Terms and Concepts
1.1.3.3.
Answer.
\(\text{False}\)
Appendix A Answers to Selected Exercises
1 Limits
1.1 An Introduction To Limits
1.1.3 Exercises
Problems
1.1.3.7.
Answer.
\(5\)
1.1.3.9.
Answer.
\(\text{DNE}\)
1.1.3.11.
Answer.
\(-4\)
1.1.3.13.
Answer.
\(\text{DNE}\)
1.1.3.15.
Answer.
\(1\)
1.1.3.17.
Answer.
\(1\)
1.1.3.19.
Answer.
\(\text{DNE}\)
1.1.3.21.
Answer.
\(-7\)
1.1.3.23.
Answer.
\(5\)
1.1.3.25.
Answer.
\(29\)
1.1.3.27.
Answer.
\(-1\)
1.2 Epsilon-Delta Definition of a Limit
Exercises
Terms and Concepts
1.2.3.
Answer.
\(\text{True}\)
1.3 Finding Limits Analytically
Exercises
Problems
1.3.7.
Answer.
\(9\)
1.3.9.
Answer.
\(0\)
1.3.11.
Answer.
\(3\)
1.3.13.
Answer.
\(3\)
1.3.15.
Answer.
\(0\)
1.3.17.
Answer.
\(\pi \)
1.3.19.
Answer.
\(23\)
1.3.21.
Answer.
\(\frac{\sqrt{3}}{4}\)
1.3.23.
Answer.
\(\text{DNE}\)
1.3.25.
Answer.
\(\frac{2\sqrt{3}}{3}\)
1.3.27.
Answer.
\(\frac{\pi ^{2}-4\pi -2}{2\pi ^{2}-2\pi +1}\)
1.3.29.
Answer.
\({\frac{1}{4}}\)
1.3.31.
Answer.
\({\frac{17}{4}}\)
1.3.33.
Answer.
\({\frac{4}{9}}\)
1.3.35.
Answer.
\(0\)
1.3.37.
Answer.
\(1\)
1.3.39.
Answer.
\(8\)
1.3.41.
Answer.
\(1\)
1.4 One-Sided Limits
Exercises
Terms and Concepts
1.4.3.
Answer.
\(\text{False}\)
Problems
1.4.5.
Answer.
\(2\)
Answer.
\(2\)
Answer.
\(2\)
Answer.
\(1\)
Answer.
\(\text{DNE}\)
Answer.
\(4\)
1.4.7.
Answer.
\(\text{DNE}\hbox{ or }\infty \)
Answer.
\(\text{DNE}\hbox{ or }\infty \)
Answer.
\(\text{DNE}\hbox{ or }\infty \)
Answer.
\(\text{DNE}\)
Answer.
\(5\)
Answer.
\(4\)
1.4.9.
Answer.
\(1\)
Answer.
\(1\)
Answer.
\(1\)
Answer.
\(1\)
1.4.11.
Answer.
\(2\)
Answer.
\(2\)
Answer.
\(2\)
Answer.
\(0\)
Answer.
\(2\)
Answer.
\(2\)
Answer.
\(2\)
Answer.
\(\text{DNE}\)
1.4.13.
Answer.
\(2\)
Answer.
\(6\)
Answer.
\(\text{DNE}\)
Answer.
\(2\)
1.4.15.
Answer.
\(9\)
Answer.
\(9\)
Answer.
\(9\)
Answer.
\(9\)
Answer.
\(126\)
Answer.
\(126\)
Answer.
\(126\)
Answer.
\(126\)
1.4.17.
Answer.
\(1-\cos^{2}\mathopen{}\left(a\right)\)
Answer.
\(\sin^{2}\mathopen{}\left(a\right)\)
Answer.
\(1-\cos^{2}\mathopen{}\left(a\right)\hbox{ or }\sin^{2}\mathopen{}\left(a\right)\)
Answer.
\(\sin^{2}\mathopen{}\left(a\right)\)
1.4.19.
Answer.
\(-4\)
Answer.
\(-4\)
Answer.
\(-4\)
Answer.
\(-2\)
1.4.21.
Answer.
\(-1\)
Answer.
\(1\)
Answer.
\(\text{DNE}\)
Answer.
\(0\)
1.5 Continuity
Exercises
Terms and Concepts
1.5.5.
Answer.
\(\text{False}\)
1.5.7.
Answer.
\(\text{True}\)
1.5.9.
Answer.
\(\text{False}\)
Problems
1.5.11.
Answer.
\(\text{No.}\)
1.5.13.
Answer.
\(\text{No.}\)
1.5.15.
Answer.
\(\text{Yes.}\)
1.5.17.
Answer 1.
\(\text{No.}\)
Answer 2.
\(\text{Yes.}\)
Answer 3.
\(\text{No.}\)
1.5.19.
Answer.
\(\text{Yes.}\)
Answer.
\(\text{Yes.}\)
1.5.21.
Answer.
\(\text{Yes.}\)
Answer.
\(\text{Yes.}\)
1.5.23.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.25.
Answer.
\(\left[-2,2\right]\)
1.5.27.
Answer.
\(\left(-\infty ,-1.73205\right], \left[1.73205,\infty \right)\)
1.5.29.
Answer.
\(\left(-\infty ,\infty \right)\)
1.5.31.
Answer.
\(\left(0,\infty \right)\)
1.5.33.
Answer.
\(\left(-\infty ,1.09861\right]\)
1.5.39.
Answer.
\(1.23633\)
1.5.41.
Answer.
\(0.693164\)
1.6 Limits Involving Infinity
1.6.4 Exercises
Terms and Concepts
1.6.4.1.
Answer.
\(\text{False}\)
1.6.4.3.
Answer.
\(\text{False}\)
1.6.4.5.
Answer.
\(\text{True}\)
Problems
1.6.4.9.
Answer.
\(-\infty \)
Answer.
\(\infty \)
1.6.4.11.
Answer.
\(0\)
Answer.
\(3\)
Answer.
\(1.5\)
Answer.
\(1.5\)
1.6.4.13.
Answer.
\(\text{DNE}\)
Answer.
\(\text{DNE}\)
1.6.4.15.
Answer.
\(-\infty \)
Answer.
\(\infty \)
Answer.
\(\text{DNE}\)
1.6.4.17.
Answer.
\(\infty \)
Answer.
\(\infty \)
Answer.
\(\infty \)
1.6.4.19.
Answer.
\(y = 2, x = -2, x = 9\)
1.6.4.21.
Answer.
\(y = 0, x = 0, x = 4\)
1.6.4.23.
Answer.
\(\text{NONE}\)
1.6.4.25.
Answer.
\(\infty \)
1.6.4.27.
Answer.
\(\infty \)
2 Derivatives
2.1 Instantaneous Rates of Change: The Derivative
2.1.3 Exercises
Terms and Concepts
2.1.3.1.
Answer.
\(\text{True}\)
Problems
2.1.3.7.
Answer.
\(0\)
2.1.3.9.
Answer.
\(-3\)
2.1.3.11.
Answer.
\(3x^{2}\)
2.1.3.13.
Answer.
\(\frac{-1}{x^{2}}\)
2.1.3.15.
Answer 1.
\(y = 6\)
Answer 2.
\(x = -2\)
2.1.3.17.
Answer 1.
\(3x+y = 4\)
Answer 2.
\(y-0.333333x = -19.3333\)
2.1.3.19.
Answer 1.
\(y-48x = -128\)
Answer 2.
\(0.0208333x+y = 64.0833\)
2.1.3.21.
Answer 1.
\(0.25x+y = -1\)
Answer 2.
\(y-4x = 7.5\)
2.1.3.23.
Answer.
\(5.9x+y = 1.2\)
2.1.3.25.
Answer.
\(y-0.0192627x = 0.0953664\)
2.1.3.27.
Answer.
\(-2, 0, 4\)
Answer.
\(2x\)
Answer.
\(-2, 0, 4\)
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.
\(\left\{-1,1\right\}\)
2.1.3.35.
Answer.
\(\text{no}\)
2.2 Interpretations of the Derivative
2.2.5 Exercises
Terms and Concepts
2.2.5.1.
Answer.
\(\text{velocity}\)
2.2.5.3.
Answer.
\(\text{linear functions}\)
Problems
2.2.5.5.
Answer.
\(-89\)
2.2.5.7.
Answer.
\(\text{f(10.1)}\)
2.2.5.9.
Answer.
\(7\)
2.2.5.11.
Answer.
\(\text{foot per second squared}\)
2.2.5.15.
Answer.
\(\text{f is the derivative of g.}\)
2.2.5.17.
Answer.
\(\text{g is the derivative of f.}\)
2.3 Basic Differentiation Rules
2.3.3 Exercises
Terms and Concepts
2.3.3.1.
Answer.
\(\text{the power rule}\)
2.3.3.3.
Answer.
\(e^{x}\)
2.3.3.5.
Answer.
\(\text{Choice 1, Choice 2, Choice 5, Choice 6}\)
2.3.3.7.
Answer.
\(17x-205\)
2.3.3.9.
Answer 1.
\(\text{a velocity function}\)
Answer 2.
\(\text{an acceleration function}\)
Problems
2.3.3.11.
Answer.
\(-\left(14x+8\right)\)
2.3.3.13.
Answer.
\(9-\left(20t^{4}+{\frac{3}{4}}t^{2}\right)\)
2.3.3.15.
Answer.
\(3e^{r}\)
2.3.3.17.
Answer.
\(\frac{6}{x}+9\)
2.3.3.19.
Answer.
\(\sin\mathopen{}\left(t\right)-\left(e^{t}+\cos\mathopen{}\left(t\right)\right)\)
2.3.3.21.
Answer.
\(0\)
2.3.3.23.
Answer.
\(24x^{2}+96x+96\)
2.3.3.25.
Answer.
\(8x+28\)
2.3.3.27.
Answer 1.
\(9x^{8}\)
Answer 2.
\(9\cdot 8x^{7}\)
Answer 3.
\(9\cdot 8\cdot 7x^{6}\)
Answer 4.
\(9\cdot 8\cdot 7\cdot 6x^{5}\)
2.3.3.29.
Answer 1.
\(-\left(4\cdot 2t+3+e^{t}\right)\)
Answer 2.
\(-\left(8+e^{t}\right)\)
Answer 3.
\(-e^{t}\)
Answer 4.
\(-e^{t}\)
2.3.3.31.
Answer 1.
\(-\left(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\right)\)
Answer 2.
\(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\)
Answer 3.
\(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\)
Answer 4.
\(-\left(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\right)\)
2.3.3.33.
Answer 1.
\(y = 20\mathopen{}\left(x-2\right)+24\)
Answer 2.
\(y = -{\frac{1}{20}}\mathopen{}\left(x-2\right)+24\)
2.3.3.35.
Answer 1.
\(y = x-1\)
Answer 2.
\(y = -\left(x-1\right)\)
2.3.3.37.
Answer 1.
