive \(x\) axis, and it extends a distance of 2 units into the first quadrant, and one unit into the third quadrant.
9.4.4.13.
Answer.
A cardioid, symmetric about the x axis, with x intercepts at -2 and 0.
The curve is a cardioid that is symmetric about the \(x\) axis. The cusp is at the origin, and the other \(x\) intercept is at \((-2,0)\text{.}\) (It is in the opposite direction of the example in the gallery of polar curves.)
9.4.4.14.
Answer.
A convex limaçon, symmetric about the y axis.
The curve is a convex limaçon. This is the fourth type of limaçon in the gallery of polar curves. In this case, the limaçon is symmetric about the \(y\) axis, with the flattened part of the curve at the bottom.
9.4.4.15.
Answer.
A convex limaçon, symmetric about the y axis.
The curve is a convex limaçon. This is the fourth type of limaçon in the gallery of polar curves. In this case, the limaçon is symmetric about the \(y\) axis, with the flattened part of the curve at the top.
9.4.4.16.
Answer.
A limaçon with an inner loop, symmetric about the y axis.
The curve is a limaçon with an inner loop. It is symmetric about the \(y\) axis. The inner loop lies below the \(x\) axis, with \(y\) intercepts at \(y=0\) and \(y=-1\text{.}\) The outer loop has its other \(y\) intercept at \(y=-3\text{.}\)
9.4.4.17.
Answer.
A limaçon with an inner loop, symmetric about the y axis.
The curve is a limaçon with an inner loop. It is symmetric about the \(y\) axis. The inner loop lies above the \(x\) axis, with \(y\) intercepts at \(y=0\) and \(y=1\text{.}\) The outer loop has its other \(y\) intercept at \(y=3\text{.}\)
9.4.4.18.
Answer.
A rose curve with four petals.
The curve is a rose curve with four loops that all pass through the origin. The loops are symmetric about the two coordinate axes, with their second intercepts at \((\pm 1,0)\) and \((0,\pm 1)\text{.}\)
9.4.4.19.
Answer.
A rose curve with three loops, symmetric about the y axis.
A rose curve with three loops that all pass through the origin. One loop is along the negative \(y\) axis, with a \(y\) intercept at \((-1,0)\text{.}\) The other two loops lie in the first and second quadrants.
9.4.4.20.
Answer.
A limaçon with an inner loop.
The curve is a limaçon with an inner loop, symmetric about the \(x\) axis, but it is shifted horizontally, to the left, relative to the example in the gallery of polar curves.
The point of self-intersection is at \((-1/2, 0)\text{.}\) The other end of the inner loop is at the origin, and the far end of the outer loop is at the point \((1,0)\text{.}\)
9.4.4.21.
Answer.
An elaborate rose curve with many self-intersections and four primary loops.
This is a more complicated curve. It passes several times through the origin, and has eight other points of self-intersection. The largest loops in the curve are similar to cardioids; there are four of these passing through the origin, with a second intercept at one of the four points \((\pm 1, 0)\text{,}\) \((0,\pm 1)\text{.}\) As these loops intersect each other, they create four other loops of intermediate size, and four smaller loops in the center.
9.4.4.22.
Answer.
A counter-clockwise spiral
The curve is a counter-clockwise spiral that begins at the origin. It makes two full revolutions, and ends at the point \((2\pi, 0)\text{.}\)
9.4.4.23.
Answer.
A circle passing through the origin with its center on the positive y axis.
A circle of radius \(3/2\) with its center at \((0,3/2)\text{.}\) It passes through the origin and the point \((0,3)\text{.}\)
9.4.4.24.
Answer.
A semi-circle in the first quadrant.
The curve is a semi-circle in the first quadrant. Its endpoints are \((0,0)\) and \((2,0)\text{,}\) making the center \((1,0)\) and radius 1.
9.4.4.25.
Answer.
A four-leaf rose with one petal in each quadrant.
The curve is a four-leafed rose that lies within the circle \(r=1/2\text{.}\) One leaf lies in each of the four quadrants.
9.4.4.26.
Answer.
A curve with two loops: a circle inside a larger leaf shape.
The curve for this exercise is a fairly strange shape. It is symmetric about the \(x\) axis. There is a large, outer loop that looks like a leaf or a raindrop. It has a cusp at \((-7,0)\text{,}\) and also passes through the origin. There is a smaller, inner loop that looks almost like a circle. It passes through the origin and the point \((-3,0)\text{.}\)
9.4.4.27.
Answer.
A straight line with positive slope.
The curve is a straight line with \(x\) intercept at \((-3,0)\) and \(y\) intercept \((0,3/5)\text{.}\)
9.4.4.28.
Answer.
A straight line with positive slope.
The curve is a straight line with \(x\) intercept at \((-2/3,0)\) and \(y\) intercept \((0,1)\text{.}\)
9.4.4.29.
Answer.
A vertical line with x=3.
The curve is the vertical line \(x=3\text{.}\)
9.4.4.30.
Answer.
The horizontal line y=4.
The curve is the horizontal line \(y=4\text{.}\)
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

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.

10.3 Integral and Comparison Tests
10.3.4 Exercises

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

10.4.3.3.
Answer.
Integral Test, Limit Comparison Test, and Root Test

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
10.5.3.
Answer.
Many examples exist; one common example is \(a_n = (-1)^n/n\text{.}\)

11 Vectors
11.1 Introduction to Cartesian Coordinates in Space
11.1.7 Exercises

Terms and Concepts

11.1.7.5.
Answer.
A hyperboloid of two sheets
11.1.7.6.
Answer.
A hyperboloid of one sheet

Problems

11.1.7.7.
Answer 1.
\(\sqrt{6}\)
Answer 2.
\(\sqrt{17}\)
Answer 3.
\(\sqrt{11}\)
Answer 4.
\(\text{do}\)
11.1.7.9.
Answer 1.
\(\left(4,-1,0\right)\)
Answer 2.
\(3\)
11.1.7.10.
Answer 1.
\(\left(-2,1,2\right)\)
Answer 2.
\(\sqrt{5}\)
11.1.7.19.
Answer.
\(x^{2}+z^{2} = \left(\frac{1}{1+y^{2}}\right)^{2}\)
11.1.7.20.
Answer.
\(y^{2}+z^{2} = x^{4}\)
11.1.7.21.
Answer.
\(x^{2}+y^{2} = z\)
11.1.7.22.
Answer.
\(x^{2}+y^{2} = \frac{1}{z^{2}}\)
11.1.7.23.
Answer.
(a)\(\ds x=y^2+\frac{z^2}{9}\)
11.1.7.24.
Answer.
(b) \(x^2-y^2+z^2=0\)
11.1.7.25.
Answer.
(b) \(\ds x^2+\frac{y^2}9+\frac{z^2}4=1\)
11.1.7.26.
Answer.
(a) \(y^2-x^2-z^2=1\)

11.2 An Introduction to Vectors

Exercises

Terms and Concepts
11.2.2.
Answer.
\((1,2)\) is a point; \(\la 1,2\ra\) is a vector that describes a displacement of 1 unit in the \(x\)-direction and 2 units in the \(y\)-direction.
11.2.3.
Answer.
A vector with magnitude 1.
11.2.4.
Answer.
Direction
Problems
11.2.7.
11.2.7.a
Answer.
\(\left<1,6\right>\)
11.2.7.b
Answer.
\(\boldsymbol{i}+6\boldsymbol{j}\)
11.2.8.
11.2.8.a
Answer.
\(\left<4,-4\right>\)
11.2.8.b
Answer.
\(4\boldsymbol{i}-4\boldsymbol{j}\)
11.2.9.
11.2.9.a
Answer.
\(\left<6,-1,6\right>\)
11.2.9.b
Answer.
\(6\boldsymbol{i}-\boldsymbol{j}+6\boldsymbol{k}\)
11.2.10.
11.2.10.a
Answer.
\(\left<2,2,0\right>\)
11.2.10.b
Answer.
\(2\boldsymbol{i}+2\boldsymbol{j}\)
11.2.11.
11.2.11.a
Answer.
\(\vec u+\vec v = \la 2,-1\ra\text{;}\) \(\vec u -\vec v = \la0,-3\ra\text{;}\) \(2\vec u-3\vec v = \la -1,-7\ra\text{.}\)
11.2.11.c
Answer.
\(\vec x = \la 1/2,2\ra\text{.}\)
11.2.12.
11.2.12.a
Answer.
\(\vec u+\vec v = \la 3,2,1\ra\text{;}\) \(\vec u -\vec v = \la-1,0,-3\ra\text{;}\) \(\pi\vec u-\sqrt{2}\vec v = \la \pi-2\sqrt{2},\pi-\sqrt{2},-\pi-2\sqrt{2}\ra\text{.}\)
11.2.12.c
Answer.
\(\vec x = \la-1,0,-3\ra\text{.}\)
11.2.17.
Answer 1.
\(\sqrt{5}\)
Answer 2.
\(\sqrt{13}\)
Answer 3.
\(\sqrt{26}\)
Answer 4.
\(\sqrt{10}\)
11.2.18.
Answer 1.
\(\sqrt{17}\)
Answer 2.
\(\sqrt{3}\)
Answer 3.
\(\sqrt{14}\)
Answer 4.
\(\sqrt{26}\)
11.2.19.
Answer 1.
\(\sqrt{5}\)
Answer 2.
\(3\sqrt{5}\)
Answer 3.
\(2\sqrt{5}\)
Answer 4.
\(4\sqrt{5}\)
11.2.20.
Answer 1.
\(7\)
Answer 2.
\(35\)
Answer 3.
\(42\)
Answer 4.
\(28\)
11.2.22.
Answer.
\(\left<\frac{3}{\sqrt{58}},\frac{7}{\sqrt{58}}\right>\)
11.2.23.
Answer.
\(\left<0.6,0.8\right>\)
11.2.24.
Answer.
\(\left<\frac{1}{3},\frac{-2}{3},\frac{2}{3}\right>\)
11.2.25.
Answer.
\(\left<\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right>\)
11.2.26.
Answer.
\(\left<\cos\mathopen{}\left(\frac{50\pi }{180}\right),\sin\mathopen{}\left(\frac{50\pi }{180}\right)\right>\)
11.2.27.
Answer.
\(\left<\frac{-1}{2},\frac{\sqrt{3}}{2}\right>\)

11.3 The Dot Product
11.3.2 Exercises

Problems

11.3.2.5.
Answer.
\(-22\)
11.3.2.6.
Answer.
\(33\)
11.3.2.7.
Answer.
\(3\)
11.3.2.8.
Answer.
\(0\)
11.3.2.9.
Answer.
not defined
11.3.2.10.
Answer.
\(0\)
11.3.2.11.
Answer.
Answers will vary.
11.3.2.12.
Answer.
Answers will vary.
11.3.2.13.
Answer.
\(\cos^{-1}\mathopen{}\left(\frac{3}{\sqrt{10}}\right)\)
11.3.2.14.
Answer.
\(\cos^{-1}\mathopen{}\left(\frac{-1}{\sqrt{170}}\right)\)
11.3.2.15.
Answer.
\(\frac{\pi }{4}\)
11.3.2.16.
Answer.
\(\frac{\pi }{2}\)
11.3.2.17.
Answer 1.
\(\left<-7,4\right>\)
Answer 2.
\(\left<4,7\right>\)
11.3.2.18.
Answer 1.
\(\left<5,3\right>\)
Answer 2.
\(\left<-3,5\right>\)
11.3.2.19.
Answer 1.
\(\left<1,0,-1\right>\)
Answer 2.
\(\left<1,1,1\right>\)
11.3.2.20.
Answer 1.
\(\left<2,1,0\right>\)
Answer 2.
\(\left<1,-2,3\right>\)
11.3.2.21.
Answer.
\(\left<\frac{-5}{10},\frac{15}{10}\right>\)
11.3.2.22.
Answer.
\(\left<\frac{20}{10},\frac{60}{10}\right>\)
11.3.2.23.
Answer.
\(\left<\frac{-1}{2},\frac{-1}{2}\right>\)
11.3.2.24.
Answer.
\(\left<\frac{0}{13},\frac{0}{13}\right>\)
11.3.2.25.
Answer.
\(\left<\frac{14}{14},\frac{28}{14},\frac{42}{14}\right>\)
11.3.2.26.
Answer.
\(\left<\frac{12}{9},\frac{12}{9},\frac{6}{9}\right>\)
11.3.2.27.
Answer 1.
\(\left<\frac{-5}{10},\frac{15}{10}\right>\)
Answer 2.
\(\left<\frac{15}{10},\frac{5}{10}\right>\)
11.3.2.28.
Answer 1.
\(\left<\frac{20}{10},\frac{60}{10}\right>\)
Answer 2.
\(\left<\frac{30}{10},\frac{-10}{10}\right>\)
11.3.2.29.
Answer 1.
\(\left<\frac{-1}{2},\frac{-1}{2}\right>\)
Answer 2.
\(\left<\frac{-5}{2},\frac{5}{2}\right>\)
11.3.2.30.
Answer 1.
\(\left<\frac{0}{13},\frac{0}{13}\right>\)
Answer 2.
\(\left<\frac{-39}{13},\frac{26}{13}\right>\)
11.3.2.31.
Answer 1.
\(\left<\frac{14}{14},\frac{28}{14},\frac{42}{14}\right>\)
Answer 2.
\(\left<\frac{0}{14},\frac{42}{14},\frac{-28}{14}\right>\)
11.3.2.32.
Answer 1.
\(\left<\frac{12}{9},\frac{12}{9},\frac{6}{9}\right>\)
Answer 2.
\(\left<\frac{15}{9},\frac{-21}{9},\frac{12}{9}\right>\)
11.3.2.33.
Answer.
1.96lb
11.3.2.34.
Answer.
5lb
11.3.2.35.
Answer.
\(141.42\)ft–lb
11.3.2.36.
Answer.
\(196.96\)ft–lb
11.3.2.37.
Answer.
\(500\)ft–lb
11.3.2.39.
Answer.
\(500\)ft–lb