\(y = \frac{2\cdot 1}{2}\mathopen{}\left(x-\frac{\pi }{6}\right)+\frac{-2\sqrt{3}}{2}\)
Answer 2.
\(y = -\left({\frac{1}{2}}\cdot 2\right)\mathopen{}\left(x-\frac{\pi }{6}\right)+\frac{-2\sqrt{3}}{2}\)
2.4 The Product and Quotient Rules
Exercises
Terms and Concepts
2.4.1.
Answer.
\(\text{False}\)
2.4.3.
Answer.
\(\text{True}\)
2.4.5.
Answer.
\(\text{False}\)
Problems
2.4.15.
Answer.
\(\sin\mathopen{}\left(y\right)+y\cos\mathopen{}\left(y\right)\)
2.4.17.
Answer.
\(e^{q}\ln\mathopen{}\left(q\right)+e^{q}\frac{1}{q}\)
2.4.19.
Answer.
\(\frac{t-4-\left(t+8\right)}{\left(t-4\right)^{2}}\)
2.4.21.
Answer.
\(-\left(\csc\mathopen{}\left(y\right)\cot\mathopen{}\left(y\right)+e^{y}\right)\)
2.4.23.
Answer.
\(7\cdot 2q-6\)
2.4.25.
Answer.
\(\left(5r^{2}+17r+10\right)e^{r}\)
2.4.27.
Answer.
\(3\)
2.4.29.
Answer.
\(\frac{\csc\mathopen{}\left(z\right)\sin\mathopen{}\left(z\right)-\csc\mathopen{}\left(z\right)\cot\mathopen{}\left(z\right)\mathopen{}\left(\cos\mathopen{}\left(z\right)+2\right)}{\left(\cos\mathopen{}\left(z\right)+2\right)^{2}}\)
2.4.31.
Answer.
\(\frac{\tan\mathopen{}\left(r\right)-r\sec^{2}\mathopen{}\left(r\right)}{\tan^{2}\mathopen{}\left(r\right)}-\frac{\csc^{2}\mathopen{}\left(r\right)r+\cot\mathopen{}\left(r\right)}{r^{2}}\)
2.4.33.
Answer.
\(7\cdot 5x^{4}e^{x}+7x^{5}e^{x}-\left(\cos\mathopen{}\left(x\right)\cos\mathopen{}\left(x\right)-\sin\mathopen{}\left(x\right)\sin\mathopen{}\left(x\right)\right)\)
2.4.35.
Answer.
\(\left(4z^{3}\ln\mathopen{}\left(z\right)+z^{4}\frac{1}{z}\right)\cos\mathopen{}\left(z\right)-z^{4}\ln\mathopen{}\left(z\right)\sin\mathopen{}\left(z\right)\)
2.4.37.
Answer 1.
\(y = -\left(7x+7\right)\)
Answer 2.
\(y = \left({\frac{1}{7}}\right)x-7\)
2.4.39.
Answer 1.
\(y = -\left(15\mathopen{}\left(x+5\right)+25\right)\)
Answer 2.
\(y = \left({\frac{1}{15}}\right)\mathopen{}\left(x+5\right)-25\)
2.4.41.
Answer.
\({\frac{17}{2}}\)
2.4.43.
Answer.
\(\text{NONE}\)
2.4.45.
Answer.
\(2\cos\mathopen{}\left(x\right)-x\sin\mathopen{}\left(x\right)\)
2.4.47.
Answer.
\(\csc\mathopen{}\left(x\right)\cot\mathopen{}\left(x\right)\cot\mathopen{}\left(x\right)+\csc^{2}\mathopen{}\left(x\right)\csc\mathopen{}\left(x\right)\)
2.5 The Chain Rule
Exercises
Terms and Concepts
2.5.1.
Answer.
\(\text{True}\)
2.5.3.
Answer.
\(\text{False}\)
2.5.5.
Answer.
\(\text{True}\)
Problems
2.5.7.
Answer.
\(10\mathopen{}\left(4x^{3}-x\right)^{9}\mathopen{}\left(12x^{2}-1\right)\)
2.5.9.
Answer.
\(3\mathopen{}\left(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\right)^{2}\mathopen{}\left(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\right)\)
2.5.11.
Answer.
\(4\mathopen{}\left(\ln\mathopen{}\left(x\right)-x^{4}\right)^{3}\mathopen{}\left(\frac{1}{x}-4x^{3}\right)\)
2.5.13.
Answer.
\(5\mathopen{}\left(y+\frac{1}{y}\right)^{4}\mathopen{}\left(1-\frac{1}{y^{2}}\right)\)
2.5.15.
Answer.
\(2\sec^{2}\mathopen{}\left(2q\right)\)
2.5.17.
Answer.
\(\left(6t^{5}-\frac{3t^{2}}{\left(t^{3}\right)^{2}}\right)\cos\mathopen{}\left(t^{6}+\frac{1}{t^{3}}\right)\)
2.5.19.
Answer.
\(-3\cos^{2}\mathopen{}\left(y^{2}+3y-3\right)\mathopen{}\left(2y+3\right)\sin\mathopen{}\left(y^{2}+3y-3\right)\)
2.5.21.
Answer.
\(\frac{1}{q^{8}}\cdot 8q^{7}\)
2.5.23.
Answer.
\(1.79176\cdot 6^{t}\)
2.5.25.
Answer.
\(0\)
2.5.27.
Answer.
\(\frac{1.79176\cdot 6^{w}\mathopen{}\left(5^{w}+6\right)-\left(6^{w}+5\right)\cdot 1.60944\cdot 5^{w}}{\left(5^{w}+6\right)^{2}}\)
2.5.29.
Answer.
\(\frac{\left(1.60944\cdot 5^{r^{2}}\cdot 2r-1\right)\cdot 6^{r^{2}}-\left(5^{r^{2}}-r\right)\cdot 1.79176\cdot 6^{r^{2}}\cdot 2r}{\left(6^{r^{2}}\right)^{2}}\)
2.5.31.
Answer.
\(6\mathopen{}\left(x^{2}+4x\right)^{5}\mathopen{}\left(2x+4\right)\mathopen{}\left(7x^{4}+x\right)^{3}+\left(x^{2}+4x\right)^{6}\cdot 3\mathopen{}\left(7x^{4}+x\right)^{2}\mathopen{}\left(7\cdot 4x^{3}+1\right)\)
2.5.33.
Answer.
\(7\cos\mathopen{}\left(9+7w\right)\cos\mathopen{}\left(4w-5\right)-4\sin\mathopen{}\left(4w-5\right)\sin\mathopen{}\left(9+7w\right)\)
2.5.35.
Answer.
\(-\frac{6\sin\mathopen{}\left(6r+4\right)\mathopen{}\left(3r+1\right)^{3}+3\cdot 3\mathopen{}\left(3r+1\right)^{2}\cos\mathopen{}\left(6r+4\right)}{\left(\left(3r+1\right)^{3}\right)^{2}}\)
2.5.37.
Answer 1.
\(y = 0\)
Answer 2.
\(x = 0\)
2.5.39.
Answer 1.
\(y = -3\mathopen{}\left(x-\frac{\pi }{2}\right)+1\)
Answer 2.
\(y = \frac{1}{3}\mathopen{}\left(x-\frac{\pi }{2}\right)+1\)
2.5.41.
Answer.
\(\frac{1}{x}\)
2.6 Implicit Differentiation
2.6.4 Exercises
Terms and Concepts
2.6.4.3.
Answer.
\(\text{True}\)
Problems
2.6.4.5.
Answer.
\(\frac{1}{2\sqrt{w}}+\frac{\frac{1}{2\sqrt{w}}}{\left(\sqrt{w}\right)^{2}}\)
2.6.4.7.
Answer.
\(\frac{1}{2\sqrt{9+t^{2}}}\cdot 2t\)
2.6.4.9.
Answer.
\(1.2y^{0.2}\)
2.6.4.11.
Answer.
\(\frac{\sqrt{w}-\left(w-8\right)\frac{1}{2\sqrt{w}}}{\left(\sqrt{w}\right)^{2}}\)
2.6.4.13.
Answer.
\(\frac{-4x^{3}}{2y+1}\)
2.6.4.15.
Answer.
\(\sin\mathopen{}\left(x\right)\sec\mathopen{}\left(y\right)\)
2.6.4.17.
Answer.
\(\frac{y}{x}\)
2.6.4.19.
Answer.
\(\frac{-2\sin\mathopen{}\left(y\right)\cos\mathopen{}\left(y\right)}{x}\)
2.6.4.21.
Answer.
\(\frac{1}{2y+2}\)
2.6.4.23.
Answer.
\(\frac{1-\cos\mathopen{}\left(x\right)}{\sin\mathopen{}\left(y\right)+1}\)
2.6.4.25.
Answer.
\(\frac{-\left(2x+y\right)}{2y+x}\)
2.6.4.27.
Answer.
\(y = 0\)
Answer.
\(y = -1.859\mathopen{}\left(x-0.1\right)+0.2811\)
2.6.4.29.
Answer.
\(y = 4\)
Answer.
\(y = \frac{3}{108^{\frac{1}{4}}}\mathopen{}\left(x-2\right)-108^{\frac{1}{4}}\)
2.6.4.31.
Answer.
\(y = \frac{-1}{\sqrt{3}}\mathopen{}\left(x-\frac{7}{2}\right)+\frac{6+3\sqrt{3}}{2}\)
Answer.
\(y = \frac{\sqrt{3}\mathopen{}\left(x-\left(4+3\sqrt{3}\right)\right)}{2}+\frac{3}{2}\)
2.6.4.33.
Answer.
\(\frac{-\left(\left(2y+1\right)\cdot 12x^{2}-4x^{3}\frac{2\mathopen{}\left(-\left(4x^{3}\right)\right)}{2y+1}\right)}{\left(2y+1\right)^{2}}\)
2.6.4.35.
Answer.
\(\sin^{2}\mathopen{}\left(x\right)\sec^{2}\mathopen{}\left(y\right)\tan\mathopen{}\left(y\right)+\cos\mathopen{}\left(x\right)\sec\mathopen{}\left(y\right)\)
2.6.4.37.
Answer 1.
\(\left(1+x\right)^{\frac{1}{x}}\mathopen{}\left(\frac{1}{x\mathopen{}\left(x+1\right)}-\frac{\ln\mathopen{}\left(1+x\right)}{x^{2}}\right)\)
Answer 2.
\(y = \left(1-2\ln\mathopen{}\left(2\right)\right)\mathopen{}\left(x-1\right)+2\)
2.6.4.39.
Answer 1.
\(\frac{x^{x}}{x+1}\mathopen{}\left(\ln\mathopen{}\left(x\right)+1-\frac{1}{x+1}\right)\)
Answer 2.
\(y = \frac{1}{4}\mathopen{}\left(x-1\right)+\frac{1}{2}\)
2.6.4.41.
Answer 1.
\(\frac{x+1}{x+2}\mathopen{}\left(\frac{1}{x+1}-\frac{1}{x+2}\right)\)
Answer 2.
\(y = \frac{1}{9}\mathopen{}\left(x-1\right)+\frac{2}{3}\)
2.7 Derivatives of Inverse Functions
Exercises
Terms and Concepts
2.7.1.
Answer.
\(\text{False}\)
Problems
2.7.9.
Answer.
\({\frac{1}{7}}\)
2.7.11.
Answer.
\(-0.5\)
2.7.13.
Answer.
\(-{\frac{25}{4}}\)
2.7.15.
Answer.
\(-\frac{1}{\sqrt{1-\left(4w\right)^{2}}}\cdot 4\)
2.7.17.
Answer.
\(\frac{1}{1+\left(2r\right)^{2}}\cdot 2\)
2.7.19.
Answer.
\(\left(\sec\mathopen{}\left(x\right)\right)^{2}\cos^{-1}\mathopen{}\left(x\right)-\frac{1}{\sqrt{1-x^{2}}}\tan\mathopen{}\left(x\right)\)
2.7.21.
Answer.
\(\frac{\frac{1}{1+z^{2}}\sin^{-1}\mathopen{}\left(z\right)-\frac{1}{\sqrt{1-z^{2}}}\tan^{-1}\mathopen{}\left(z\right)}{\left(\sin^{-1}\mathopen{}\left(z\right)\right)^{2}}\)
2.7.23.
Answer.
\(\csc\mathopen{}\left(\frac{1}{q^{3}}\right)\cot\mathopen{}\left(\frac{1}{q^{3}}\right)\frac{3q^{2}}{\left(q^{3}\right)^{2}}\)
2.7.29.
Answer.
\(y = 2\mathopen{}\left(x-\frac{-\sqrt{3}}{2}\right)+\left(-\frac{\pi }{3}\right)\)
3 The Graphical Behavior of Functions
3.1 Extreme Values
Exercises
Terms and Concepts
3.1.5.
Answer.
\(\text{False}\)
Problems
3.1.7.
Answer 1.
\(\text{B}\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\text{B}, \text{G}\)
Answer 4.
\(\text{C}, \text{F}\)
3.1.9.
Answer.
\(0\)
3.1.11.
Answer 1.
\(0\)
Answer 2.
\(0\)
3.1.13.
Answer 1.
\(\text{DNE}\)
Answer 2.
\(0\)
3.1.15.
Answer.
\(0\)
3.1.17.
Answer 1.
\(14\)
Answer 2.
\(-2\)
3.1.19.
Answer 1.
\(-2.82843\)
Answer 2.
\(-4\)
3.1.21.
Answer 1.
\({\frac{9}{2}}\)
Answer 2.
\(2.82843\)
3.1.23.
Answer 1.
\(\frac{e^{\frac{\pi }{4}}}{\sqrt{2}}\)
Answer 2.
\(-e^{\pi }\)
3.1.25.
Answer 1.
\(\frac{1}{2e}\)
Answer 2.
\(0\)
3.2 The Mean Value Theorem
Exercises
Problems
3.2.3.
Answer.
\(\left(-1,1\right)\)
3.2.5.
Answer.
\(-{\frac{1}{2}}\)
3.2.7.
Answer.
\(\text{does not apply}\)
3.2.9.
Answer.
\(\text{does not apply}\)
3.2.11.
Answer.
\(0\)
3.2.13.
Answer.
\(3\frac{\sqrt{2}}{2}\)
3.2.15.
Answer.
\(\text{does not apply}\)
3.2.17.
Answer.
\(-\sec^{-1}\mathopen{}\left(\frac{2}{\sqrt{\pi }}\right), \sec^{-1}\mathopen{}\left(\frac{2}{\sqrt{\pi }}\right)\)
3.2.19.
Answer.
\(5+7\frac{\sqrt{7}}{6}, 5-7\frac{\sqrt{7}}{6}\)
3.3 Increasing and Decreasing Functions
Exercises
Terms and Concepts
3.3.3.
Answer.
Answers will vary; graphs should be steeper near \(x=0\) than near \(x=2\text{.}\)
3.3.5.
Answer.
\(\text{False}\)
Problems
3.3.15.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(-2\)
Answer 3.
\(\left[-2,\infty \right)\)
Answer 4.
\(\left(-\infty ,-2\right]\)
Answer 5.
\(\text{NONE}\)
Answer 6.
\(-2\)
3.3.17.
Answer 1.
\(\left(-\infty ,\infty \right)\)
Answer 2.
\(-{\frac{5}{7}}, {\frac{7}{3}}\)
Answer 3.
\(\left(-\infty ,-0.714286\right], \left[2.33333,\infty \right)\)
Answer 4.
\(\left[-0.714286,2.33333\right]\)
Answer 5.
\(-{\frac{5}{7}}\)
Answer 6.
\({\frac{7}{3}}\)
3.3.19.
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.
\(\text{NONE}\)
3.3.21.
Answer 1.
\(\left(-\infty ,-7\right)\cup \left(-7,-5\right)\cup \left(-5,\infty \right)\)
Answer 2.
\(-5.91608, 5.91608\)
Answer 3.
\(\left[-5.91608,-5\right), \left(-5,5.91608\right]\)
Answer 4.
\(\left(-\infty ,-7\right), \left(-7,-5.91608\right], \left[5.91608,\infty \right)\)
Answer 5.
\(5.91608\)
Answer 6.
\(-5.91608\)
3.3.23.
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.
\(-0.785398, 2.35619\)
3.4 Concavity and the Second Derivative
3.4.3 Exercises
Terms and Concepts
3.4.3.1.
Answer.
Answers will vary.
3.4.3.3.
Answer.
Yes; Answers will vary.
Problems
3.4.3.15.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\left(-\infty ,\infty \right)\)
Answer 3.
\(\text{NONE}\)
3.4.3.17.
Answer 1.
\(0\)
Answer 2.
\(\left[0,\infty \right)\)
Answer 3.
\(\left(-\infty ,0\right]\)
3.4.3.19.
Answer 1.
\(-{\frac{32}{3}}, 0\)
Answer 2.
\(\left(-\infty ,-10.6667\right], \left[0,\infty \right)\)
Answer 3.
\(\left[-10.6667,0\right]\)
3.4.3.21.
Answer 1.
\(-2\)
Answer 2.
\(\left(-\infty ,\infty \right)\)
Answer 3.
\(\text{NONE}\)
3.4.3.23.
Answer 1.
\(-0.57735, 0.57735\)
Answer 2.
\(\left(-\infty ,-0.57735\right], \left[0.57735,\infty \right)\)
Answer 3.
\(\left[-0.57735,0.57735\right]\)
3.4.3.25.
Answer 1.
\(-0.785398, 2.35619\)
Answer 2.
\(\left(-3.14159,-0.785398\right], \left[2.35619,3.14159\right)\)
Answer 3.
\(\left[-0.785398,2.35619\right]\)
3.4.3.27.
Answer 1.
\(0.22313\)
Answer 2.
\(\left[0.22313,\infty \right)\)
Answer 3.
\(\left(0,0.22313\right]\)
3.4.3.29.
Answer 1.
\(-7\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(-7\)
3.4.3.31.
Answer 1.
\(-1.1547, 1.1547\)
Answer 2.
\(-1.1547\)
Answer 3.
\(1.1547\)
3.4.3.33.
Answer 1.
\(-4\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(-4\)
3.4.3.35.
Answer 1.
\(3\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(\text{NONE}\)
3.4.3.37.
Answer 1.
\(-9\)
Answer 2.
\(-9\)
Answer 3.
\(\text{NONE}\)
3.4.3.39.
Answer 1.
\(-2.35619, 0.785398\)
Answer 2.
\(0.785398\)
Answer 3.
\(-2.35619\)
3.4.3.41.
Answer 1.
\(0.606531\)
Answer 2.
\(\text{NONE}\)
Answer 3.
\(0.606531\)
3.4.3.43.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
3.4.3.45.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(0\)
3.4.3.47.
Answer 1.
\(-{\frac{28}{3}}\)
Answer 2.
\(0\)
3.4.3.49.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(\text{NONE}\)
3.4.3.51.
Answer 1.
\(0\)
Answer 2.
\(2\)
3.4.3.53.
Answer 1.
\(-0.785398\)
Answer 2.
\(2.35619\)
3.4.3.55.
Answer 1.
\(\text{NONE}\)
Answer 2.
\(0.22313\)
3.5 Curve Sketching
Exercises
Terms and Concepts
3.5.3.
Answer.
\(\text{True}\)
3.5.5.
Answer.
\(\text{True}\)
4 Applications of the Derivative
4.1 Newton’s Method
Exercises
Terms and Concepts
4.1.1.
Answer.
\(\text{False}\)
Problems
4.1.3.
Answer 1.
\(1.57091\)
Answer 2.
\(1.5708\)
Answer 3.
\(1.5708\)
Answer 4.
\(1.5708\)
Answer 5.
\(1.5708\)
4.1.5.
Answer 1.
\(2\)
Answer 2.
\(1.2\)
Answer 3.
\(1.01176\)
Answer 4.
\(1.00005\)
Answer 5.
\(1\)
4.1.7.
Answer 1.
\(0.613706\)
Answer 2.
\(0.913341\)
Answer 3.
\(0.996132\)
Answer 4.
\(0.999993\)
Answer 5.
\(1\)
4.1.9.
Answer.
\(\left\{-5.15633,-0.369102,0.525428\right\}\)
4.1.11.
Answer.
\(\left\{-1.0134,0.988312,1.39341\right\}\)
4.1.13.
Answer.
\(\left\{-0.824132,0.824132\right\}\)
4.1.15.
Answer.
\(\left\{0\right\}\)
4.2 Related Rates
Exercises
Terms and Concepts
4.2.1.
Answer.
\(\text{True}\)
Problems
4.2.3.
Answer.
\(0.198944\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
Answer.
\(0.0198944\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
Answer.
\(0.00198944\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.5.
Answer.
\(51.066\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut h}}\)
4.2.7.
Answer.
\(258.537\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
Answer.
\(413.417\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
Answer.
\(424\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
4.2.9.
Answer.
\(0.0417029\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(0.458349\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(3.35489\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(\infty \)
4.2.11.
Answer.
\(19.1658\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(0.191658\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(0.0395988\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(381.791\ {\rm s}\)
4.2.13.
Answer.
\(80\ {\rm ft}\)
Answer.
\(1.71499\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(1.83829\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(74.162\ {\rm ft}\)
4.2.15.
Answer.
\(0.00230973\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.3 Optimization
Exercises
Terms and Concepts
4.3.1.
Answer.
\(\text{True}\)
Problems
4.3.3.
Answer.
\(5625\)
4.3.5.
Answer.
\(\text{DNE}\)
4.3.7.
Answer.
\(1\)
4.3.9.
Answer 1.
\(3.83722\ {\rm cm}\)
Answer 2.
\(7.67443\ {\rm cm}\)
4.3.11.