11.4 The Cross Product
11.4.3 Exercises

Problems

11.4.3.7.
Answer.
\(\left<12,-15,3\right>\)
11.4.3.8.
Answer.
\(\left<11,1,-17\right>\)
11.4.3.9.
Answer.
\(\left<-5,-31,27\right>\)
11.4.3.10.
Answer.
\(\left<47,-36,-44\right>\)
11.4.3.11.
Answer.
\(\left<0,-2,0\right>\)
11.4.3.12.
Answer.
\(\left<0,0,0\right>\)
11.4.3.13.
Answer.
\(\vec u\times \vec v = \langle 0,0,ad-bc\rangle\)
11.4.3.14.
Answer.
\(\boldsymbol{k}\)
11.4.3.15.
Answer.
\(-\boldsymbol{j}\)
11.4.3.16.
Answer.
\(\boldsymbol{i}\)
11.4.3.17.
Answer.
Answers will vary.
11.4.3.18.
Answer.
Answers will vary.
11.4.3.19.
Answer.
\(5\)
11.4.3.20.
Answer.
\(21\)
11.4.3.21.
Answer.
\(0\)
11.4.3.22.
Answer.
\(5\)
11.4.3.23.
Answer.
\(\sqrt{14}\)
11.4.3.24.
Answer.
\(\sqrt{230}\)
11.4.3.25.
Answer.
\(3\)
11.4.3.26.
Answer.
\(6\)
11.4.3.27.
Answer.
\(\frac{5\sqrt{2}}{2}\)
11.4.3.28.
Answer.
\(3\sqrt{30}\)
11.4.3.29.
Answer.
\(1\)
11.4.3.30.
Answer.
\(\frac{5}{2}\)
11.4.3.31.
Answer.
\(7\)
11.4.3.32.
Answer.
\(4\sqrt{14}\)
11.4.3.33.
Answer.
\(2\)
11.4.3.34.
Answer.
\(15\)
11.4.3.35.
Answer.
\(\left<0.408248,0.408248,-0.816497\right>\hbox{ or }\left<-0.408248,-0.408248,0.816497\right>\)
11.4.3.36.
Answer.
\(\left<-0.436436,0.218218,0.872872\right>\hbox{ or }\left<0.436436,-0.218218,-0.872872\right>\)
11.4.3.37.
Answer.
\(\left<0,1,0\right>\hbox{ or }\left<0,-1,0\right>\)
11.4.3.38.
Answer.
\(\left<\frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}},0\right>\)
11.4.3.39.
Answer.
\(87.5\)ft–lb
11.4.3.40.
Answer.
\(43.75\sqrt{3}\approx 75.78\)ft–lb
11.4.3.41.
Answer.
\(200/3\approx 66.67\)ft–lb
11.4.3.42.
Answer.
\(11.58\)ft–lb

11.5 Lines
11.5.4 Exercises

Terms and Concepts

11.5.4.1.
Answer.
A point on the line and the direction of the line.

Problems

11.5.4.6.
Answer 1.
\(\left(6,1,7\right)+t\mathopen{}\left<-3,2,5\right>\)
Answer 2.
\(x = 6-3t, y = 1+2t, z = 7+5t\)
Answer 3.
\(\frac{x-6}{-3} = \frac{y-1}{2} = \frac{z-7}{5}\)
11.5.4.8.
Answer 1.
\(\left(1,-2,3\right)+t\mathopen{}\left<4,7,2\right>\)
Answer 2.
\(x = 1+4t, y = -2+7t, z = 3+2t\)
Answer 3.
\(\frac{x-1}{4} = \frac{y+2}{7} = \frac{z-3}{2}\)
11.5.4.10.
Answer 1.
\(\left(5,1,9\right)+t\mathopen{}\left<0,-1,0\right>\)
Answer 2.
\(x = 5, y = 1-t, z = 9\)
Answer 3.
\(\text{DNE}\)
11.5.4.11.
Answer 1.
\(\left(7,2,-1\right)+t\mathopen{}\left<1,-1,2\right>\)
Answer 2.
\(x = 7+t, y = 2-t, z = -1+2t\)
Answer 3.
\(x-7 = 2-y = \frac{z+1}{2}\)
11.5.4.12.
Answer 1.
\(\left(2,2,3\right)+t\mathopen{}\left<5,-1,-3\right>\)
Answer 2.
\(x = 2+5t, y = 2-t, z = 3-3t\)
Answer 3.
\(\frac{x-2}{5} = -\left(y-2\right) = \frac{-\left(z-3\right)}{3}\)
11.5.4.14.
Answer 1.
\(\left(-2,5\right)+t\mathopen{}\left<0,1\right>\)
Answer 2.
\(x = -2, y = 5+t\)
Answer 3.
\(\text{DNE}\)
11.5.4.15.
Answer.
\(\text{parallel}\)
11.5.4.16.
Answer.
\(\left(12,3,7\right)\)
11.5.4.18.
Answer.
\(\text{same}\)
11.5.4.19.
Answer.
\(\text{skew}\)
11.5.4.20.
Answer.
\(\text{parallel}\)
11.5.4.22.
Answer.
\(\text{skew}\)
11.5.4.23.
Answer.
\(\sqrt{41}/3\)
11.5.4.24.
Answer.
\(3\sqrt{2}\)
11.5.4.25.
Answer.
\(5\sqrt{2}/2\)
11.5.4.26.
Answer.
\(5\)
11.5.4.27.
Answer.
\(3/\sqrt{2}\)
11.5.4.28.
Answer.
\(2\)

11.6 Planes
11.6.2 Exercises

Terms and Concepts

11.6.2.1.
Answer.
A point in the plane and a normal vector (i.e., a direction orthogonal to the plane).
11.6.2.2.
Answer.
A normal vector is orthogonal to the plane.

Problems

11.6.2.3.
Answer.
Answers will vary.
11.6.2.4.
Answer.
\(\left(-2,9,0\right), \left(2,9,3\right)\)
11.6.2.5.
Answer.
Answers will vary.
11.6.2.6.
Answer.
\(\left(0,-2,6\right), \left(1,-2,6\right)\)
11.6.2.8.
Answer 1.
\(2\mathopen{}\left(y-3\right)+4\mathopen{}\left(z-5\right) = 0\)
Answer 2.
\(2y+4z = 26\)
11.6.2.10.
Answer 1.
\(-5\mathopen{}\left(x-5\right)+3\mathopen{}\left(y-3\right)+2\mathopen{}\left(z-8\right) = 0\)
Answer 2.
\(-5x+3y+2z = 0\)
11.6.2.12.
Answer 1.
\(3\mathopen{}\left(x-5\right)+3\mathopen{}\left(z-3\right) = 0\)
Answer 2.
\(3x+3z = 24\)
11.6.2.14.
Answer 1.
\(2\mathopen{}\left(x-1\right)+y-1-3\mathopen{}\left(z-1\right) = 0\)
Answer 2.
\(2x+y-3z = 0\)
11.6.2.16.
Answer 1.
\(4\mathopen{}\left(x-5\right)-2\mathopen{}\left(y-7\right)-2\mathopen{}\left(z-3\right) = 0\)
Answer 2.
\(4x-2y-2z = 0\)
11.6.2.17.
Answer 1.
\(x-5+y-7+z-3 = 0\)
Answer 2.
\(x+y+z = 15\)
11.6.2.18.
Answer 1.
\(4\mathopen{}\left(x-4\right)+y-1+z-1 = 0\)
Answer 2.
\(4x+y+z = 18\)
11.6.2.19.
Answer 1.
\(3\mathopen{}\left(x+4\right)+8\mathopen{}\left(y-7\right)-10\mathopen{}\left(z-2\right) = 0\)
Answer 2.
\(3x+8y-10z = 24\)
11.6.2.20.
Answer 1.
\(x-1 = 0\)
Answer 2.
\(x = 1\)
11.6.2.22.
Answer.
\(\left(1,3,3.5\right)+t\mathopen{}\left<20,2,-26\right>\)
11.6.2.24.
Answer.
\(\left(3,1,1\right)\)
11.6.2.26.
Answer.
\(\text{the entire line}\)
11.6.2.27.
Answer.
\(\sqrt{5/7}\)
11.6.2.28.
Answer.
\(\frac{8}{\sqrt{21}}\)
11.6.2.29.
Answer.
\(1/\sqrt{3}\)
11.6.2.30.
Answer.
\(3\)

12 Vector Valued Functions
12.1 Vector-Valued Functions
12.1.4 Exercises

Problems

12.1.4.14.
Answer.
Graph of the vector valued function from the example.
Graph of the function \(\vec r(t) = \la 3\cos(t) , \sin(t) , t/\pi\ra\) on \([0,2\pi]\text{.}\) The graph of the function is an oval-shaped spiral centered about the \(z\)-axis. Ignoring the \(z\)-axis, the curve is simply an oval having a horizontal width of \(6\) and a height of \(2\) in the \(xy\)-plane. Incorporating the \(z\) coordinate then creates the linearly increasing oval spiral, which begins at the point \((3,0,0)\text{,}\) completes precisely one full revolution and ends at the point \((3,0,2)\text{.}\)
12.1.4.15.
Answer.
Graph of the vector valued function from the example.
Graph of the function \(\vec r(t) = \la \cos(t) , \sin(t) ,\sin(t) \ra\) on \([0,2\pi]\text{.}\) The graph of the function is an oval lying in the plane coming from rotating the \(xy\) plane \(45\) degrees towards the \(z\)-axis. The oval lying in this plane has a horizontal width of \(\sqrt{2}\) and a height of \(1\text{.}\) Ignoring the \(z\) coordinate, the curve is a unit circle in the \(xy\) plane. Similarly ignoring the \(y\) coordinate, the curve is a unit circle in the \(xz\) plane. If we now ignore the \(x\) coordinate, the resulting curve is a diagonal line given by \(z=y\) in the \(yz\) plane. This line turns back on itself, which can be seen in the image of the oval when considering all three coordinate axes.
12.1.4.16.
Answer.
Graph of the vector valued function from the example.
Graph of the function \(\vec r(t) = \la \cos(t) , \sin(t) ,\sin(2t)\ra\) on \([0,2\pi]\text{.}\) The graph of the function resembles a saddle centered at the origin whose height is defined by the \(z\)-axis. The two sides of the saddle that taper off fall into negative \(z\) and lie in the second and third quadrants in the \(xy\) plane. Ignoring the \(z\) coordinate, the curve is a unit circle in the \(xy\) plane. Ignoring the \(x\) or \(y\) coordinates individually, the curve looks like the \(\infty\) symbol in the \(yz\) and the \(xz\) planes, respectively. We now describe the \(z\) coordinate with respect to travelling along the unit circle in the \(xy\) plane. Starting at \(t=\text{,}\) the function begins at the point \((1,0,0)\text{.}\) As \(t\) increases and we travel along the unit circle in the \(x\) and \(y\) coordinates, \(z\) increases until we get to \(t=\frac{\pi}{2}\) at which \(z=1\text{.}\) Then, continuing along the unit circle, \(z\) decreases until it reaches a minimum of \(z=-1\) when \(t=\frac{3\pi}{4}\text{.}\) Continuing along the circle, \(z\) begins to increase once again, reaching one more maximum of \(z=1\) when \(t=\frac{5\pi}{4}\text{.}\) Finally, \(z\) begins to decrease, reaching its last minimum of \(z=1\) when \(t=\frac{7\pi}{4}\text{,}\) after which \(z\) increases, and the curve ends where it began, at the point \((1,0,0)\text{.}\)
12.1.4.17.
Answer.
\(\left|t\right|\sqrt{1+t^{2}}\)
12.1.4.19.
Answer.
\(\sqrt{4+t^{2}}\)
12.1.4.21.
Answer.
\(\left<2\cos\mathopen{}\left(t\right)+1,2\sin\mathopen{}\left(t\right)+2\right>\)
12.1.4.25.
Answer.
\(\left<t+2,5t+3\right>\)
12.1.4.27.
Answer.
Specific forms may vary, though most direct solutions are
\(\vec r(t) = \la 1,2,3\ra +t\la 3,3,3\ra\) and
\(\vec r(t) = \la 3t+1, 3t+2, 3t+3\ra\text{.}\)
12.1.4.28.
Answer.
Specific forms may vary, though most direct solutions are
\(\vec r(t) = \la 1,2\ra +t\la 3,2\ra\) and
\(\vec r(t) = \la 3t+1, 2t+2\ra\text{.}\)
12.1.4.29.
Answer.
\(\left<2\cos\mathopen{}\left(t\right),2\sin\mathopen{}\left(t\right),2t\right>\)
12.1.4.31.
Answer.
\(\left<1,0\right>\)
12.1.4.33.
Answer.
\(\left<0,0,1\right>\)