Answer 1.
\(3.0456\ {\rm cm}\)
Answer 2.
\(12.1824\ {\rm cm}\)
4.3.13.
Answer.
\(10.3923\ {\rm in};\,14.6969\ {\rm in}\)
4.3.15.
Answer 1.
\(0\ {\rm mi}\)
Answer 2.
\(\$474{,}341.65\)
4.3.17.
Answer.
\(23.7599\ {\rm ft}\)
4.4 Differentials
Exercises
Terms and Concepts
4.4.1.
Answer.
\(\text{True}\)
4.4.3.
Answer.
\(\text{False}\)
Problems
4.4.7.
Answer.
\(4.28\)
4.4.9.
Answer.
\(83.2\)
4.4.11.
Answer.
\(5.05\)
4.4.13.
Answer.
\(4.98667\)
4.4.15.
Answer.
\(0.141593\)
4.4.17.
Answer.
\(\left(2x-5\right)dx\)
4.4.19.
Answer.
\(-\frac{24x^{5}}{\left(4x^{6}\right)^{2}}dx\)
4.4.21.
Answer.
\(\left(7x^{6}+8e^{8x}\right)dx\)
4.4.23.
Answer.
\(\frac{9\mathopen{}\left(\tan\mathopen{}\left(x\right)+2\right)-9x\sec^{2}\mathopen{}\left(x\right)}{\left(\tan\mathopen{}\left(x\right)+2\right)^{2}}dx\)
4.4.25.
Answer.
\(\left(e^{x}\sin\mathopen{}\left(x\right)+e^{x}\cos\mathopen{}\left(x\right)\right)dx\)
4.4.27.
Answer.
\(\frac{x+5-\left(x-4\right)}{\left(x+5\right)^{2}}dx\)
4.4.29.
Answer.
\(\tan^{-1}\mathopen{}\left(x\right)dx\)
4.4.31.
Answer.
\(5.02655\ {\rm cm^{3}}\)
4.4.33.
Answer.
\(3.92699\)
4.4.35.
Answer.
\(297.717\ {\rm ft}\)
Answer.
\(62.3155\ {\rm ft}\)
Answer.
\(20.9\%\)
4.4.37.
Answer.
\(298.868\ {\rm ft}\)
Answer.
\(8.66751\ {\rm ft}\)
Answer.
\(2.9\%\)
4.4.39.
Answer.
\(1\%\)
5 Integration
5.1 Antiderivatives and Indefinite Integration
Exercises
Terms and Concepts
5.1.7.
Answer.
\(\text{velocity}\)
Problems
5.1.9.
Answer.
\(\left({\frac{7}{6}}\right)x^{6}+C\)
5.1.11.
Answer.
\(\left({\frac{7}{4}}\right)x^{8}+6x+C\)
5.1.13.
Answer.
\(s+C\)
5.1.15.
Answer.
\(C-\frac{1}{t^{4}}\)
5.1.17.
Answer.
\(\sec\mathopen{}\left(\theta\right)+C\)
5.1.19.
Answer.
\(\sec\mathopen{}\left(x\right)+\csc\mathopen{}\left(x\right)+C\)
5.1.21.
Answer.
\(\frac{9^{t}}{\ln\mathopen{}\left(9\right)}+C\)
5.1.23.
Answer.
\(3t^{3}+3t^{2}+t+\left({\frac{1}{9}}\right)+C\)
5.1.25.
Answer.
\(\frac{x^{15}}{15}+C\)
5.1.27.
Answer.
\(px+C\)
5.1.31.
Answer.
\(8e^{x}+5\)
5.1.33.
Answer.
\(\tan\mathopen{}\left(x\right)+4\)
5.1.35.
Answer.
\(2x^{2}+2x+9\)
5.1.37.
Answer.
\(6e^{x}-9x+2\)
5.1.39.
Answer.
\(\frac{5x^{6}}{6}+\frac{7^{x}}{3.78657}-\sin\mathopen{}\left(x\right)+1.4861x+5.73591\)
5.2 The Definite Integral
Exercises
Terms and Concepts
5.2.3.
Answer.
\(0\)
Problems
5.2.5.
Answer.
\(3\)
Answer.
\(4\)
Answer.
\(3\)
Answer.
\(0\)
Answer.
\(-4\)
Answer.
\(9\)
5.2.7.
Answer.
\(4\)
Answer.
\(2\)
Answer.
\(4\)
Answer.
\(2\)
Answer.
\(1\)
Answer.
\(2\)
5.2.9.
Answer.
\(\pi \)
Answer.
\(\pi \)
Answer.
\(2\pi \)
Answer.
\(10\pi \)
5.2.11.
Answer.
\(-59\)
Answer.
\(-48\)
Answer.
\(-27\)
Answer.
\(-33\)
5.2.13.
Answer.
\(4\)
Answer.
\(4\)
Answer.
\(-4\)
Answer.
\(-2\)
5.2.15.
Answer.
\(2\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(2\ {\rm ft}\)
Answer.
\(1.5\ {\rm ft}\)
5.2.17.
Answer.
\(64\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
Answer.
\(64\ {\rm ft}\)
Answer.
\(2\ {\rm s}\)
Answer.
\(4.64575\ {\rm s}\)
5.2.19.
Answer.
\(2\)
5.2.21.
Answer.
\(16\)
5.2.23.
Answer.
\(22\)
5.2.25.
Answer.
\(0\)
5.3 Riemann Sums
5.3.4 Exercises
Terms and Concepts
5.3.4.1.
Answer.
\(\text{limits}\)
5.3.4.3.
Answer.
\(\text{rectangles}\)
Problems
5.3.4.5.
Answer 1.
\(4+9+16+25\)
Answer 2.
\(54\)
5.3.4.7.
Answer 1.
\(0+\left(-1\right)+0+1\)
Answer 2.
\(0\)
5.3.4.9.
Answer 1.
\(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\)
Answer 2.
\({\frac{137}{60}}\)
5.3.4.11.
Answer 1.
\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}\)
Answer 2.
\({\frac{5}{6}}\)
5.3.4.13.
Answer.
\(1;\,4;\,3i\)
5.3.4.15.
Answer.
\(1;\,4;\,\frac{i}{i+3}\)
5.3.4.17.
Answer.
\(56\)
5.3.4.19.
Answer.
\(946\)
5.3.4.21.
Answer.
\(-29582\)
5.3.4.23.
Answer.
\(3655\)
5.3.4.25.
Answer.
\(195\)
5.3.4.27.
Answer.
\(49\)
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.
\({\frac{1}{4}}\)
5.3.4.37.
Answer 1.
\(12\)
Answer 2.
\(12\)
Answer 3.
\(12\)
Answer 4.
\(12\)
Answer 5.
\(12\)
5.3.4.39.
Answer 1.
\(88-\frac{242}{n}\)
Answer 2.
\(63.8\)
Answer 3.
\(85.58\)
Answer 4.
\(87.758\)
Answer 5.
\(88\)
5.4 The Fundamental Theorem of Calculus
5.4.6 Exercises
Terms and Concepts
5.4.6.3.
Answer.
\(\text{True}\)
Problems
5.4.6.5.
Answer.
\(16\)
5.4.6.7.
Answer.
\(0\)
5.4.6.9.
Answer.
\(\sqrt{3}-1\)
5.4.6.11.
Answer.
\(\frac{\left({\frac{1295}{216}}\right)}{\ln\mathopen{}\left(6\right)}\)
5.4.6.13.
Answer.
\(-8\)
5.4.6.15.
Answer.
\({\frac{128}{3}}\)
5.4.6.17.
Answer.
\({\frac{525}{4}}\)
5.4.6.19.
Answer.
\({\frac{4}{5}}\)
5.4.6.21.
Answer.
\({\frac{1}{2}}\)
5.4.6.23.
Answer.
\({\frac{1}{4}}\)
5.4.6.25.
Answer.
\(16\)
5.4.6.27.
Answer.
\(0\)
5.4.6.31.
Answer.
\(2.3094\)
5.4.6.33.
Answer.
\(0.541325\)
5.4.6.35.
Answer.
\(\frac{\frac{1}{\pi -\frac{\pi }{2}}\cdot 3.14159}{\pi }\)
5.4.6.37.
Answer.
\({\frac{5}{2}}\)
5.4.6.39.
Answer.
\({\frac{343}{4}}\)
5.4.6.41.
Answer.
\(-848\ {\rm ft}\)
5.4.6.43.
Answer.
\(57\ {\rm ft}\)
5.4.6.45.
Answer.
\(2\ {\rm ft}\)
5.4.6.47.
Answer.
\(-192\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.49.
Answer.
\(32\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.55.
Answer.
\(\frac{3x^{2}-7}{x^{3}-7x}\)
5.4.6.57.
Answer.
\(3x^{2}\mathopen{}\left(x^{3}-5\right)-\left(x-5\right)\)
5.4.6.59.
Answer.
\(3x^{2}\sin\mathopen{}\left(4x^{6}\right)\)
5.5 Numerical Integration
5.5.6 Exercises
Terms and Concepts
5.5.6.1.
Answer.
\(\text{False}\)
Problems
5.5.6.5.
Answer.
\(0.75\)
Answer.
\(0.666667\)
Answer.
\(0.666667\)
5.5.6.7.
Answer.
\(1.89612\)
Answer.
\(2.00456\)
Answer.
\(2\)
5.5.6.9.
Answer.
\(38.5781\)
Answer.
\(36.75\)
Answer.
\(36.75\)
5.5.6.11.
Answer.
\(0\)
Answer.
\(0\)
Answer.
\(0\)
5.5.6.13.
Answer.
\(0.900628\)
Answer.
\(0.904523\)
5.5.6.15.
Answer.
\(13.9604\)
Answer.
\(13.9066\)
5.5.6.17.
Answer.
\(1.17029\)
Answer.
\(1.18728\)
5.5.6.19.
Answer.
\(1.08025\)
Answer.
\(1.07699\)
5.5.6.21.
Answer.
\(161\)
Answer.
\(12\)
5.5.6.23.
Answer.
\(994\)
Answer.
\(62\)
5.5.6.25.
Answer 1.
\(30.8667\ {\rm cm^{2}}\)
Answer 2.
\(308667\ {\rm ft^{2}}\)
6 Techniques of Antidifferentiation
6.1 Substitution
6.1.5 Exercises
Terms and Concepts
6.1.5.1.
Answer.
\(\text{the Chain Rule}\)
Problems
6.1.5.3.
Answer.
\({\frac{1}{9}}\mathopen{}\left(x^{4}+3\right)^{9}+C\)
6.1.5.5.
Answer.
\({\frac{1}{20}}\mathopen{}\left(x^{2}+8\right)^{10}+C\)
6.1.5.7.
Answer.
\({\frac{1}{3}}\ln\mathopen{}\left(\left|3x+4\right|\right)+C\)
6.1.5.9.
Answer.
\({\frac{2}{3}}\mathopen{}\left(x+6\right)\sqrt{x-3}+C\)
6.1.5.11.