12.2 Calculus and Vector-Valued Functions
12.2.5 Exercises

Terms and Concepts

12.2.5.4.
Answer.
A scalar-vector product, a dot product and a cross product.

Problems

12.2.5.5.
Answer.
\(\left<11,74,\sin\mathopen{}\left(5\right)\right>\)
12.2.5.6.
Answer.
\(\la e^3,0\ra\)
12.2.5.7.
Answer.
\(\left<1,e\right>\)
12.2.5.8.
Answer.
\(\la2t,1,0\ra\)
12.2.5.9.
Answer.
\(\left(-\infty ,0\right)\cup \left(0,\infty \right)\)
12.2.5.10.
Answer.
\((0,\infty)\)
12.2.5.11.
Answer.
\(\left<-\sin\mathopen{}\left(t\right),e^{t},\frac{1}{t}\right>\)
12.2.5.12.
Answer.
\(\vrp(t) = \la -1/t^2, 5/(3t+1)^2, \sec^2(t) \ra\)
12.2.5.13.
Answer.
\(\left<2t\sin\mathopen{}\left(t\right)+t^{2}\cos\mathopen{}\left(t\right),6t^{2}+10t\right>\)
12.2.5.14.
Answer.
\(\left(t^{2}+1\right)\cos\mathopen{}\left(t\right)+2t\sin\mathopen{}\left(t\right)+4t+3\)
12.2.5.15.
Answer.
\(\left<-1,\cos\mathopen{}\left(t\right)-2t,6t^{2}+10t+2+\cos\mathopen{}\left(t\right)-\sin\mathopen{}\left(t\right)-t\cos\mathopen{}\left(t\right)\right>\)
12.2.5.16.
Answer.
\(\vrp(t) = \la \sinh t,\cosh t\ra\)
12.2.5.21.
Answer.
\(\left<2+3t,t\right>\)
12.2.5.22.
Answer.
\(\left(\frac{3\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)+t\mathopen{}\left<\frac{-3\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right>\)
12.2.5.23.
Answer.
\(\ell(t) = \la -3,0,\pi\ra + t\la0,-3,1\ra\)
12.2.5.24.
Answer.
\(\left(1,0,0\right)+t\mathopen{}\left<1,1,1\right>\)
12.2.5.26.
Answer.
\(1\)
12.2.5.28.
Answer.
\(1, -1\)
12.2.5.32.
Answer.
Both derivatives return \(\la 6t^5,3t^2,0\ra\)
12.2.5.33.
Answer.
\(\la \frac14t^4,\sin(t) ,te^t-e^t\ra + \vec C\)
12.2.5.34.
Answer.
\(\la \tan^{-1}(t) ,\tan(t) \ra + \vec C\)
12.2.5.35.
Answer.
\(\left<-2,0\right>\)
12.2.5.36.
Answer.
\(\la4,-4 \ra\)
12.2.5.37.
Answer.
\(\left<\frac{t^{2}}{2}+2,-\cos\mathopen{}\left(t\right)+3\right>\)
12.2.5.39.
Answer.
\(\left<\frac{t^{4}}{12}+t+4,\frac{t^{3}}{6}+2t+5,\frac{t^{2}}{2}+3t+6\right>\)
12.2.5.41.
Answer.
\(2\cdot 3.60555\pi \)
12.2.5.42.
Answer.
\(10\pi\)
12.2.5.43.
Answer.
\(\frac{1}{54}\mathopen{}\left(22^{\frac{3}{2}}-8\right)\)
12.2.5.44.
Answer.
\(\sqrt{2}(1-e^{-1})\)

12.3 The Calculus of Motion
12.3.3 Exercises

Problems

12.3.3.7.
Answer.
\(\vvt = \la 2,5,0\ra\text{,}\) \(\vat = \la 0,0,0\ra\)
12.3.3.8.
Answer 1.
\(\left<6t-2,-2t+1\right>\)
Answer 2.
\(\left<6,-2\right>\)
12.3.3.10.
Answer 1.
\(\left<\frac{1}{10},\sin\mathopen{}\left(t\right),\cos\mathopen{}\left(t\right)\right>\)
Answer 2.
\(\left<0,\cos\mathopen{}\left(t\right),-\sin\mathopen{}\left(t\right)\right>\)
12.3.3.16.
Answer 1.
\(\left|t\right|\sqrt{9t^{2}-12t+8}\)
Answer 2.
\(0\)
Answer 3.
\(-1\)
12.3.3.19.
Answer 1.
\(\left|\sec\mathopen{}\left(t\right)\right|\sqrt{\tan^{2}\mathopen{}\left(t\right)+\sec^{2}\mathopen{}\left(t\right)}\)
Answer 2.
\(0\)
Answer 3.
\(\frac{\pi }{4}\)
12.3.3.22.
Answer 1.
\(\sqrt{8t^{2}+3}\)
Answer 2.
\(0\)
Answer 3.
\(1\)
12.3.3.30.
Answer.
\(\left<t^{2}-t+5,\frac{3t^{2}}{2}-t-\frac{5}{2}\right>\)
12.3.3.32.
Answer.
\(\left<10t,-16t^{2}+50t\right>\)
12.3.3.34.
Answer 1.
\(\left<-10,0\right>\)
Answer 2.
\(5\pi \)
Answer 3.
\(\left<\frac{-10}{\pi },0\right>\)
Answer 4.
\(5\)
12.3.3.36.
Answer 1.
\(\left<10,20,-10\right>\)
Answer 2.
\(10\sqrt{6}\)
Answer 3.
\(\left<1,2,-1\right>\)
Answer 4.
\(\sqrt{6}\)
12.3.3.38.
12.3.3.38.a
Answer.
\(t=\sin^{-1}(3/20)/(8\pi) + n/4 \approx 0.006 + n/4\text{,}\) where \(n\) is an integer
12.3.3.38.b
Answer.
\(\norm{\vrp(t)} = 24\pi\approx 51.4\) ft/s
12.3.3.38.c
Answer.
\(0.27\) radians, or \(15.69^\circ\)
12.3.3.39.
12.3.3.39.a
Answer.
\(0.013\ {\rm radians}\)
12.3.3.39.b
Answer.
\(11.7\ {\rm ft}\)

12.4 Unit Tangent and Normal Vectors
12.4.4 Exercises

Terms and Concepts

12.4.4.3.
Answer.
\(\unittangent(t)\) and \(\unitnormal(t)\text{.}\)

Problems

12.4.4.5.
Answer.
\(\unittangent(t) = \la\frac{4 t}{\sqrt{20 t^2-4t+1}},\frac{2 t-1}{\sqrt{20 t^2-4t+1}}\ra\text{;}\) \(\unittangent(1) = \la 4/\sqrt{17},1/\sqrt{17}\ra\)
12.4.4.6.
Answer 1.
\(\left<\frac{1}{\sqrt{1+\sin^{2}\mathopen{}\left(t\right)}},\frac{-\sin\mathopen{}\left(t\right)}{\sqrt{1+\sin^{2}\mathopen{}\left(t\right)}}\right>\)
Answer 2.
\(\left<\sqrt{\frac{2}{3}},\frac{-1}{\sqrt{3}}\right>\)
12.4.4.8.
Answer 1.
\(\left<-\sin\mathopen{}\left(t\right),\cos\mathopen{}\left(t\right)\right>\)
Answer 2.
\(\left<0,-1\right>\)
12.4.4.9.
Answer.
\(\left(2,0\right)+t\mathopen{}\left<\frac{4}{\sqrt{17}},\frac{1}{\sqrt{17}}\right>\)
12.4.4.10.
Answer.
\(\left(\frac{\pi }{4},\frac{\sqrt{2}}{2}\right)+t\mathopen{}\left<\sqrt{\frac{2}{3}},\frac{-1}{\sqrt{3}}\right>\)
12.4.4.12.
Answer.
\(\left(-1,0\right)+t\mathopen{}\left<0,-1\right>\)
12.4.4.13.
Answer.
\(\unittangent(t) = \la -\sin(t) ,\cos(t) \ra\text{;}\) \(\unitnormal(t) = \la -\cos(t) ,-\sin(t) \ra\)
12.4.4.14.
Answer.
\(\unittangent(t) = \la \frac{1}{\sqrt{1+4t^2}},\frac{2t}{\sqrt{1+4t^2}}\ra\text{;}\) \(\unitnormal(t) = \la -\frac{2t}{\sqrt{1+4t^2}},\frac{1}{\sqrt{1+4t^2}}\ra\)
12.4.4.15.
Answer.
\(\unittangent(t) = \la -\frac{\sin(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }},\frac{2\cos(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }}\ra\text{;}\) \(\unitnormal(t) = \la -\frac{2\cos(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }},-\frac{\sin(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }}\ra\)
12.4.4.16.
Answer.
\(\unittangent(t) = \la \frac{e^t}{\sqrt{e^{2t}+e^{-2t}}},-\frac{e^{-t}}{\sqrt{e^{2t}+e^{-2t}}}\ra\text{;}\) \(\unitnormal(t) = \la \frac{e^{-t}}{\sqrt{e^{2t}+e^{-2t}}},\frac{e^{t}}{\sqrt{e^{2t}+e^{-2t}}}\ra\)
12.4.4.22.
Answer.
\(\left<-\cos\mathopen{}\left(t\right),\frac{-3}{5}\sin\mathopen{}\left(t\right),\frac{-4}{5}\sin\mathopen{}\left(t\right)\right>\)
12.4.4.24.
Answer.
\(\left<-\cos\mathopen{}\left(at\right),-\sin\mathopen{}\left(at\right),0\right>\)
12.4.4.26.
Answer 1.
\(\frac{\frac{-2}{t^{5}}}{\sqrt{1+\frac{1}{t^{4}}}}\)
Answer 2.
\(\frac{\frac{2}{t^{3}}}{\sqrt{1+\frac{1}{t^{4}}}}\)
Answer 3.
\(-\sqrt{2}\)
Answer 4.
\(\sqrt{2}\)
Answer 5.
\(\frac{-1}{4\sqrt{17}}\)
Answer 6.
\(\frac{1}{\sqrt{17}}\)
12.4.4.28.
Answer 1.
\(2\)
Answer 2.
\(4t^{2}\)
Answer 3.
\(2\)
Answer 4.
\(2\pi \)
Answer 5.
\(2\)
Answer 6.
\(4\pi \)
12.4.4.30.
Answer 1.
\(0\)
Answer 2.
\(5\)
Answer 3.
\(0\)
Answer 4.
\(5\)
Answer 5.
\(0\)
Answer 6.
\(5\)

12.5 The Arc Length Parameter and Curvature
12.5.4 Exercises

Terms and Concepts

12.5.4.3.
Answer.
Answers may include lines, circles, helixes
12.5.4.4.
Answer.
Answer should mention the circle is tangent to the curve and has the same curvature as the curve at that point.
12.5.4.6.
Answer.
\(a_\text{T}\) is not affected by curvature; the greater the curvature, the larger \(a_\text{N}\) becomes.

Problems

12.5.4.8.
Answer 1.
\(7t\)
Answer 2.
\(\left<7\cos\mathopen{}\left(\frac{s}{7}\right),7\sin\mathopen{}\left(\frac{s}{7}\right)\right>\)
12.5.4.10.
Answer 1.
\(13t\)
Answer 2.
\(\left<5\cos\mathopen{}\left(\frac{s}{13}\right),13\sin\mathopen{}\left(\frac{s}{13}\right),12\cos\mathopen{}\left(\frac{s}{13}\right)\right>\)
12.5.4.12.
Answer 1.
\(\text{greater than}\)
Answer 2.
\(\frac{\left|\frac{6x^{2}-2}{\left(x^{2}+1\right)^{3}}\right|}{\left(1+\frac{4x^{2}}{\left(x^{2}+1\right)^{4}}\right)^{\frac{3}{2}}}\)
Answer 3.
\(2\)
Answer 4.
\(\frac{2750}{641^{\frac{3}{2}}}\)
12.5.4.15.
Answer 1.
\(\text{less than}\)
Answer 2.
\(\frac{\left|2\cos\mathopen{}\left(t\right)\cos\mathopen{}\left(2t\right)+4\sin\mathopen{}\left(t\right)\sin\mathopen{}\left(2t\right)\right|}{\left(4\cos^{2}\mathopen{}\left(2t\right)+\sin^{2}\mathopen{}\left(t\right)\right)^{\frac{3}{2}}}\)
Answer 3.
\(\frac{1}{4}\)
Answer 4.
\(8\)
12.5.4.18.
Answer 1.
\(\text{greater than}\)
Answer 2.
\(\frac{\left|\sec^{3}\mathopen{}\left(t\right)\right|}{\left(\sec^{4}\mathopen{}\left(t\right)+\sec^{2}\mathopen{}\left(t\right)\tan^{2}\mathopen{}\left(t\right)\right)^{\frac{3}{2}}}\)
Answer 3.
\(1\)
Answer 4.
\(\frac{3\sqrt{3}}{5\sqrt{5}}\)
12.5.4.20.
Answer 1.
\(\text{greater than}\)
Answer 2.
\(\frac{2\sqrt{18t^{4}+15t^{2}+1}}{\left(18t^{4}-2t^{2}+1\right)^{\frac{3}{2}}}\)
Answer 3.
\(2\)
Answer 4.
\(\frac{2\sqrt{2}}{17}\)
12.5.4.22.
Answer 1.
\(\text{equal to}\)
Answer 2.
\(\frac{1}{13}\)
Answer 3.
\(\frac{1}{13}\)
Answer 4.
\(\frac{1}{13}\)
12.5.4.23.
Answer.
\(\frac{\sqrt{2}}{\sqrt[4]{5}}, \frac{-\sqrt{2}}{\sqrt[4]{5}}\)
12.5.4.25.
Answer.
\(\frac{1}{4}\)
12.5.4.26.
Answer.
\(\sqrt{5}, -\sqrt{5}\)
12.5.4.28.
Answer.
\(5\sqrt{10}\)
12.5.4.30.
Answer.
\(\frac{1}{45}\)
12.5.4.32.
Answer.
\(\left(x-\frac{8}{3}\right)^{2}+y^{2} = \frac{1}{9}\)
12.5.4.34.
Answer.
\(\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2} = \frac{1}{2}\)

13 Introduction to Functions of Several Variables
13.2 Limits and Continuity of Multivariable Functions
13.2.5 Exercises

Terms and Concepts

13.2.5.2.
Answer.
Answers will vary. One answer is “As \((x,y)\) gets close to \((1,2)\text{,}\) \(f(x,y)\) gets close to 17.”
13.2.5.3.
Answer.
Answers will vary.
One possible answer: \(\{(x,y) | x^2+y^2\leq 1\}\)
13.2.5.4.
Answer.
Answers will vary.
One possible answer: \(\{(x,y) | y\geq x^2 \}\)
13.2.5.5.
Answer.
Answers will vary.
One possible answer: \(\{(x,y) | x^2+y^2\lt 1 \}\)
13.2.5.6.
Answer.
Answers will vary.
One possible answer: \(\{(x,y) | y \gt x^2 \}\)

Problems

13.2.5.7.
Answer.
  1. Answers will vary. interior point: \((1,3)\) boundary point: \((3,3)\)
  2. \(S\) is a closed set
  3. \(S\) is bounded
13.2.5.8.
Answer.
  1. Answers will vary. Interior point: \((1,0)\) (any point with \(y\neq x^2\) will do). Boundary point: \((1,1)\) (any point with \(y=x^2\) will do).
  2. \(S\) is an open set.
  3. \(S\) is unbounded.
13.2.5.11.
Answer.
  1. \(D = \left\{(x,y)\, |\, 9-x^2-y^2\geq 0\right\}\text{.}\)
  2. \(D\) is a closed set.
  3. \(D\) is bounded.
13.2.5.12.
Answer.
  1. \(D = \left\{(x,y)\, |\, y\geq x^2\right\}\text{.}\)
  2. \(D\) is a closed set.
  3. \(D\) is unbounded.
13.2.5.13.
Answer.
  1. \(D = \left\{(x,y)\, |\, y \gt x^2\right\}\text{.}\)
  2. \(D\) is an open set.
  3. \(D\) is unbounded.
13.2.5.14.
Answer.
  1. \(D = \left\{(x,y)\, |\, (x,y)\neq (0,0) \right\}\text{.}\)
  2. \(D\) is an open set.
  3. \(D\) is unbounded.

13.3 Partial Derivatives
13.3.7 Exercises

Problems

13.3.7.6.
Answer 1.
\(0\)
Answer 2.
\(0\)
13.3.7.8.
Answer 1.
\(-{\frac{1}{2}}\)
Answer 2.
\(-{\frac{1}{3}}\)
13.3.7.10.
Answer 1.
\(3x^{2}+6xy+3y^{2}\)
Answer 2.
\(3x^{2}+6xy+3y^{2}\)
Answer 3.
\(6x+6y\)
Answer 4.
\(6x+6y\)
Answer 5.
\(6x+6y\)
Answer 6.
\(6x+6y\)
13.3.7.12.
Answer 1.
\(\frac{-4}{x^{2}y}\)
Answer 2.
\(\frac{-4}{xy^{2}}\)
Answer 3.
\(\frac{8}{x^{3}y}\)
Answer 4.
\(\frac{4}{x^{2}y^{2}}\)
Answer 5.
\(\frac{4}{x^{2}y^{2}}\)
Answer 6.
\(\frac{8}{xy^{3}}\)
13.3.7.14.
Answer 1.
\(e^{x+2y}\)
Answer 2.
\(2e^{x+2y}\)
Answer 3.
\(e^{x+2y}\)
Answer 4.
\(2e^{x+2y}\)
Answer 5.
\(2e^{x+2y}\)
Answer 6.
\(4e^{x+2y}\)
13.3.7.16.
Answer 1.
\(3\mathopen{}\left(x+y\right)^{2}\)
Answer 2.
\(3\mathopen{}\left(x+y\right)^{2}\)
Answer 3.
\(6\mathopen{}\left(x+y\right)\)
Answer 4.
\(6\mathopen{}\left(x+y\right)\)
Answer 5.
\(6\mathopen{}\left(x+y\right)\)
Answer 6.
\(6\mathopen{}\left(x+y\right)\)
13.3.7.18.
Answer 1.
\(10x\cos\mathopen{}\left(5x^{2}+2y^{3}\right)\)
Answer 2.
\(6y^{2}\cos\mathopen{}\left(5x^{2}+2y^{3}\right)\)
Answer 3.
\(10\cos\mathopen{}\left(5x^{2}+2y^{3}\right)-100x^{2}\sin\mathopen{}\left(5x^{2}+2y^{3}\right)\)
Answer 4.
\(-60xy^{2}\sin\mathopen{}\left(5x^{2}+2y^{3}\right)\)
Answer 5.
\(-60xy^{2}\sin\mathopen{}\left(5x^{2}+2y^{3}\right)\)
Answer 6.
\(12y\cos\mathopen{}\left(5x^{2}+2y^{3}\right)-36y^{4}\sin\mathopen{}\left(5x^{2}+2y^{3}\right)\)
13.3.7.19.
Answer 1.
\(\frac{2y^{2}}{\sqrt{4xy^{2}+1}}\)
Answer 2.
\(\frac{4xy}{\sqrt{4xy^{2}+1}}\)
Answer 3.
\(\frac{-4y^{4}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}\)
Answer 4.
\(\frac{-8xy^{3}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4y}{\sqrt{4xy^{2}+1}}\)
Answer 5.
\(\frac{-8xy^{3}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4y}{\sqrt{4xy^{2}+1}}\)
Answer 6.
\(\frac{-16x^{2}y^{2}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4x}{\sqrt{4xy^{2}+1}}\)
13.3.7.22.
Answer 1.
\(5\)
Answer 2.
\(-17\)
Answer 3.
\(0\)
Answer 4.
\(0\)
Answer 5.
\(0\)
Answer 6.
\(0\)
13.3.7.24.
Answer 1.
\(\frac{2x}{x^{2}+y}\)
Answer 2.
\(\frac{1}{x^{2}+y}\)
Answer 3.
\(\frac{-4x^{2}}{\left(x^{2}+y\right)^{2}}+\frac{2}{x^{2}+y}\)
Answer 4.
\(\frac{-2x}{\left(x^{2}+y\right)^{2}}\)
Answer 5.
\(\frac{-2x}{\left(x^{2}+y\right)^{2}}\)
Answer 6.
\(\frac{-1}{\left(x^{2}+y\right)^{2}}\)
13.3.7.26.
Answer 1.
\(5e^{x}\sin\mathopen{}\left(y\right)\)
Answer 2.
\(5e^{x}\cos\mathopen{}\left(y\right)\)
Answer 3.
\(5e^{x}\sin\mathopen{}\left(y\right)\)
Answer 4.
\(5e^{x}\cos\mathopen{}\left(y\right)\)
Answer 5.
\(5e^{x}\cos\mathopen{}\left(y\right)\)
Answer 6.
\(-5e^{x}\sin\mathopen{}\left(y\right)\)
13.3.7.28.
Answer.
\(\frac{1}{2}x^{2}+xy+\frac{1}{2}y^{2}\)
13.3.7.30.
Answer.
\(\ln\mathopen{}\left(x^{2}+y^{2}\right)\)
13.3.7.32.
Answer 1.
\(3x^{2}y^{2}+3x^{2}z\)
Answer 2.
\(2x^{3}y+2yz\)
Answer 3.
\(x^{3}+y^{2}\)
Answer 4.
\(2y\)
Answer 5.
\(2y\)
13.3.7.34.
Answer 1.
\(\frac{1}{x}\)
Answer 2.
\(\frac{1}{y}\)
Answer 3.
\(\frac{1}{z}\)
Answer 4.
\(0\)
Answer 5.
\(0\)

IV Math 2580: Calculus IV
14 Functions of Several Variables, Continued
14.2 The Multivariable Chain Rule
14.2.3 Exercises