Answer.
\(2e^{\sqrt{x}}+C\)
6.1.5.13.
Answer.
\(C-{\frac{1}{2}}\mathopen{}\left(\frac{1}{x}+6\right)^{2}\)
6.1.5.15.
Answer.
\(\frac{\left(\sin\mathopen{}\left(x\right)\right)^{3}}{3}+C\)
6.1.5.17.
Answer.
\(C-\frac{\sin\mathopen{}\left(7-7x\right)}{7}\)
6.1.5.19.
Answer.
\({\frac{1}{5}}\ln\mathopen{}\left(\left|\sec\mathopen{}\left(5x\right)+\tan\mathopen{}\left(5x\right)\right|\right)+C\)
6.1.5.21.
Answer.
\(C-{\frac{1}{7}}\cos\mathopen{}\left(x^{7}\right)\)
6.1.5.23.
Answer.
\(\ln\mathopen{}\left(\left|\sin\mathopen{}\left(x\right)\right|\right)+C\)
6.1.5.25.
Answer.
\({\frac{1}{3}}e^{3x+9}+C\)
6.1.5.27.
Answer.
\({\frac{1}{2}}e^{\left(x-3\right)^{2}}+C\)
6.1.5.29.
Answer.
\(\ln\mathopen{}\left(e^{x}+6\right)+C\)
6.1.5.31.
Answer.
\(\frac{8^{8x}}{16.6355}+C\)
6.1.5.33.
Answer.
\(\frac{\ln^{2}\mathopen{}\left(x\right)}{2}+C\)
6.1.5.35.
Answer.
\(\left({\frac{3}{2}}\right)\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)^{2}+C\)
6.1.5.37.
Answer.
\(\frac{x^{2}}{2}-x+6\ln\mathopen{}\left(\left|x\right|\right)+C\)
6.1.5.39.
Answer.
\({\frac{1}{3}}\mathopen{}\left(x+1\right)^{3}+\left({\frac{3}{2}}\right)\mathopen{}\left(x+1\right)^{2}+3\mathopen{}\left(x+1\right)-6\ln\mathopen{}\left(\left|x+1\right|\right)+C\)
6.1.5.41.
Answer.
\(4\mathopen{}\left(x+4\right)^{2}-66\mathopen{}\left(x+4\right)+137\ln\mathopen{}\left(\left|x+4\right|\right)+C\)
6.1.5.43.
Answer.
\(1.73205\tan^{-1}\mathopen{}\left(\frac{x}{1.73205}\right)+C\)
6.1.5.45.
Answer.
\(2\sin^{-1}\mathopen{}\left(\frac{x}{2.44949}\right)+C\)
6.1.5.47.
Answer.
\(\left({\frac{2}{3}}\right)\sec^{-1}\mathopen{}\left(\frac{\left|x\right|}{6}\right)+C\)
6.1.5.49.
Answer.
\(0.301511\tan^{-1}\mathopen{}\left(\frac{x+5}{11}\right)+C\)
6.1.5.51.
Answer.
\(6\sin^{-1}\mathopen{}\left(\frac{x-9}{9}\right)+C\)
6.1.5.53.
Answer.
\(C-\frac{1}{4\mathopen{}\left(x^{4}-5\right)}\)
6.1.5.55.
Answer.
\(\sqrt{5+x^{2}}+C\)
6.1.5.57.
Answer.
\(C-{\frac{2}{3}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{\left({\frac{3}{2}}\right)}\)
6.1.5.59.
Answer.
\(\ln\mathopen{}\left(\left|x+7\right|\right)+C\)
6.1.5.61.
Answer.
\(2x^{2}-7x+\ln\mathopen{}\left(\left|x^{2}-7x-1\right|\right)+C\)
6.1.5.63.
Answer.
\(4\ln\mathopen{}\left(\left|6x-3x^{2}+4\right|\right)+C\)
6.1.5.65.
Answer.
\({\frac{1}{12}}\tan^{-1}\mathopen{}\left(\frac{x^{2}}{6}\right)+C\)
6.1.5.67.
Answer.
\(\sec^{-1}\mathopen{}\left(\left|8x\right|\right)+C\)
6.1.5.69.
Answer.
\(\left({\frac{9}{2}}\right)\ln\mathopen{}\left(\left|x^{2}+18x+130\right|\right)-\left({\frac{5}{7}}\right)\tan^{-1}\mathopen{}\left(\frac{x+9}{7}\right)+C\)
6.1.5.71.
Answer.
\(x+0.774597\tan^{-1}\mathopen{}\left(\frac{x-5}{3.87298}\right)+\left({\frac{9}{2}}\right)\ln\mathopen{}\left(\left|x^{2}-10x+40\right|\right)+C\)
6.1.5.73.
Answer.
\({\frac{1}{2}}x^{2}+2x+\left({\frac{17}{2}}\right)\ln\mathopen{}\left(\left|x^{2}-2x+6\right|\right)+20.1246\tan^{-1}\mathopen{}\left(\frac{x-1}{2.23607}\right)+C\)
6.1.5.75.
Answer.
\(\tan^{-1}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)+C\)
6.1.5.77.
Answer.
\(6\sqrt{x^{2}-14x+48}+C\)
6.1.5.79.
Answer.
\(\ln\mathopen{}\left(\left({\frac{5}{8}}\right)\right)\)
6.1.5.81.
Answer.
\({\frac{1}{5}}\)
6.1.5.83.
Answer.
\({\frac{1}{2}}\mathopen{}\left(e^{4}-e^{4}\right)\)
6.1.5.85.
Answer.
\(\frac{\pi }{2}\)
6.2 Integration by Parts
Exercises
Terms and Concepts
6.2.1.
Answer.
\(\text{True}\)
Problems
6.2.5.
Answer.
\(\sin\mathopen{}\left(x\right)-x\cos\mathopen{}\left(x\right)+C\)
6.2.7.
Answer.
\(-x^{2}\cos\mathopen{}\left(x\right)+2x\sin\mathopen{}\left(x\right)+2\cos\mathopen{}\left(x\right)+C\)
6.2.9.
Answer.
\({\frac{1}{2}}e^{x^{2}}+C\)
6.2.11.
Answer.
\(-{\frac{1}{2}}xe^{-2x}-\frac{e^{-2x}}{4}+C\)
6.2.13.
Answer.
\({\frac{1}{5}}e^{2x}\mathopen{}\left(\sin\mathopen{}\left(x\right)+2\cos\mathopen{}\left(x\right)\right)+C\)
6.2.15.
Answer.
\(\left({\frac{1}{12}}\right)e^{6x}\mathopen{}\left(\sin\mathopen{}\left(6x\right)+\cos\mathopen{}\left(6x\right)\right)+C\)
6.2.17.
Answer.
\(\sqrt{1-x^{2}}+x\sin^{-1}\mathopen{}\left(x\right)+C\)
6.2.19.
Answer.
\(0.5x^{2}\tan^{-1}\mathopen{}\left(x\right)-\frac{x}{2}+0.5\tan^{-1}\mathopen{}\left(x\right)+C\)
6.2.21.
Answer.
\(0.5x^{2}\ln\mathopen{}\left(x\right)-\frac{x^{2}}{4}+C\)
6.2.23.
Answer.
\({\frac{1}{2}}x^{2}\ln\mathopen{}\left(x+1\right)-{\frac{1}{4}}\mathopen{}\left(x+1\right)^{2}+x-\left({\frac{1}{2}}\right)\ln\mathopen{}\left(x+1\right)+C\)
6.2.25.
Answer.
\(0.333333x^{3}\ln\mathopen{}\left(x\right)-\frac{x^{3}}{9}+C\)
6.2.27.
Answer.
\(2\mathopen{}\left(x+7\right)+\left(x+7\right)\mathopen{}\left(\ln\mathopen{}\left(x+7\right)\right)^{2}-2\mathopen{}\left(x+7\right)\ln\mathopen{}\left(x+7\right)+C\)
6.2.29.
Answer.
\(\ln\mathopen{}\left(\left|\sin\mathopen{}\left(x\right)\right|\right)-x\cot\mathopen{}\left(x\right)+C\)
6.2.31.
Answer.
\({\frac{1}{3}}\mathopen{}\left(x^{2}-4\right)^{\left({\frac{3}{2}}\right)}+C\)
6.2.33.
Answer.
\(x\sec\mathopen{}\left(x\right)-\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)+C\)
6.2.35.
Answer.
\(\frac{x}{2}\mathopen{}\left(\sin\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)+\cos\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)\right)+C\)
6.2.37.
Answer.
\(2\cos\mathopen{}\left(\sqrt{x}\right)+2\sqrt{x}\sin\mathopen{}\left(\sqrt{x}\right)+C\)
6.2.39.
Answer.
\(2\sqrt{x}e^{\sqrt{x}}-2e^{\sqrt{x}}+C\)
6.2.41.
Answer.
\(\pi \)
6.2.43.
Answer.
\(0\)
6.2.45.
Answer.
\({\frac{7}{2}}\)
6.2.47.
Answer.
\(\left(-{\frac{7}{4}}\right)e^{-6}-\left(-{\frac{3}{4}}\right)e^{-2}\)
6.2.49.
Answer.
\(0.4\mathopen{}\left(-e^{2\pi }+e^{-2\pi }\right)\)
6.3 Trigonometric Integrals
6.3.4 Exercises
Terms and Concepts
6.3.4.1.
Answer.
\(\text{False}\)
6.3.4.3.
Answer.
\(\text{False}\)
Problems
6.3.4.5.
Answer.
\(-0.2\cos^{5}\mathopen{}\left(x\right)+C\)
6.3.4.7.
Answer.
\({\frac{1}{5}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{5}-{\frac{1}{3}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{3}+C\)
6.3.4.9.
Answer.
\({\frac{1}{9}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{9}-{\frac{2}{7}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{7}+{\frac{1}{5}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{5}+C\)
6.3.4.11.
Answer.
\(\frac{x}{8}-0.03125\sin\mathopen{}\left(4x\right)+C\)
6.3.4.13.
Answer.
\(C-\left(\left({\frac{1}{4}}\right)\cos\mathopen{}\left(2x\right)+\left({\frac{1}{8}}\right)\cos\mathopen{}\left(4x\right)\right)\)
6.3.4.15.
Answer.
\(\frac{1}{8\pi }\sin\mathopen{}\left(4\pi x\right)-\frac{1}{12\pi }\sin\mathopen{}\left(6\pi x\right)+C\)
6.3.4.17.
Answer.
\(\frac{3}{5\pi }\cos\mathopen{}\left(\frac{5\pi }{6}\pi x\right)+\frac{3}{7\pi }\cos\mathopen{}\left(\frac{7\pi }{6}\pi x\right)+C\)
6.3.4.19.
Answer.
\(\frac{\tan^{5}\mathopen{}\left(x\right)}{5}+\frac{\tan^{3}\mathopen{}\left(x\right)}{3}+C\)
6.3.4.21.