Problems

14.2.3.7.
Answer.
  1. \(\frac{dz}{dt} = 3(2t)+4(2) = 6t+8\text{.}\)
  2. At \(t=1\text{,}\) \(\frac{dz}{dt} = 14\text{.}\)
14.2.3.8.
Answer 1.
\(2x-4yt\)
Answer 2.
\(2\)
14.2.3.9.
Answer.
  1. \(\displaystyle \frac{dz}{dt} = 5(-2\sin(t) )+2(\cos(t) ) = -10\sin(t) +2\cos(t)\)
  2. At \(t=\pi/4\text{,}\) \(\frac{dz}{dt} = -4\sqrt{2}\text{.}\)
14.2.3.10.
Answer 1.
\(\frac{-\sin\mathopen{}\left(t\right)}{1+y^{2}}-\frac{2xy\cos\mathopen{}\left(t\right)}{\left(y^{2}+1\right)^{2}}\)
Answer 2.
\(-{\frac{1}{2}}\)
14.2.3.11.
Answer.
  1. \(\ds\frac{dz}{dt} = 2x(\cos(t) ) + 4y(3\cos(t) )\text{.}\)
  2. At \(t=\pi/4\text{,}\) \(x=\sqrt{2}/2\text{,}\) \(y=3\sqrt{2}/2\text{,}\) and \(\frac{dz}{dt} = 19\text{.}\)
14.2.3.12.
Answer 1.
\(-\sin\mathopen{}\left(x\right)\sin\mathopen{}\left(y\right)\pi +\cos\mathopen{}\left(x\right)\cos\mathopen{}\left(y\right)\cdot 2\pi \)
Answer 2.
\(0\)
14.2.3.14.
Answer.
\(-\sqrt{\frac{3}{2}}, 0, \sqrt{\frac{3}{2}}\)
14.2.3.16.
Answer.
\(0, \pi \)
14.2.3.18.
Answer.
\(0, \frac{1}{\pi }\tan^{-1}\mathopen{}\left(\sqrt{5}\right), 1-\frac{1}{\pi }\tan^{-1}\mathopen{}\left(\sqrt{5}\right), 1, 1+\frac{1}{\pi }\tan^{-1}\mathopen{}\left(\sqrt{5}\right), 2-\frac{1}{\pi }\tan^{-1}\mathopen{}\left(\sqrt{5}\right)\)
14.2.3.20.
Answer 1.
\(-\pi \sin\mathopen{}\left(\pi x+\frac{\pi y}{2}\right)t^{2}-\frac{1}{2}\pi \sin\mathopen{}\left(\pi x+\frac{\pi y}{2}\right)\cdot 2st\)
Answer 2.
\(-\pi \sin\mathopen{}\left(\pi x+\frac{\pi y}{2}\right)\cdot 2st-\frac{1}{2}\pi \sin\mathopen{}\left(\pi x+\frac{\pi y}{2}\right)s^{2}\)
Answer 3.
\(2\pi \)
Answer 4.
\(\frac{5\pi }{2}\)
14.2.3.21.
Answer 1.
\(2x\cos\mathopen{}\left(t\right)+2y\sin\mathopen{}\left(t\right)\)
Answer 2.
\(-2xs\sin\mathopen{}\left(t\right)+2ys\cos\mathopen{}\left(t\right)\)
Answer 3.
\(4\)
Answer 4.
\(0\)
14.2.3.22.
Answer 1.
\(-2yt^{2}e^{-\left(x^{2}+y^{2}\right)}\)
Answer 2.
\(-2xe^{-\left(x^{2}+y^{2}\right)}-4stye^{-\left(x^{2}+y^{2}\right)}\)
Answer 3.
\(\frac{-2}{e^{2}}\)
Answer 4.
\(\frac{-6}{e^{2}}\)
14.2.3.24.
Answer.
\(\frac{-x}{y^{2}}\)
14.2.3.26.
Answer.
\(\frac{-\left(2x+y\right)}{2y+x}\)
14.2.3.28.
Answer.
\(0\)
14.2.3.30.
Answer 1.
\(-2\)
Answer 2.
\(5\)

14.3 Directional Derivatives
14.3.3 Exercises

Problems

14.3.3.8.
Answer.
\(\left<\cos\mathopen{}\left(x\right)\cos\mathopen{}\left(y\right),-\sin\mathopen{}\left(x\right)\sin\mathopen{}\left(y\right)\right>\)
14.3.3.10.
Answer.
\(\left<-4,3\right>\)
14.3.3.12.
Answer.
\(\left<2xy^{3}-2,3x^{2}y^{2}\right>\)
14.3.3.13.
14.3.3.13.a
Answer.
\(2/5\)
14.3.3.13.b
Answer.
\(-2/\sqrt{5}\)
14.3.3.14.
14.3.3.14.a
Answer.
\(\frac{1}{4}\mathopen{}\left(1-\sqrt{3}\right)\)
14.3.3.14.b
Answer.
\(\frac{4\sqrt{3}-3}{10\sqrt{2}}\)
14.3.3.15.
14.3.3.15.a
Answer.
\(0\)
14.3.3.15.b
Answer.
\(2\sqrt{2}/9\)
14.3.3.16.
14.3.3.16.a
Answer.
\(\frac{-9}{\sqrt{10}}\)
14.3.3.16.b
Answer.
\(\frac{27}{\sqrt{34}}\)
14.3.3.17.
14.3.3.17.a
Answer.
\(0\)
14.3.3.17.b
Answer.
\(0\)
14.3.3.18.
14.3.3.18.a
Answer.
\(\frac{3}{\sqrt{2}}\)
14.3.3.18.b
Answer.
\(3\)
14.3.3.19.
14.3.3.19.a
Answer.
\(\nabla f(2,1) = \la -2,2\ra\)
14.3.3.19.b
Answer.
\(\sqrt{8}\)
14.3.3.19.c
Answer.
\(\la 2, -2\ra\)
14.3.3.19.d
Answer.
\(\vec u = \la 1/\sqrt{2},1/\sqrt{2}\ra\)
14.3.3.20.
14.3.3.20.a
Answer.
\(\left<\frac{1}{2\sqrt{2}},\frac{-1}{2}\sqrt{\frac{3}{2}}\right>\)
14.3.3.20.b
Answer.
\(\frac{1}{\sqrt{2}}\)
14.3.3.20.c
Answer.
\(\left<\frac{-1}{2\sqrt{2}},\frac{1}{2}\sqrt{\frac{3}{2}}\right>\)
14.3.3.20.d
Answer.
\(\left<\frac{1}{2}\sqrt{\frac{3}{2}},\frac{1}{2\sqrt{2}}\right>\)
14.3.3.21.
14.3.3.21.a
Answer.
\(\nabla f(1,1) = \la -2/9,-2/9\ra\)
14.3.3.21.b
Answer.
\(2\sqrt{2}/9\)
14.3.3.21.c
Answer.
\(\la 2/9,2/9\ra\)
14.3.3.21.d
Answer.
\(\vec u = \la 1/\sqrt{2},-1/\sqrt{2}\ra\)
14.3.3.22.
14.3.3.22.a
Answer.
\(\left<-4,3\right>\)
14.3.3.22.b
Answer.
\(5\)
14.3.3.22.c
Answer.
\(\left<4,-3\right>\)
14.3.3.22.d
Answer.
\(\left<3,4\right>\)
14.3.3.23.
14.3.3.23.a
Answer.
No such direction
14.3.3.23.b
Answer.
\(0\)
14.3.3.23.c
Answer.
No such direction
14.3.3.23.d
Answer.
All directions
14.3.3.24.
14.3.3.24.a
Answer.
\(\left<0,3\right>\)
14.3.3.24.b
Answer.
\(3\)
14.3.3.24.c
Answer.
\(\left<0,-3\right>\)
14.3.3.24.d
Answer.
\(\left<1,0\right>\)
14.3.3.25.
14.3.3.25.a
Answer.
\(\nabla F(x,y,z) = \la 6xz^3+4y, 4x, 9x^2z^2-6z\ra\)
14.3.3.25.b
Answer.
\(113/\sqrt{3}\)
14.3.3.26.
14.3.3.26.a
Answer.
\(\left<\cos\mathopen{}\left(x\right)\cos\mathopen{}\left(y\right)e^{z},-\sin\mathopen{}\left(x\right)\sin\mathopen{}\left(y\right)e^{z},\sin\mathopen{}\left(x\right)\cos\mathopen{}\left(y\right)e^{z}\right>\)
14.3.3.26.b
Answer.
\(\frac{2}{3}\)
14.3.3.27.
14.3.3.27.a
Answer.
\(\nabla F(x,y,z) = \la 2xy^2, 2y(x^2-z^2), -2y^2z\ra\)
14.3.3.27.b
Answer.
\(0\)
14.3.3.28.
14.3.3.28.a
Answer.
\(\left<\frac{-4x}{\left(x^{2}+y^{2}+z^{2}\right)^{2}},\frac{-4y}{\left(x^{2}+y^{2}+z^{2}\right)^{2}},\frac{-4z}{\left(x^{2}+y^{2}+z^{2}\right)^{2}}\right>\)
14.3.3.28.b
Answer.
\(0\)

14.4 Tangent Lines, Normal Lines, and Tangent Planes
14.4.5 Exercises

Problems

14.4.5.6.
Answer 1.
\(x = \frac{\pi }{3}+t, y = \frac{\pi }{6}, z = \frac{3}{4}-\frac{3\sqrt{3}}{4}t\)
Answer 2.
\(x = \frac{\pi }{3}, y = \frac{\pi }{6}+t, z = \frac{3}{4}+\frac{3\sqrt{3}}{4}t\)
Answer 3.
\(x = \frac{\pi }{3}+\frac{t}{\sqrt{5}}, y = \frac{\pi }{6}+\frac{2t}{\sqrt{5}}, z = \frac{3}{4}+\frac{3\sqrt{\frac{3}{5}}}{4}t\)
14.4.5.8.
Answer 1.
\(x = 1+t, y = 2, z = 3\)
Answer 2.
\(x = 1, y = 2+t, z = 3\)
Answer 3.
\(x = 1+\frac{t}{\sqrt{2}}, y = 2+\frac{t}{\sqrt{2}}, z = 3\)
14.4.5.10.
Answer.
\(x = \frac{\pi }{3}-\frac{3\sqrt{3}}{4}t, y = \frac{\pi }{6}+\frac{3\sqrt{3}}{4}t, z = \frac{3}{4}-t\)
14.4.5.12.
Answer.
\(x = 1, y = 2, z = 3-t\)
14.4.5.18.
Answer.
\(1.29904y-1.29904x-z = -1.43017\)
14.4.5.20.
Answer.
\(z = 3\)
14.4.5.22.
Answer 1.
\(x = 4-2t, y = -3+\frac{2}{3}t, z = \sqrt{5}+2\sqrt{5}t\)
Answer 2.
\(0.666667y-2x+4.47214z = 0\)
14.4.5.24.
Answer 1.
\(x = 2+\frac{\pi }{8\sqrt{3}}t, y = \frac{\pi }{12}-\sqrt{3}t, z = 4-\frac{\pi }{8\sqrt{3}}t\)
Answer 2.
\(0.226725x-1.73205y-0.226725z = -0.9069\)

14.5 Extreme Values
14.5.3 Exercises

Problems

14.5.3.6.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\left(7,-6\right)\)
Answer 4.
\(\text{NONE}\)
14.5.3.8.
Answer 1.
\(\left(0,0\right)\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\text{NONE}\)
Answer 4.
\(\text{NONE}\)
14.5.3.10.
Answer 1.
\(\left(-1,-2\right)\)
Answer 2.
\(\left(1,2\right)\)
Answer 3.
\(\left(1,-2\right), \left(-1,2\right)\)
Answer 4.
\(\text{NONE}\)
14.5.3.12.
Answer 1.
\(\left(0,-3\right)\)
Answer 2.
\(\left(-1,3\right), \left(1,3\right)\)
Answer 3.
\(\left(-1,-3\right), \left(0,3\right), \left(1,-3\right)\)
Answer 4.
\(\text{NONE}\)
14.5.3.14.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\text{NONE}\)
Answer 4.
\(\left(0,0\right)\)
14.5.3.15.
Answer 1.
\(3\)
Answer 2.
\(\left(0,1\right)\)
Answer 3.
\(\frac{3}{4}\)
Answer 4.
\(\left(0,\frac{-1}{2}\right)\)
14.5.3.16.
Answer 1.
\(\frac{25}{28}\)
Answer 2.
\(\left(\frac{5}{14},\frac{25}{196}\right)\)
Answer 3.
\(-12\)
Answer 4.
\(\left(-1,1\right)\)

15 Multiple Integration
15.1 Iterated Integrals and Area
15.1.4 Exercises

Problems

15.1.4.5.
15.1.4.5.a
Answer.
\(18x^2+42x-117\)
15.1.4.5.b
Answer.
\(-108\)
15.1.4.6.
15.1.4.6.a
Answer.
\(2+\pi ^{2}\cos\mathopen{}\left(y\right)\)
15.1.4.6.b
Answer.
\(\pi ^{2}+\pi \)
15.1.4.7.
15.1.4.7.a
Answer.
\(x^4/2-x^2+2x-3/2\)
15.1.4.7.b
Answer.
\(23/15\)
15.1.4.8.
15.1.4.8.a
Answer.
\(\frac{y^{4}}{2}-y^{3}+\frac{y^{2}}{2}\)
15.1.4.8.b
Answer.
\(\frac{8}{15}\)
15.1.4.9.
15.1.4.9.a
Answer.
\(\sin^2(y)\)
15.1.4.9.b
Answer.
\(\pi/2\)
15.1.4.10.
15.1.4.10.a
Answer.
\(\frac{x}{1+x^{2}}\)
15.1.4.10.b
Answer.
\(\frac{1}{2}\ln\mathopen{}\left(\frac{5}{2}\right)\)

15.3 Double Integration with Polar Coordinates

Exercises

Problems
15.3.3.
Answer.
\(4\pi \)
15.3.5.
Answer.
\(16\pi \)
15.3.8.
Answer.
\(\frac{\pi }{2}\)
15.3.12.
Answer.
\(\frac{128}{3}\)

15.5 Surface Area

Exercises

Problems
15.5.7.
Answer.
\(\ds SA = \int_0^{2\pi}\int_0^{2\pi} \sqrt{1+ \cos^2(x) \cos^2(y) +\sin^2(x) \sin^2(y) }\, dx\, dy\)
15.5.8.
Answer.
\(\ds SA = \int_{-3}^{3}\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \sqrt{1+ \frac{4x^2+4y^2}{(1+x^2+y^2)^4}}\, dx\, dy\)
Polar offers simpler bounds:
\(\ds SA = \int_0^{2\pi}\int_0^3 r\sqrt{1+\frac{4r^2}{(1+r^2)^4}}\, dr\, d\theta\)
15.5.9.
Answer.
\(\ds SA = \int_{-1}^{1}\int_{-1}^{1} \sqrt{1+ 4x^2+4y^2}\, dx\, dy\)
15.5.10.
Answer.
\(\ds SA = \int_{-5}^{5}\int_{0}^{1} \sqrt{1+ \frac{4x^2e^{2x^2}}{\big(1+e^{x^2}\big)^4}}\, dy\, dx\)

15.6 Volume Between Surfaces and Triple Integration
15.6.4 Exercises

Problems

15.6.4.6.
Answer.
\(52\)
15.6.4.8.
Answer.
\(\frac{3\pi }{2}\)
15.6.4.9.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^3\int_0^{1-x/3}\int_0^{2-2x/3-2y}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_0^1\int_0^{3-3y}\int_0^{2-2x/3-2y}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^3\int_0^{2-2x/3}\int_0^{1-x/3-z/2}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^2\int_0^{3-3z/2}\int_0^{1-x/3-z/2}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_0^1\int_0^{2-2y}\int_0^{3-3y-3z/2}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^2\int_0^{1-z/2}\int_0^{3-3y-3z/2}\, dx\, dy\, dz\)
\(\ds V = \int_0^3\int_0^{1-x/3}\int_0^{2-2x/3-2y}\, dz\, dy\, dx =1\text{.}\)
15.6.4.10.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_1^3\int_0^{2}\int_0^{(3-x)/2}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_0^2\int_1^{3}\int_0^{(3-x)/2}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_1^3\int_0^{(3-x)/2}\int_0^{2}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^1\int_1^{3-2z}\int_0^{2}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_0^2\int_0^{1}\int_1^{3-2z}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^1\int_0^{2}\int_1^{3-2z}\, dx\, dy\, dz\)
\(\ds V = \int_0^1\int_0^{2}\int_1^{3-2z}\, dx\, dy\, dz =2\text{.}\)
15.6.4.11.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^2\int_{-2}^{0}\int_{y^2/2}^{-y}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{-2}^0\int_0^{2}\int_{y^2/2}^{-y}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{-2}^0\int_{y^2/2}^{-y}\int_0^{2}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^2\int_{-\sqrt{2z}}^{-z}\int_0^{2}\, dx\, dy\, dz\) \(\ds V = \int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dz\, dx =4/3\text{.}\)
15.6.4.12.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^3\int_{3x}^{9}\int_{0}^{\sqrt{y^2-9x^2}}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^9\int_0^{y/3}\int_{0}^{\sqrt{y^2-9x^2}}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^3\int_0^{\sqrt{81-9x^2}}\int_{\sqrt{z^2+9x^2}}^{9}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^9\int_0^{\sqrt{9-z^2/9}}\int_{\sqrt{z^2+9x^2}}^{9}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^9\int_{0}^{y}\int_0^{\frac13\sqrt{y^2-z^2}}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^9\int_{z}^{9}\int_0^{\frac13\sqrt{y^2-z^2}}\, dx\, dy\, dz\)
15.6.4.13.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^2\int_{1-x/2}^{1}\int_{0}^{2x+4y-4}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^1\int_{2-2y}^{2}\int_{0}^{2x+4y-4}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^2\int_0^{2x}\int_{z/4-x/2+1}^{1}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^4\int_{z/2}^{2}\int_{z/4-x/2+1}^{1}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{4y}\int_{z/2-2y+2}^2\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^4\int_{z/4}^{1}\int_{z/2-2y+2}^2\, dx\, dy\, dz\) \(\ds V = \int_0^4\int_{z/4}^{1}\int_{z/2-2y-2}^2\, dx\, dy\, dz = 4/3\text{.}\)
15.6.4.14.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_{-2}^2\int_{0}^{4-x^2}\int_{0}^{2y}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^4\int_{-\sqrt{4-y}}^{\sqrt{4-y}}\int_{0}^{2x+4y-4}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_{-2}^2\int_0^{8-2x^2}\int_{z/2}^{4-x^2}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^8\int_{-\sqrt{4-z/2}}^{\sqrt{4-z/2}}\int_{z/2}^{4-x^2}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^4\int_{0}^{2y}\int_{-\sqrt{4-y}}^{\sqrt{4-y}}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^8\int_{z/2}^{4}\int_{-\sqrt{4-y}}^{\sqrt{4-y}}\, dx\, dy\, dz\) \(\ds V = \int_{-2}^2\int_{0}^{4-x^2}\int_{0}^{2y}\, dz\, dy\, dx = 512/15\text{.}\)
15.6.4.15.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_{0}^1\int_{0}^{1-x^2}\int_{0}^{\sqrt{1-y}}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{\sqrt{1-y}}\int_{0}^{\sqrt{1-y}}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_{0}^1\int_0^{x}\int_{0}^{1-x^2}\, dy\, dz\, dx + \int_{0}^1\int_x^{1}\int_{0}^{1-z^2}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^1\int_{0}^{z}\int_{0}^{1-z^2}\, dy\, dx\, dz+\int_0^1\int_{z}^{1}\int_{0}^{1-x^2}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{\sqrt{1-y}}\int_{0}^{\sqrt{1-y}}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^1\int_{0}^{1-z^2}\int_{0}^{\sqrt{1-y}}\, dx\, dy\, dz\) Answers will vary. Neither order is particularly “hard.” The order \(dz\, dy\, dx\) requires integrating a square root, so powers can be messy; the order \(dy\, dz\, dx\) requires two triple integrals, but each uses only polynomials.
15.6.4.16.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_{0}^1\int_{0}^{3x}\int_{0}^{1-x}\, dz\, dy\, dx+\int_{0}^1\int_{3x}^{3}\int_{0}^{1-y/3}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^3\int_{0}^{y/3}\int_{0}^{1-y/3}\, dz\, dy\, dx+\int_{0}^3\int_{y/3}^{1}\int_{0}^{1-x}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_{0}^1\int_0^{1-x}\int_{0}^{3-3z}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^1\int_{0}^{1-z}\int_{0}^{3-3z}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^3\int_{0}^{1-y/3}\int_{0}^{1-z}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^1\int_{0}^{3-3z}\int_{0}^{1-z}\, dx\, dy\, dz\) \(\ds V = \int_0^1\int_{0}^{3-3z}\int_{0}^{1-z}\, dx\, dy\, dz = 1\text{.}\)
15.6.4.18.
Answer.
\(\frac{7}{8}\)
15.6.4.20.
Answer.
\(0\)

15.7 Triple Integration with Cylindrical and Spherical Coordinates
15.7.3 Exercises

Problems

15.7.3.11.
Answer.
\(\ds\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}\int_{z_1}^{z_2}h(r,\theta,z)r\, dz\, dr\, d\theta\)
15.7.3.12.
Answer.
\(\ds\int_{\varphi_1}^{\varphi_2}\int_{\theta_1}^{\theta_2}\int_{\rho_1}^{\rho_2}h(\rho,\theta,\varphi)\rho^2\sin(\varphi)\, d\rho\, d\theta\, d\varphi\)
15.7.3.19.
Answer.
Describes the portion of the unit ball that resides in the first octant.
15.7.3.20.
Answer.
Describes half of a spherical shell (i.e., \(y\geq 0\)) with inner radius of \(1\) and outer radius of \(1.1\) centered at the origin.

16 Vector Analysis
16.1 Introduction to Line Integrals
16.1.4 Exercises

Terms and Concepts

16.1.4.1.
Answer.
When \(C\) is a curve in the plane and \(f\) is a function defined over \(C\text{,}\) then \(\int_C f(s)\, ds\) describes the area under the spatial curve that lies on \(f\text{,}\) over \(C\text{.}\)
16.1.4.2.
Answer.
The evaluation is the same. The \(\oint\) notation signifies that the curve \(C\) is a closed curve, though the evaluation is the same.
16.1.4.3.
Answer.
The variable \(s\) denotes the arc-length parameter, which is generally difficult to use. Theorem 16.1.4 allows one to parametrize a curve using another, ideally easier-to-use, parameter.

Problems

16.1.4.5.
Answer.
\(12\sqrt{2}\)
16.1.4.6.
Answer.
\(41\sqrt{10}/2\)
16.1.4.7.
Answer.
\(40\pi\)
16.1.4.8.
Answer.
\(10\pi^2\)
16.1.4.9.
Answer.
Over the first subcurve of \(C\text{,}\) the line integral has a value of \(3/2\text{;}\) over the second subcurve, the line integral has a value of \(4/3\text{.}\) The total value of the line integral is thus \(17/6\text{.}\)
16.1.4.10.
Answer.
Over the first subcurve of \(C\text{,}\) the line integral has a value of \(2\sqrt{2}/3\text{;}\) over the second subcurve, the line integral has a value of \(\pi-2\text{.}\) The total value of the line integral is thus \(\pi+2\sqrt{2}/3-2\text{.}\)
16.1.4.11.
Answer.
\(\int_0^1(5t^2+_2t+2)\sqrt{(4t+1)^2+1}\, dt \approx 17.071\)
16.1.4.12.
Answer.
\(\int_0^\pi t\sqrt{1+\cos^2t}\, dt \approx 6.001\)
16.1.4.13.
Answer.
\(\oint_0^{2\pi} \big(10-4\cos^2t-\sin^2t\big)\sqrt{\cos^2t+4\sin^2t}\, dt \approx 74.986\)
16.1.4.14.
Answer.
\(\int_{-1}^{1} \big(3t^3+2t+5\big)\sqrt{9t^4+1}\, dt \approx 15.479\)
16.1.4.15.
Answer.
\(7\sqrt{26}/3\)
16.1.4.16.
Answer.
\(2\pi\)
16.1.4.17.
Answer.
\(8\pi^3\)
16.1.4.18.
Answer.
\(5/2\)
16.1.4.19.
Answer.
\(M=8\sqrt{2}\pi^2\text{;}\) center of mass is \((0,-1/(2\pi), 8\pi/3)\text{.}\)
16.1.4.20.
Answer.
\(M\approx 0.237\text{;}\) center of mass is approximately \((0.173, 0.099,0.065)\text{.}\)

16.2 Vector Fields
16.2.3 Exercises

Terms and Concepts

16.2.3.1.
Answer.
Answers will vary. Appropriate answers include velocities of moving particles (air, water, etc.); gravitational or electromagnetic forces.
16.2.3.2.
Answer.
Specific answers will vary, though should relate to the idea that “more of the vector field is moving into that point than out of that point.”
16.2.3.3.
Answer.
Specific answers will vary, though should relate to the idea that the vector field is spinning clockwise at that point.
16.2.3.4.
Answer.
No; to be incompressible, the divergence needs to be 0 everywhere, not just at one point.