Answer.
\({\frac{1}{8}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{8}+C\)
6.3.4.23.
Answer.
\({\frac{1}{12}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{12}-{\frac{1}{5}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{10}+{\frac{1}{8}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{8}+C\)
6.3.4.25.
Answer.
\(0.25\tan\mathopen{}\left(x\right)\sec^{3}\mathopen{}\left(x\right)+0.375\mathopen{}\left(\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)+\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)\right)+C\)
6.3.4.27.
Answer.
\(0.25\tan\mathopen{}\left(x\right)\sec^{3}\mathopen{}\left(x\right)-0.125\mathopen{}\left(\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)+\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)\right)+C\)
6.3.4.29.
Answer.
\(0\)
6.3.4.31.
Answer.
\({\frac{1}{2}}\)
6.3.4.33.
Answer.
\({\frac{2}{5}}\)
6.4 Trigonometric Substitution
Exercises
Terms and Concepts
6.4.1.
Answer.
\(\text{backward}\)
6.4.3.
Answer 1.
\(\tan^{2}\mathopen{}\left(\theta\right)+1 = \sec^{2}\mathopen{}\left(\theta\right)\)
Answer 2.
\(5\sec^{2}\mathopen{}\left(\theta\right)\)
Problems
6.4.5.
Answer.
\({\frac{1}{2}}\mathopen{}\left(x\sqrt{x^{2}+1}+\ln\mathopen{}\left(\sqrt{x^{2}+1}+x\right)\right)+C\)
6.4.7.
Answer.
\({\frac{1}{2}}\sin^{-1}\mathopen{}\left(x\right)+\frac{x}{2}\sqrt{1-x^{2}}+C\)
6.4.9.
Answer.
\({\frac{1}{2}}x\sqrt{x^{2}-1}-{\frac{1}{2}}\ln\mathopen{}\left(\left|x+\sqrt{x^{2}-1}\right|\right)+C\)
6.4.11.
Answer.
\(\frac{x}{2}\sqrt{16x^{2}+1}+{\frac{1}{8}}\ln\mathopen{}\left(4x+\sqrt{16x^{2}+1}\right)+C\)
6.4.13.
Answer.
\(\frac{x}{2}\sqrt{36x^{2}-1}-{\frac{1}{12}}\ln\mathopen{}\left(\left|6x+\sqrt{36x^{2}-1}\right|\right)+C\)
6.4.15.
Answer.
\(7\sin^{-1}\mathopen{}\left(\frac{x}{3.60555}\right)+C\)
6.4.17.
Answer.
\(\sqrt{x^{2}-15}-3.87298\sec^{-1}\mathopen{}\left(\frac{x}{3.87298}\right)+C\)
6.4.19.
Answer.
\(\sqrt{x^{2}-3}+C\)
6.4.21.
Answer.
\(C-\frac{1}{\sqrt{x^{2}+25}}\)
6.4.23.
Answer.
\(\left({\frac{1}{162}}\right)\frac{x-2}{x^{2}-4x+85}+\left({\frac{1}{1458}}\right)\tan^{-1}\mathopen{}\left(\frac{x-2}{9}\right)+C\)
6.4.25.
Answer.
\(C-\left(\frac{\sqrt{5-x^{2}}}{8x}+{\frac{1}{8}}\sin^{-1}\mathopen{}\left(\frac{x}{2.23607}\right)\right)\)
6.4.27.
Answer.
\(\frac{\pi }{2}\)
6.4.29.
Answer.
\(\left({\frac{3}{2}}\right)\sqrt{13}+2\ln\mathopen{}\left(\left({\frac{3}{2}}\right)+\left({\frac{1}{2}}\right)\sqrt{13}\right)\)
6.4.31.
Answer.
\(9\sin^{-1}\mathopen{}\left(\left({\frac{2}{3}}\right)\right)+2\sqrt{5}\)
6.5 Partial Fraction Decomposition
Exercises
Terms and Concepts
6.5.1.
Answer.
\(\text{rational}\)
6.5.3.
Answer.
\(\frac{A}{x}+\frac{B}{x+8}\)
6.5.5.
Answer.
\(\frac{A}{x-\sqrt{3}}+\frac{B}{x+\sqrt{3}}\)
Problems
6.5.7.
Answer.
\(2\ln\mathopen{}\left(\left|x+3\right|\right)+5\ln\mathopen{}\left(\left|x+2\right|\right)+C\)
6.5.9.
Answer.
\(\left({\frac{9}{4}}\right)\ln\mathopen{}\left(\left|x+4\right|\right)-\left({\frac{9}{4}}\right)\ln\mathopen{}\left(\left|x-4\right|\right)+C\)
6.5.11.
Answer.
\(9\ln\mathopen{}\left(\left|x-6\right|\right)-\frac{7}{x-6}+C\)
6.5.13.
Answer.
\(3\ln\mathopen{}\left(\left|x\right|\right)+7\ln\mathopen{}\left(\left|x+8\right|\right)+\frac{8}{x+8}+C\)
6.5.15.
Answer.
\(\left({\frac{1}{4}}\right)\ln\mathopen{}\left(\left|4x+3\right|\right)-\left({\frac{3}{7}}\right)\ln\mathopen{}\left(\left|7x+10\right|\right)+\frac{\left({\frac{2}{5}}\right)}{5x-60}+C\)
6.5.17.
Answer.
\({\frac{1}{2}}x^{2}+7x+\left({\frac{1}{9}}\right)\ln\mathopen{}\left(\left|x+1\right|\right)+\left({\frac{512}{9}}\right)\ln\mathopen{}\left(\left|x-8\right|\right)+C\)
6.5.19.
Answer.
\(\left({\frac{1}{6}}\right)\ln\mathopen{}\left(\left|x\right|\right)-\left({\frac{1}{12}}\right)\ln\mathopen{}\left(x^{2}-4x+6\right)+0.235702\tan^{-1}\mathopen{}\left(\frac{x-2}{1.41421}\right)+C\)
6.5.21.
Answer.
\(\ln\mathopen{}\left(\left|9x^{2}-x+9\right|\right)+\ln\mathopen{}\left(\left|x-1\right|\right)+C\)
6.5.23.
Answer.
\(\left({\frac{343}{40}}\right)\ln\mathopen{}\left(\left|x-6\right|\right)-\left({\frac{23}{80}}\right)\ln\mathopen{}\left(x^{2}+4\right)+\left({\frac{111}{40}}\right)\tan^{-1}\mathopen{}\left(\frac{x}{2}\right)+C\)
6.5.25.
Answer.
\(\ln\mathopen{}\left(x^{2}+2x+4\right)-\ln\mathopen{}\left(\left|x-3\right|\right)+2.88675\tan^{-1}\mathopen{}\left(\frac{x+1}{1.73205}\right)+C\)
6.5.27.
Answer.
\(\ln\mathopen{}\left(\left({\frac{3125}{524288}}\right)\right)\)
6.5.29.
Answer.
\(\ln\mathopen{}\left(\left({\frac{2}{3}}\right)\right)+\tan^{-1}\mathopen{}\left(-2\right)-\tan^{-1}\mathopen{}\left(-4\right)\)
6.6 Hyperbolic Functions
6.6.3 Exercises
Problems
6.6.3.11.
Answer.
\(2\cosh\mathopen{}\left(2x\right)\)
6.6.3.13.
Answer.
\(\mathop{\rm sech}\nolimits^{2}\mathopen{}\left(x^{2}\right)\cdot 2x\)
6.6.3.15.
Answer.
\(\cosh\mathopen{}\left(x\right)\cosh\mathopen{}\left(x\right)+\sinh\mathopen{}\left(x\right)\sinh\mathopen{}\left(x\right)\)
6.6.3.17.
Answer.
\(-\frac{1}{x^{2}\sqrt{1-\left(x^{2}\right)^{2}}}\cdot 2x\)
6.6.3.19.
Answer.
\(\frac{1}{\sqrt{\left(2x^{2}\right)^{2}-1}}\cdot 2\cdot 2x\)
6.6.3.21.
Answer.
\(-\frac{1}{1-\cos^{2}\mathopen{}\left(x\right)}\sin\mathopen{}\left(x\right)\)
6.6.3.23.
Answer.
\(1\mathopen{}\left(x-0\right)+0\)
6.6.3.25.
Answer.
\(0.36\mathopen{}\left(x-\left(-1.09861\right)\right)+\left(-0.8\right)\)
6.6.3.27.
Answer.
\(1\mathopen{}\left(x-0\right)+0\)
6.6.3.29.
Answer.
\(0.5\ln\mathopen{}\left(\cosh\mathopen{}\left(2x\right)\right)+C\)
6.6.3.31.
Answer.
\(0.5\sinh^{2}\mathopen{}\left(x\right)+C\)
6.6.3.33.
Answer.
\(x\cosh\mathopen{}\left(x\right)-\sinh\mathopen{}\left(x\right)+C\)
6.6.3.35.
Answer.
\(\cosh^{-1} x/3 +C=\ln\big(x+\sqrt{x^2-9}\big)+C\)
6.6.3.37.
Answer.
\(\cosh^{-1}\mathopen{}\left(\frac{x^{2}}{2}\right)+C\)
6.6.3.39.
Answer.
\(-0.0625\tan^{-1}\mathopen{}\left(\frac{x}{2}\right)+0.03125\ln\mathopen{}\left(\left|x-2\right|\right)-0.03125\ln\mathopen{}\left(\left|x+2\right|\right)+C\)
6.6.3.41.
Answer.
\(\tan^{-1}\mathopen{}\left(e^{x}\right)+C\)
6.6.3.43.
Answer.
\(x\tanh^{-1}\mathopen{}\left(x\right)+0.5\ln\mathopen{}\left(\left|x^{2}-1\right|\right)+C\)
6.6.3.45.
Answer.
\(0\)
6.6.3.47.
Answer.
\(0.761594\)
6.7 L’Hospital’s Rule
6.7.4 Exercises
Terms and Concepts
6.7.4.3.
Answer.
\(\text{False}\)
Problems
6.7.4.9.
Answer.
\(3\)
6.7.4.11.
Answer.
\(-1\)
6.7.4.13.
Answer.
\(5\)
6.7.4.15.
Answer.
\(0.666667\)
6.7.4.17.
Answer.
\(\infty \)
6.7.4.19.
Answer.
\(0\)
6.7.4.21.
Answer.
\(0\)
6.7.4.23.
Answer.
\(\infty \)
6.7.4.25.
Answer.
\(0\)
6.7.4.27.
Answer.
\(-2\)
6.7.4.29.
Answer.
\(0\)
6.7.4.31.
Answer.
\(0\)
6.7.4.33.
Answer.
\(\infty \)
6.7.4.35.
Answer.
\(\infty \)
6.7.4.37.
Answer.
\(0\)
6.7.4.39.
Answer.
\(1\)
6.7.4.41.
Answer.
\(1\)
6.7.4.43.
Answer.