Problems

16.2.3.9.
Answer.
\(\divv \vec F = 1+2y\)
\(\curl \vec F = 0\)
16.2.3.10.
Answer.
\(\divv \vec F = 0\)
\(\curl \vec F = 1+2y\)
16.2.3.11.
Answer.
\(\divv \vec F = x\cos(xy)-y\sin(xy)\)
\(\curl \vec F = y\cos(xy)+x\sin(xy)\)
16.2.3.12.
Answer.
\(\divv \vec F = \frac{4}{(x^2+y^2)^2}\)
\(\curl \vec F = 0\)
16.2.3.13.
Answer.
\(\divv \vec F = 3\)
\(\curl \vec F = \la -1,-1,-1\ra\)
16.2.3.14.
Answer.
\(\divv \vec F = 2x+2y+2z\)
\(\curl \vec F = \la 2y,2z,2x\ra\)
16.2.3.15.
Answer.
\(\divv \vec F = 1+2y\)
\(\curl\vec F = 0\)
16.2.3.16.
Answer.
\(\divv \vec F = 2y\)
\(\curl\vec F = 0\)
16.2.3.17.
Answer.
\(\divv \vec F = 2y-\sin(z)\)
\(\curl\vec F = \vec 0\)
16.2.3.18.
Answer.
\(\divv \vec F = \frac{2}{(x^2+y^2+z^2)^2}\)
\(\curl\vec F = \vec 0\)

16.3 Line Integrals over Vector Fields
16.3.4 Exercises

Terms and Concepts

16.3.4.2.
Answer.
The input of \(\vec F\) should be a point in the plane, not a two dimensional vector.
16.3.4.5.
Answer.
We can conclude that \(\vec F\) is conservative.
16.3.4.6.
Answer.
By the Fundamental Theorem of Line Integrals, since \(\vec F\) is conservative, \(\oint_C \vec F\cdot d\vec r = f(B) - f(A)\text{,}\) where \(f\) is a potential function for \(\vec F\) and \(A\) and \(B\) are the initial and terminal points of \(C\text{,}\) respectively. Since \(C\) is a closed curve, \(A = B\text{,}\) and hence \(f(B) - f(A) = 0\text{.}\)

Problems

16.3.4.7.
Answer.
\(11/6\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle 3t,t\rangle\) on \(0\leq t\leq 1\text{.}\))
16.3.4.8.
Answer.
\(5/3\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle t,t^2\rangle\) on \(0\leq t\leq 1\text{.}\))
16.3.4.9.
Answer.
\(0\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle \cos(t),\sin(t)\rangle\) on \(0\leq t\leq \pi\text{.}\))
16.3.4.10.
Answer.
\(2/5\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle t,t^3\rangle\) on \(-1\leq t\leq 1\text{.}\))
16.3.4.11.
Answer.
\(12\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle 1,2,3\rangle+t\langle 3,1,-1\rangle\) on \(0\leq t\leq 1\text{.}\))
16.3.4.12.
Answer.
\(1\text{.}\)
16.3.4.13.
Answer.
\(5/6\) joules. (One parametrization for \(C\) is \(\vec r(t) = \langle t,t\rangle\) on \(0\leq t\leq 1\text{.}\))
16.3.4.14.
Answer.
\(13/15\) joules. (One parametrization for \(C\) is \(\vec r(t) = \langle t,\sqrt t\rangle\) on \(0\leq t\leq 1\text{.}\))
16.3.4.15.
Answer.
\(24\) ft-lbs.
16.3.4.16.
Answer.
\(24\) ft-lbs.
16.3.4.17.
Answer.
  1. \(\displaystyle f(x,y) = xy+x\)
  2. \(\curl \vec F = 0\text{.}\)
  3. \(1\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle t,t-1\rangle\) on \(0\leq t\leq 1\text{.}\))
  4. \(1\) (with \(A = (0,1)\) and \(B = (1,0)\text{,}\) \(f(B) - f(A) = 1\text{.}\))
16.3.4.18.
Answer.
  1. \(\displaystyle f(x,y) = x^2+xy+y^2\)
  2. \(\curl \vec F = 0\text{.}\)
  3. \(0\text{.}\)
  4. \(0\) (with \(A = (0,0)\) and \(B = (0,0)\text{,}\) \(f(B) - f(A) = 0\text{.}\))
16.3.4.19.
Answer.
  1. \(\displaystyle f(x,y) = x^2yz\)
  2. \(\curl \vec F = \vec 0\text{.}\)
  3. \(250\text{.}\)
  4. \(250\) (with \(A = (1,-1,0)\) and \(B = (5,5,2)\text{,}\) \(f(B) - f(A) = 250\text{.}\))
16.3.4.20.
Answer.
  1. \(\displaystyle f(x,y) = x^2+y^2+z^2\)
  2. \(\curl \vec F = \vec 0\text{.}\)
  3. \(0\text{.}\)
  4. \(0\) (with \(A = (1,0,0)\) and \(B = (1,0,0)\text{,}\) \(f(B) - f(A) = 0\text{.}\))

16.4 Flow, Flux, Green’s Theorem and the Divergence Theorem
16.4.4 Exercises

Terms and Concepts

16.4.4.2.
Answer.
It is the measure of flow around the entirety of a closed curve \(C\text{.}\)
16.4.4.4.

Problems

16.4.4.7.
Answer.
\(12\)
16.4.4.8.
Answer.
\(12\)
16.4.4.9.
Answer.
\(-2/3\)
16.4.4.10.
Answer.
\(10/3\)
16.4.4.11.
Answer.
\(1/2\)
16.4.4.12.
Answer.
\(1/2\)
16.4.4.13.
Answer.
The line integral \(\oint_C\vec F\cdot d\vec r\text{,}\) over the parabola, is \(38/3\text{;}\) over the line, it is \(-10\text{.}\) The total line integral is thus \(38/3-10 = 8/3\text{.}\) The double integral of \(\curl \vec F = 2\) over \(R\) also has value \(8/3\text{.}\)
16.4.4.14.
Answer.
Both the line integral and double integral have value of \(2\pi\text{.}\)
16.4.4.15.
Answer.
Three line integrals need to be computed to compute \(\oint_C \vec F\cdot d\vec r\text{.}\) It does not matter which corner one starts from first, but be sure to proceed around the triangle in a counterclockwise fashion.
From \((0,0)\) to \((2,0)\text{,}\) the line integral has a value of 0. From \((2,0)\) to \((1,1)\) the integral has a value of \(7/3\text{.}\) From \((1,1)\) to \((0,0)\) the line integral has a value of \(-1/3\text{.}\) Total value is 2.
The double integral of \(\curl\vec F\) over \(R\) also has value 2.
16.4.4.16.
Answer.
Two line integrals need to be computed to compute \(\oint_C \vec F\cdot d\vec r\text{.}\) Along the parabola, the line integral has value \(25.5\text{.}\) Along the line, the line integral has value \(-21\text{.}\) Together, the total value is \(4.5\)
The double integral of \(\curl\vec F\) over \(R\) also has value \(4.5\text{.}\)
16.4.4.17.
Answer.
Any choice of \(\vec F\) is appropriate as long as \(\curl \vec F = 1\text{.}\) When \(\vec F = \langle -y/2,x/2\rangle\text{,}\) the integrand of the line integral is simply 6. The area of \(R\) is \(12\pi\text{.}\)
16.4.4.18.
Answer.
Any choice of \(\vec F\) is appropriate as long as \(\curl \vec F = 1\text{.}\) The choices of \(\vec F = \langle -y,0\rangle\) and \(\langle 0,x\rangle\) each lead to reasonable integrands. The area of \(R\) is \(4/3\text{.}\)
16.4.4.19.
Answer.
Any choice of \(\vec F\) is appropriate as long as \(\curl \vec F = 1\text{.}\) The choices of \(\vec F = \langle -y,0\rangle\text{,}\) \(\langle 0,x\rangle\) and \(\langle -y/2,x/2\rangle\) each lead to reasonable integrands. The area of \(R\) is \(16/15\text{.}\)
16.4.4.20.
Answer.
Any choice of \(\vec F\) is appropriate as long as \(\curl \vec F = 1\text{.}\) The choice of \(\vec F = \langle -y/2,x/2\rangle\) leads to a reasonable integrand after simplification. The area of \(R\) is \(41\pi/10\text{.}\)
16.4.4.21.
Answer.
The line integral \(\oint_C\vec F\cdot \vec n\, ds\text{,}\) over the parabola, is \(-22/3\text{;}\) over the line, it is \(10\text{.}\) The total line integral is thus \(-22/3+10 = 8/3\text{.}\) The double integral of \(\divv \vec F = 2\) over \(R\) also has value \(8/3\text{.}\)
16.4.4.22.
Answer.
Both the line integral and double integral have value of \(0\text{.}\)
16.4.4.23.
Answer.
Three line integrals need to be computed to compute \(\oint_C \vec F\cdot \vec n\, ds\text{.}\) It does not matter which corner one starts from first, but be sure to proceed around the triangle in a counterclockwise fashion.
From \((0,0)\) to \((2,0)\text{,}\) the line integral has a value of 0. From \((2,0)\) to \((1,1)\) the integral has a value of \(1/3\text{.}\) From \((1,1)\) to \((0,0)\) the line integral has a value of \(1/3\text{.}\) Total value is \(2/3\text{.}\)
The double integral of \(\divv\vec F\) over \(R\) also has value \(2/3\text{.}\)
16.4.4.24.
Answer.
Two line integrals need to be computed to compute \(\oint_C \vec F\cdot \vec n\, ds\text{.}\) Along the parabola, the line integral has value \(159/20\text{.}\) Along the line, the line integral has value \(6\text{.}\) Together, the total value is \(279/20\text{.}\)
The double integral of \(\divv\vec F\) over \(R\) also has value \(279/20\text{.}\)

16.5 Parametrized Surfaces and Surface Area
16.5.3 Exercises

Terms and Concepts

16.5.3.1.
Answer.
Answers will vary, though generally should meaningfully include terms like “two sided”.
16.5.3.2.
Answer.
Many possible answers exist; the one given by the book is the Möbius band.