\(1\)
6.7.4.45.
Answer.
\(1\)
6.7.4.47.
Answer.
\(1\)
6.7.4.49.
Answer.
\(2\)
6.7.4.51.
Answer.
\(-\infty \)
6.7.4.53.
Answer.
\(0\)
6.8 Improper Integration
6.8.4 Exercises
Terms and Concepts
6.8.4.5.
Answer.
\(p\gt 1\)
Problems
6.8.4.7.
Answer.
\(\frac{e^{5}}{2}\)
6.8.4.9.
Answer.
\(\frac{1}{3}\)
6.8.4.11.
Answer.
\(\frac{1}{\ln\mathopen{}\left(2\right)}\)
6.8.4.13.
Answer.
\(\infty \)
6.8.4.15.
Answer.
\(1\)
6.8.4.17.
Answer.
\(\infty \)
6.8.4.19.
Answer.
\(\infty \)
6.8.4.21.
Answer.
\(\infty \)
6.8.4.23.
Answer.
\(1\)
6.8.4.25.
Answer.
\(0\)
6.8.4.27.
Answer.
\(\frac{-1}{4}\)
6.8.4.29.
Answer.
\(\infty \)
6.8.4.31.
Answer.
\(1\)
6.8.4.33.
Answer.
\(\frac{1}{2}\)
6.8.4.35.
Answer 1.
\(\text{Limit Comparison Test}\)
Answer 2.
\(\text{diverges}\)
Answer 3.
\(\frac{1}{x}\)
6.8.4.37.
Answer 1.
\(\text{Limit Comparison Test}\)
Answer 2.
\(\text{diverges}\)
Answer 3.
\(\frac{1}{x}\)
6.8.4.39.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(e^{-x}\)
6.8.4.41.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(\frac{1}{x^{2}-1}\)
6.8.4.43.
Answer 1.
\(\text{Direct Comparison Test}\)
Answer 2.
\(\text{converges}\)
Answer 3.
\(\frac{1}{e^{x}}\)
7 Applications of Integration
7.1 Area Between Curves
Exercises
Terms and Concepts
7.1.1.
Answer.
\(\text{True}\)
Problems
7.1.5.
Answer.
\(22.436\)
7.1.7.
Answer.
\(3.14159\)
7.1.9.
Answer.
\(0.5\)
7.1.11.
Answer.
\(0.721354\)
7.1.13.
Answer.
\(4.5\)
7.1.15.
Answer.
\(0.429204\)
7.1.17.
Answer.
\(0.166667\)
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 \(2\sqrt{2}\text{.}\)
7.1.21.
Answer.
\(1\)
7.1.23.
Answer.
\(4.5\)
7.1.25.
Answer.
\(0.514298\)
7.1.27.
Answer.
\(1\)
7.1.29.
Answer.
\(4\)
7.1.31.
Answer.
\(262800\ {\rm ft^{2}}\)
7.2 Volume by Cross-Sectional Area; Disk and Washer Methods
Exercises
Terms and Concepts
7.2.1.
Answer.
T
Problems
7.2.5.
Answer.
\(175\pi/3\) units\(^3\)
7.2.7.
Answer.
\(\pi/6\) units\(^3\)
7.2.9.
Answer.
\(35\pi/3\) units\(^3\)
7.2.11.
Answer.
\(2\pi/15\) units\(^3\)
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.
\(128\pi/3\)
7.2.15.
7.2.15.a
Answer.
\(104\pi/15\)
7.2.15.b
Answer.
\(64\pi/15\)
7.2.15.c
Answer.
\(32\pi/5\)
7.2.17.
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.
\(8\pi/3\)
7.2.19.
Answer.
\(250\pi/3\)
7.2.21.
Answer.
\(187.5\)
7.3 The Shell Method
Exercises
Terms and Concepts
7.3.1.
Answer.
T
7.3.3.
Answer.
F
Problems
7.3.5.
Answer.
\(9\pi/2\) units\(^3\)
7.3.7.
Answer.
\(\pi^2-2\pi\) units\(^3\)
7.3.9.
Answer.
\(48\pi\sqrt{3}/5\) units\(^3\)
7.3.11.
Answer.
\(\pi^2/4\) 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.
\(5\pi/6\)
7.3.15.
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.
\(2\pi/3\)
7.3.17.
7.3.17.a
Answer.
\(2\pi(\sqrt{2}-1)\)
7.3.17.b
Answer.
\(2\pi(1-\sqrt{2}+\sinh^{-1}(1))\)
7.4 Arc Length and Surface Area
7.4.3 Exercises
Problems
7.4.3.3.
Answer.
\(\sqrt{2}\)
7.4.3.5.
Answer.
\(\frac{10}{3}\)
7.4.3.7.
Answer.
\(\frac{157}{3}\)
7.4.3.9.
Answer.
\(\frac{12}{5}\)
7.4.3.11.
Answer.
\(-\ln(2-\sqrt{3}) \approx 1.31696\)
7.4.3.13.
Answer.
\(\int_0^1 \sqrt{1+4x^2}\, dx\)
7.4.3.15.
Answer.
\(\int_1^e \sqrt{1+\frac1{x^2}}\, dx\)
7.4.3.17.
Answer.
\(\int_0^{\pi/2}\sqrt{1+\sin^2(x)}\,dx\)
7.4.3.19.
Answer.
\(1.4790\)
7.4.3.21.
Answer.
\(2.1300\)
7.4.3.23.
Answer.
\(1.00013\)
7.4.3.25.
Answer.
\(2\pi\int_0^1 2x\sqrt{5}\, dx = 2\pi\sqrt{5}\)
7.4.3.27.
Answer.
\(2\pi\int_0^1 x\sqrt{1+4x^2}\, dx = \pi/6(5\sqrt{5}-1)\)
7.4.3.29.
Answer.
\(\int_0^1 \sqrt{1+\frac{1}{4x}}\, dx\)
7.4.3.31.
Answer.
\(\int_{-3}^3 \sqrt{1+\frac{x^2}{81-9x^2}}\, dx\)
7.4.3.33.
Answer.
\(2\pi\int_0^1 \sqrt{1-x^2}\sqrt{1+x/(1-x^2)}\, dx = 4\pi\)
7.5 Work
7.5.4 Exercises
Terms and Concepts
7.5.4.1.
Answer.
In SI units, it is one joule, i.e., one newton–meter, or kg·m⁄s2m In Imperial Units, it is ft–lb.
7.5.4.3.
Answer.
Smaller.
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
7.5.4.7.
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.
\(\ell(1-\sqrt{2}/2) \approx 0.2929\ell\)
7.5.4.9.
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.
7.5.4.11.
Answer.
575 ft–lb
7.5.4.13.
Answer.
0.05 J
7.5.4.15.
Answer.
5/3 ft–lb
7.5.4.17.
Answer.
\(f\cdot d/2\) J
7.5.4.19.
Answer.
5 ft–lb
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.
7.5.4.23.
Answer.
212,135 ft–lb
7.5.4.25.
Answer.
187,214 ft–lb
7.5.4.27.
Answer.
4,917,150 J
7.6 Fluid Forces
Exercises
Terms and Concepts
7.6.1.
Answer.
Answers will vary.
Problems
7.6.3.
Answer.
499.2 lb
7.6.5.
Answer.
6739.2 lb
7.6.7.
Answer.
3920.7 lb
7.6.9.
Answer.
2496 lb
7.6.11.
Answer.
602.59 lb
7.6.13.
Answer.
- 2340 lb
- 5625 lb
7.6.15.
Answer.
- 1597.44 lb
- 3840 lb
7.6.17.
Answer.
- 56.42 lb
- 135.62 lb
7.6.19.
Answer.
5.1 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.
8.1.4.5.
Answer.
Many differential equations are impossible to solve analytically.
Problems
8.1.4.7.
Answer.
Answers will vary.
8.1.4.9.
Answer.
Answers will vary.
8.1.4.11.
Answer.
\(C = 2\)
8.1.4.13.
Answer.
The \(x\) and \(y\) axes are uncalibrated.In the first quadrant in the top left, the field lines are north-east facing and in the bottom right they are southeast facing. In the second quadrant the field lines are all north-east facing. In the third quadrant like in the first quadrant in the top left the field lines are northeast facing and in the bottom right they are southeast facing. In the fourth quadrant all lines are southeast facing.
8.1.4.15.
Answer.
The \(x\) and \(y\) axes are uncalibrated. There are five instances where the field lines run parallel to the \(x\) axis. One of them is on the \(x\) axis itself, other two pairs of such field lines are above and below the \(x\) axis. In between the \(x\) axis and the first horizontal field line for some positive \(y\) value, the field lines are all northeast facing. Above the horizontal field line for some \(y\) value until another with a higher \(y\) value, the field lines in between are southeast facing.
Similarly below the \(x\) axis till the first horizontal line with some negative \(y\) value, the field lines in between are southeast facing. In between this horizontal line and another horizontal line with a higher negative \(y\) value, the field lines are northeast facing.
8.1.4.17.
Answer.
b
8.1.4.19.
Answer.
d
8.1.4.21.
Answer.
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.
8.1.4.23.
Answer.
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.
8.1.4.25.
Answer.
\begin{align*}
x_i \amp \quad \amp \quad \amp y_i\\
0.00 \amp \quad \amp \quad \amp 1.0000 \\
0.25 \amp \quad \amp \quad \amp 1.5000 \\
0.50 \amp \quad \amp \quad \amp 2.3125 \\
0.75 \amp \quad \amp \quad \amp 3.5938\\
1.00 \amp \quad \amp \quad \amp 5.5781
\end{align*}
8.1.4.27.
Answer.
\begin{align*}
x_i \amp \quad \amp \quad \amp y_i \\
0.0 \amp \quad \amp \quad \amp 2.0000 \\
0.2 \amp \quad \amp \quad \amp 2.4000 \\
0.4 \amp \quad \amp \quad \amp 2.9197 \\
0.6 \amp \quad \amp \quad \amp 3.5816 \\
0.8 \amp \quad \amp \quad \amp 4.4108 \\
1.0 \amp \quad \amp \quad \amp 5.4364
\end{align*}
8.1.4.29.
Answer.
| \(x\) | \(0.0\) | \(0.2\) | \(0.4\) | \(0.6\) | \(0.8\) | \(1.0\) |
| \(y(x)\) | 1.0000 | 1.0204 | 1.0870 | 1.2195 | 1.4706 | 2.0000 |
| \(h = 0.2\) | 1.0000 | 1.0000 | 1.0400 | 1.1265 | 1.2788 | 1.5405 |
| \(h = 0.1\) | 1.0000 | 1.0100 | 1.0623 | 1.1687 | 1.3601 | 1.7129 |
8.2 Separable Differential Equations
8.2.2 Exercises
Problems
8.2.2.1.
Answer.
Separable. \(\displaystyle \frac{1}{y^2-y}\,dy = dx\)
8.2.2.3.
Answer.
Not separable.
8.2.2.5.