Problems

16.5.3.3.
Answer.
  1. \(\vec r(u,v) = \langle u, v, 3u^2v\rangle\) on \(-1\leq u\leq 1\text{,}\) \(0\leq v\leq 2\text{.}\)
  2. \(\vec r(u,v) = \langle 3v\cos(u)+1, 3v\sin(u)+2, 3(3v\cos(u)+1)^2(3v\sin(u)+2)\rangle\text{,}\) on \(0\leq u\leq 2\pi\text{,}\) \(0\leq v\leq 1\text{.}\)
  3. \(\vec r(u,v) = \langle u, v(2-2u), 3u^2v(2-2u)\rangle\) on \(0\leq u, v\leq 1\text{.}\)
  4. \(\vec r(u,v) = \langle u, v(1-u^2), 3u^2v(1-u^2)\rangle\) on \(-1\leq u\leq 1\text{,}\) \(0\leq v\leq 1\text{.}\)
16.5.3.4.
Answer.
  1. \(\vec r(u,v) = \langle u, v, 4u+2u^2\rangle\) on \(1\leq u\leq 4\text{,}\) \(5\leq v\leq 7\text{.}\)
  2. \(\vec r(u,v) = \langle 4v\cos(u), 3v\sin(u), 16v\cos(u)+2(3v\sin(u))^2\rangle\text{,}\) on \(0\leq u\leq 2\pi\text{,}\) \(0\leq v\leq 1\text{.}\)
  3. \(\vec r(u,v) = \langle u, u+v(4-2u), 4u+2\big(u+v(4-2u)\big)^2\rangle\) on \(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\text{.}\)
  4. \(\vec r(u,v) = \langle v\cos(u), v\sin(u), 4v\cos(u) + 2(v\sin(u))^2\rangle\) on \(0\leq u\leq 2\pi\text{,}\) \(2\leq v\leq 5\text{.}\)
16.5.3.5.
Answer.
\(\vec r(u,v) = \langle 0, u, v\rangle\) with \(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\text{.}\)
16.5.3.6.
Answer.
\(\vec r(u,v) = \langle u, 0, 1-u+vu\rangle\) with \(0\leq u\leq 1\text{,}\) \(0\leq v\leq 1\text{.}\)
16.5.3.7.
Answer.
\(\vec r(u,v) = \langle 3\sin(u)\cos(v), 2\sin(u)\sin(v), 4\cos(u)\rangle\) with \(0\leq u\leq \pi\text{,}\) \(0\leq v\leq 2\pi\text{.}\)
16.5.3.8.
Answer.
Answers may vary; one solution is \(\vec r(u,v) = \langle v\cos(u), v, 4v\sin(u)\rangle\) with \(0\leq u\leq 2\pi\text{,}\) \(-1\leq v\leq 5\text{.}\)
16.5.3.9.
Answer.
Answers may vary.
For \(z = \frac12(3-x)\text{:}\) \(\vec r(u,v) = \langle u, v , \frac12(3-u)\rangle\text{,}\) with \(1\leq u\leq 3\) and \(0\leq v\leq 2\text{.}\)
For \(x=1\text{:}\) \(\vec r(u,v) = \langle 1,u,v\rangle\text{,}\) with \(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\)
For \(y=0\text{:}\) \(\vec r(u,v) = \langle u,0,v/2(3-u)\rangle\text{,}\) with \(1\leq u\leq 3\text{,}\) \(0\leq v\leq 1\)
For \(y=2\text{:}\) \(\vec r(u,v) = \langle u,2,v/2(3-u)\rangle\text{,}\) with \(1\leq u\leq 3\text{,}\) \(0\leq v\leq 1\)
For \(z=0\text{:}\) \(\vec r(u,v) = \langle u,v,0\rangle\text{,}\) with \(1\leq u\leq 3\text{,}\) \(0\leq v\leq 2\)
16.5.3.10.
Answer.
Answers may vary.
For \(z=2x+4y-4\text{:}\) \(\vec r(u,v) = \langle u, 1-u/2+uv/2, 2u+4(1-u/2+uv/2)-4\rangle\text{,}\) with \(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\text{.}\)
For \(x=2\text{:}\) \(\vec r(u,v) = \langle 2,u,4uv\rangle\text{,}\) with \(0\leq u\leq 1\text{,}\) \(0\leq v\leq 1\)
For \(y=1\text{:}\) \(\vec r(u,v) = \langle u,1,2uv\rangle\text{,}\) with \(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\)
For \(z=0\text{:}\) \(\vec r(u,v) = \langle u, 1-u/2+uv/2,0\rangle\text{,}\) with \(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\)
16.5.3.11.
Answer.
Answers may vary.
For \(z=2y: \vec r(u,v) = \langle u, v(4-u^2), 2v(4-u^2)\rangle\) with \(-2\leq u\leq 2\) and \(0\leq v\leq 1\text{.}\)
For \(y=4-x^2: \vec r(u,v) = \langle u, 4-u^2, 2v(4-u^2)\rangle\) with \(-2\leq u\leq 2\) and \(0\leq v\leq 1\text{.}\)
For \(z=0\text{:}\) \(\vec r(u,v) = \langle u, v(4-u^2), 0\rangle\) with \(-2\leq u\leq 2\) and \(0\leq v\leq 1\text{.}\)
16.5.3.12.
Answer.
Answers may vary.
For \(y=1-z^2\text{:}\) \(\vec r(u,v) = \langle u, v(1-u^2), \sqrt{1-v(1-u^2)}\rangle\) with \(0\leq u\leq 1\) and \(0\leq v\leq 1\text{.}\)
For \(y=1-x^2\text{:}\) \(\vec r(u,v) = \langle u, 1-u^2, uv\rangle\) with \(0\leq u\leq 1\) and \(0\leq v\leq 1\text{.}\)
For \(x=0\text{:}\) \(\vec r(u,v) = \langle 0, v(1-u^2),u\rangle\) with \(0\leq u\leq 1\) and \(0\leq v\leq 1\text{.}\)
For \(y=0\text{:}\) \(\vec r(u,v) = \langle u, 0,v\rangle\) with \(0\leq u\leq 1\) and \(0\leq v\leq 1\text{.}\)
For \(z=0\text{:}\) \(\vec r(u,v) = \langle u, v(1-u^2), 0rangle\) with \(0\leq u\leq 1\) and \(0\leq v\leq 1\text{.}\)
16.5.3.13.
Answer.
Answers may vary.
For \(x^2+y^2/9=1\text{:}\) \(\vec r(u,v) = \langle \cos(u), 3\sin(u), v\rangle\) with \(0\leq u\leq 2\pi\) and \(1\leq v\leq 3\text{.}\)
For \(z=1\text{:}\) \(\vec r(u,v) = \langle v\cos(u), 3v\sin(u), 1\rangle\) with \(0\leq u\leq 2\pi\) and \(0\leq v\leq 1\text{.}\)
For \(z=3\text{:}\) \(\vec r(u,v) = \langle v\cos(u), 3v\sin(u), 3\rangle\) with \(0\leq u\leq 2\pi\) and \(0\leq v\leq 1\text{.}\)
16.5.3.14.
Answer.
Answers may vary.
For \(x^2+y^2=(z-1)^2\text{:}\) \(\vec r(u,v) = \langle v\cos(u), v\sin(u), 1-v\rangle\) with \(0\leq u\leq 2\pi\) and \(0\leq v\leq 1\text{.}\)
For \(z=0\text{:}\) \(\vec r(u,v) = \langle v\cos(u), v\sin(u), 0\rangle\) with \(0\leq u\leq 2\pi\) and \(0\leq v\leq 1\text{.}\)
16.5.3.15.
Answer.
Answers may vary.
For \(z=1-x^2\text{:}\) \(\vec r(u,v) = \langle u,v,1-u^2\rangle\) with \(-1\leq u\leq 1\) and \(-1\leq v\leq 2\text{.}\)
For \(y=-1\text{:}\) \(\vec r(u,v) = \langle u,-1,v(1-u^2)\rangle\) with \(-1\leq u\leq 1\) and \(0\leq v\leq 1\text{.}\)
For \(y=2\text{:}\) \(\vec r(u,v) = \langle u,2,v(1-u^2)\rangle\) with \(-1\leq u\leq 1\) and \(0\leq v\leq 1\text{.}\)
For \(z=0\text{:}\) \(\vec r(u,v) = \langle u,v,0\rangle\) with \(-1\leq u\leq 1\) and \(-1\leq v\leq 2\text{.}\)
16.5.3.16.
Answer.
Answers may vary.
For \(z=4-x^2-4y^2\text{:}\) \(\vec r(u,v) = \langle 2v\cos(u),v\sin(u),4-(2v\cos(u))^2-4(v\sin(u))^2\rangle\) with \(0\leq u\leq 2\pi\) and \(0\leq v\leq 1\text{.}\)
For \(z=0\text{:}\) \(\vec r(u,v) = \langle 2v\cos(u),v\sin(u),0\rangle\) with \(0\leq u\leq 2\pi\) and \(0\leq v\leq 1\text{.}\)
16.5.3.17.
Answer.
\(S = 2\sqrt{14}\text{.}\)
16.5.3.18.
Answer.
\(S = \sqrt{6}/2\text{.}\)
16.5.3.19.
Answer.
\(S = 4\sqrt{3}\pi\text{.}\)
16.5.3.20.
Answer.
\(S = 3\sqrt{3}\pi\text{.}\)
16.5.3.21.
Answer.
\(S =\int_0^3\int_0^{2\pi}\sqrt{v^2+4v^4}\, du\, dv= (37\sqrt{37}-1)\pi/6 \approx 117.319\text{.}\)
16.5.3.22.
Answer.
\(S = \int_0^1\int_0^1\sqrt{v^2+4u^2v^2+4v^4}\, du\, dv \approx 0.931\text{.}\)
16.5.3.23.
Answer.
\(S =\int_0^1\int_{-1}^{1}\sqrt{(5u^2-5)^2+2(1-u^2)^2}\, du\, dv = 4\sqrt{3}\approx 6.9282\text{.}\)
16.5.3.24.
Answer.
\(S =\int_0^1\int_{0}^{2\pi}\sqrt{v^2+4v^4}\, du\, dv = (5\sqrt{5}-1)\pi/6 \approx 5.330\text{.}\)

16.6 Surface Integrals
16.6.3 Exercises

Terms and Concepts

16.6.3.2.
Answer.
Answers will vary; in general, it means that more of the vector field passes through the surface opposite the direction of the normal vector than in the same direction of the normal vector.

Problems

16.6.3.5.
Answer.
\(240\sqrt{3}\)
16.6.3.6.
Answer.
\(40\pi\)
16.6.3.7.
Answer.
\(24\)
16.6.3.8.
Answer.
\(15\)
16.6.3.9.
Answer.
\(0\)
16.6.3.10.
Answer.
\(0\)
16.6.3.11.
Answer.
\(-1/2\)
16.6.3.12.
Answer.
\(\pi\)
16.6.3.13.
Answer.
\(0\text{;}\) the flux over \(\surfaceS_1\) is \(-45\pi\) and the flux over \(\surfaceS_2\) is \(45\pi\text{.}\)
16.6.3.14.
Answer.
\(9\pi/8\text{;}\) the flux over \(\surfaceS_1\) is \(3\pi/4\) (use \(\vec r(u,v) = \langle \sin(u)\cos(v),\sin(u)\sin(v),\cos(u)\rangle\) on \(\pi/3\leq u\leq \pi\text{,}\) \(0\leq v\leq 2\pi\)) and the flux over \(\surfaceS_2\) is \(3\pi/8\) (use \(\vec r(u,v) = \langle v\sqrt{3}\cos (u)/2, v\sqrt{3}\sin(u)/2,1/2\rangle\) for \(0\leq u\leq 2\pi\text{,}\) \(0\leq v\leq 1\text{.}\)

16.7 The Divergence Theorem and Stokes’ Theorem
16.7.4 Exercises

Terms and Concepts

16.7.4.1.
Answer.
Answers will vary; in Section 16.4, the Divergence Theorem connects outward flux over a closed curve in the plane to the divergence of the vector field, whereas in this section the Divergence Theorem connects outward flux over a closed surface in space to the divergence of the vector field.
16.7.4.2.
Answer.
Divergence.
16.7.4.3.
Answer.
Curl.
16.7.4.4.
Answer.
Green’s Theorem.

Problems

16.7.4.5.
Answer.
Outward flux across the plane \(z=2-x/2-2y/3\) is 14; across the plane \(z=0\) the outward flux is \(-8\text{;}\) across the planes \(x=0\) and \(y=0\) the outward flux is 0.
Total outward flux: \(14\text{.}\)
\(\iint_D\divv\vec F\, dV = \int_0^{4}\int_0^{3-3x/4}\int_0^{2-x/2-2y/3}(2x+2y)\, dz\, dy\, dx = 14\text{.}\)
16.7.4.6.
Answer.
Outward flux across the cylinder \(x^2+y^2=1\) is 0; across the plane \(z=3\) the outward flux is \(3\pi\text{;}\) across the plane \(z=-3\) the outward flux is \(3\pi\text{.}\)
Total outward flux: \(6\pi\text{.}\)
\(\iint_D\divv\vec F\, dV = \int_0^{2\pi}\int_0^{1}\int_{-3}^{3}r\, dz\, dr\, d\theta = 6\pi\text{.}\)
16.7.4.7.
Answer.
Outward flux across the surface \(z=xy(3-x)(3-y)\) is 252; across the plane \(z=0\) the outward flux is \(-9\text{.}\)
Total outward flux: \(243\text{.}\)
\(\iint_D\divv\vec F\, dV = \int_0^{3}\int_0^{3}\int_{0}^{xy(3-x)(3-y)}12\, dz\, dy\, dx = 243\text{.}\)
16.7.4.8.
Answer.
Outward flux across the paraboloid is \(112\pi/3\text{;}\) across the disk the outward flux is 0.
Total outward flux: \(112\pi/3\text{.}\)
\(\iint_D\divv\vec F\, dV = \int_0^{2\pi}\int_0^2\int_0^{4-r^2}(2z+2)r\, dz\, dr\, d\theta = 112\pi/3\text{.}\)
16.7.4.9.
Answer.
Circulation on \(C\text{:}\) \(\oint_C \vec F\cdot d\vec r = -\pi\)
\(\iint_\surfaceS\big(\curl \vec F\big)\cdot\vec n\, dS = -\pi\text{.}\)
16.7.4.10.
Answer.
Circulation on \(C\text{:}\) \(\oint_C \vec F\cdot d\vec r = \pi\)
\(\iint_\surfaceS\big(\curl \vec F\big)\cdot\vec n\, dS = \pi\text{.}\)
16.7.4.11.
Answer.
Circulation on \(C\text{:}\) The flow along the line from \((0,0,2)\) to \((4,0,0)\) is 0; from \((4,0,0)\) to \((0,3,0)\) it is \(-6\text{,}\) and from \((0,3,0)\) to \((0,0,2)\) it is 6. The total circulation is \(0+(-6)+6=0\text{.}\)
\(\iint_\surfaceS\big(\curl \vec F\big)\cdot\vec n\, dS = \iint_\surfaceS 0 \, dS = 0\text{.}\)
16.7.4.12.
Answer.
Circulation on \(C\text{:}\) The flow along the parabola is \(-32/15\text{;}\) the flow along the line is \(4/3\text{.}\) The total circulation is \(4/3-32/15 = -4/5\text{.}\)
\(\iint_\surfaceS\big(\curl \vec F\big)\cdot\vec n\, dS = -4/5\text{.}\)
16.7.4.13.
Answer.
\(128/225\)
16.7.4.14.
Answer.
\(8\)
16.7.4.15.
Answer.
\(8192/105\approx 78.019\)
16.7.4.16.
Answer.
\(64/3\)
16.7.4.17.
Answer.
\(5/3\)
16.7.4.18.
Answer.
\(8\pi\)
16.7.4.19.
Answer.
\(23\pi\)
16.7.4.20.
Answer.
\(0\)
16.7.4.21.
Answer.
Each field has a divergence of 1; by the Divergence Theorem, the total outward flux across \(\surfaceS\) is \(\iint_D 1\, dS\) for each field.
16.7.4.22.
16.7.4.22.a
Answer.
\(\curl\vec F = 1\text{.}\)
16.7.4.22.b
Answer.
\(\curl\vec F\cdot \vec n = 1\text{,}\) where \(\vec n\) is a unit vector normal to \(\surfaceS\text{.}\)
16.7.4.23.
Answer.
Answers will vary. Often the closed surface \(\surfaceS\) is composed of several smooth surfaces. To measure total outward flux, this may require evaluating multiple double integrals. Each double integral requires the parametrization of a surface and the computation of the cross product of partial derivatives. One triple integral may require less work, especially as the divergence of a vector field is generally easy to compute.
16.7.4.24.
Answer.
Answers will vary. Often the closed curve \(C\) is composed of several smooth curves. To measure the total circulation, one may have to evaluate line integrals along each curve. Each line integral requires the parametrization of its curve. It may be less work to evaluate one single double (i.e., surface) integral.
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