Answer.
\(\left\{ \displaystyle y = \frac{1 + Ce^{2x}}{1 - Ce^{2x}}, y = -1\right\}\)
8.2.2.7.
Answer.
\(y = Cx^4\)
8.2.2.9.
Answer.
\(\displaystyle (y-1)e^y = -e^{-x} - \frac{1}{3}e^{-3x} + C\)
8.2.2.11.
Answer.
\(\left\{ \arcsin{2y} - \arctan(x^2+1) = C, y = \pm \displaystyle \frac{1}{2} \right\}\)
8.2.2.13.
Answer.
\(\sin y + \cos(x) = 2\)
8.2.2.15.
Answer.
\(\frac{1}{2}y^2 - \ln(1+x^2) = 8\)
8.2.2.17.
Answer.
\(\displaystyle \frac{1}{2}y^2 - y = \frac{1}{2}\big ( (x^2+1)\ln(x^2+1) - (x^2 + 1)\big) + \frac{1}{2}\)
8.2.2.19.
Answer.
\(2\tan 2y = 2x + \sin 2x\)
8.3 First Order Linear Differential Equations
8.3.2 Exercises
Problems
8.3.2.1.
Answer.
\(y = \displaystyle \frac{3}{2} + Ce^{2x}\)
8.3.2.3.
Answer.
\(y = \displaystyle -\frac{1}{2x} + Cx\)
8.3.2.5.
Answer.
\(y = \sec x + C(\csc x)\)
8.3.2.7.
Answer.
\(y = \displaystyle Ce^{3x}-(x+1)e^{2x}\)
8.3.2.9.
Answer.
\(y = (x^2+2)e^x\)
8.3.2.11.
Answer.
\(y = \displaystyle 1 - \frac{2}{x} + \frac{2-e^{1-x}}{x^2}\)
8.3.2.13.
Answer.
\(y = \displaystyle \frac{x^2+1}{x+1}e^{-x}\)
8.3.2.15.
Answer.
\(y = \displaystyle \frac{(x-2)(x+1)}{x-1}\)
8.3.2.17.
Answer.
Both; \(\displaystyle y = -5e^{x + \frac{1}{3}x^3}\)
8.3.2.19.
Answer.
linear; \(\displaystyle y = \frac{x^3-3x-6}{3(x-1)}\)
8.3.2.21.
Answer.
The \(x\) and \(y\) axes are uncalibrated, the field lines in the first quadrant are shown. On the bottom right the field lines are facing northeast. On the top left the field lines transition from southeast facing to east facing moving downwards. A curve is shown that almost represents a straight line with a positive slope.
The solution will increase and begin to follow the line \(y=x-1\text{.}\)
\(y = x-1 + e^{-x}\)
8.4 Modeling with Differential Equations
8.4.3 Exercises
Problems
8.4.3.1.
Answer.
\(y = 10 + Ce^{-kx}\)
8.4.3.3.
Answer.
4.43 days
8.4.3.5.
Answer.
\(x = \begin{cases}\displaystyle\frac{ab(1 - e^{(a-b)kt})}{b-ae^{(a-b)kt}} \amp \text{ if } a \neq b\\ \displaystyle \frac{a^2kt}{1+akt} \amp \text{ if } a = b \end{cases}\)
8.4.3.7.
Answer.
\(\displaystyle y = 60 - 3.69858e^{-\frac{1}{4}t} + 43.69858e^{-0.0390169 t}\)
8.4.3.9.
Answer.
\(y = 8(1-e^{-\frac{1}{2}t})\) g/cm\(^2\)
8.4.3.11.
Answer.
11.00075 g
9 Sequences and Series
9.1 Sequences
Exercises
Terms and Concepts
9.1.1.
Answer.
Answers will vary.
9.1.3.
Answer.
Answers will vary.
Problems
9.1.5.
Answer.
\(2,\frac{8}{3},\frac{8}{3},\frac{32}{15},\frac{64}{45}\)
9.1.7.
Answer.
\(-\frac{1}{3},-2,-\frac{81}{5},-\frac{512}{3},-\frac{15625}{7}\)
9.1.9.
Answer.
\(a_n = 3n+1\)
9.1.11.
Answer.
\(a_n = 10\cdot 2^{n-1}\)
9.1.13.
Answer.
\(1/7\)
9.1.15.
Answer.
\(0\)
9.1.17.
Answer.
diverges
9.1.19.
Answer.
converges to \(0\)
9.1.21.
Answer.
diverges
9.1.23.
Answer.
converges to \(e\)
9.1.25.
Answer.
converges to 0
9.1.27.
Answer.
converges to 2
9.1.29.
Answer.
bounded
9.1.31.
Answer.
bounded
9.1.33.
Answer.
neither bounded above or below
9.1.35.
Answer.
monotonically increasing
9.1.37.
Answer.
never monotonic
9.2 Infinite Series
9.2.4 Exercises
Terms and Concepts
9.2.4.1.
Answer.
Answers will vary.
9.2.4.5.
Answer.
F
9.3 Integral and Comparison Tests
9.3.4 Exercises
Terms and Concepts
9.3.4.1.
Answer.
continuous, positive and decreasing
Problems
9.3.4.5.
Answer.
Converges
9.3.4.7.
Answer.
Diverges
9.3.4.9.
Answer.
Converges
9.3.4.11.
Answer.
Converges
9.4 Ratio and Root Tests
9.4.3 Exercises
Terms and Concepts
9.4.3.1.
Answer.
algebraic, or polynomial.
9.4.3.3.
Answer.
Integral Test, Limit Comparison Test, and Root Test
Problems
9.4.3.5.
Answer.
Converges
9.4.3.7.
Answer.
Converges
9.4.3.9.
Answer.
The Ratio Test is inconclusive; the \(p\)-Series Test states it diverges.
9.4.3.11.
Answer.
Converges
9.4.3.13.
Answer.
Converges; note the summation can be rewritten as \(\ds\infser \frac{2^nn!}{3^nn!}\text{,}\) from which the Ratio Test or Geometric Series Test can be applied.
9.4.3.15.
Answer.
Converges
9.4.3.17.
Answer.
Converges
9.4.3.19.
Answer.
Diverges
9.4.3.21.
Answer.
Diverges. The Root Test is inconclusive, but the \(n\)th-Term Test shows divergence. (The terms of the sequence approach \(e^{-2}\text{,}\) not 0, as \(n\to\infty\text{.}\))
9.4.3.23.
Answer.
Converges
9.5 Alternating Series and Absolute Convergence
Exercises
Terms and Concepts
9.5.3.
Answer.
Many examples exist; one common example is \(a_n = (-1)^n/n\text{.}\)
9.7 Taylor Polynomials
Exercises
Terms and Concepts
9.7.3.
Answer.
\(6+3x-4x^{2}\)
Problems
9.7.5.
Answer.
\(1-x+0.5x^{2}-0.166667x^{3}\)
9.7.7.
Answer.
\(x+x^{2}+0.5x^{3}+0.166667x^{4}+0.0416667x^{5}\)
9.7.9.
Answer.
\(1+2x+2x^{2}+1.33333x^{3}+0.666667x^{4}\)
9.7.11.
Answer.
\(1-x+x^{2}-x^{3}+x^{4}\)
9.7.13.
Answer.
\(1+0.5\mathopen{}\left(x-1\right)-0.125\mathopen{}\left(x-1\right)^{2}+0.0625\mathopen{}\left(x-1\right)^{3}-0.0390625\mathopen{}\left(x-1\right)^{4}\)
9.7.15.
Answer.
\(0.707107-0.707107\mathopen{}\left(x-\frac{\pi }{4}\right)-0.353553\mathopen{}\left(x-\frac{\pi }{4}\right)^{2}+0.117851\mathopen{}\left(x-\frac{\pi }{4}\right)^{3}+0.0294628\mathopen{}\left(x-\frac{\pi }{4}\right)^{4}-0.00589256\mathopen{}\left(x-\frac{\pi }{4}\right)^{5}-0.000982093\mathopen{}\left(x-\frac{\pi }{4}\right)^{6}\)
9.7.17.
Answer.
\(0.5-0.25\mathopen{}\left(x-2\right)+0.125\mathopen{}\left(x-2\right)^{2}-0.0625\mathopen{}\left(x-2\right)^{3}+0.03125\mathopen{}\left(x-2\right)^{4}+0.015625\mathopen{}\left(x-2\right)^{5}\)
9.7.19.
Answer.
\(0.5+0.5\mathopen{}\left(x+1\right)+0.25\mathopen{}\left(x+1\right)^{2}\)
9.7.31.
Answer.
The \(n\)th term is: when \(n\) even, 0; when \(n\) is odd, \(\frac{(-1)^{(n-1)/2}}{n!}x^n\text{.}\)
10 Curves in the Plane
10.1 Conic Sections
10.1.4 Exercises
Problems
10.1.4.19.
Answer.
\(\frac{(x+1)^2}{9}+\frac{(y-2)^2}{4}=1\text{;}\) foci at \((-1\pm\sqrt{5},2)\text{;}\) \(e=\sqrt{5}/3\)
10.1.4.29.
Answer.
\(x^2-\frac{y^2}{3}=1\)
10.1.4.31.
Answer.
\(\frac{(y-3)^2}{4}-\frac{(x-1)^2}{9}=1\)
10.1.4.45.
Answer.
The sound originated from a point approximately 31m to the right of \(B\) and 1390m above or below it. (Since the three points are collinear, we cannot distinguish whether the sound originated above/below the line containing the points.)
10.2 Parametric Equations
10.2.4 Exercises
Terms and Concepts
10.2.4.1.
Answer.
\(\text{True}\)
10.2.4.3.
Answer.
\({\text{rectangular}}\)
Problems
10.2.4.5.
Answer.
The sketch for this exercise is a curve that lies mostly in the fourth quadrant. It resembles part of a slingshot orbit for a comet passing around the sun: the curve passes through the origin from below, turns quickly in the second quadrant, crossing the \(y\) axis at \((0,1)\text{,}\) and then the \(x\) axis at \((2,0)\text{,}\) where it returns to the fourth quadrant.
10.2.4.7.
Answer.
The horizontal line \(y=2\text{.}\) On the line there are two arrows pointing in opposite directions. These indicate that the direction of travel is to the left when \(t\lt 0\text{,}\) and to the right when \(t\gt 0\text{.}\)
10.2.4.9.
Answer.
A curve resembling a check mark, with a cusp at the origin. Direction of travel is from the second quadrant toward the cusp, and then up from the cusp to a \(y\) intercept at \((0,4)\text{,}\) and then into the first quadrant.
10.2.4.11.
Answer.
The curve is an ellipse, centered at the origin, with counter-clockwise direction of travel.
10.2.4.13.
Answer.
The curve resembles a parabola, with vertex at \((0,-1)\text{.}\) The direction of travel is from right to left.
10.2.4.15.
Answer.
The curve resembles one branch of a hyperbola, opening to the right, with a vertex at \((2,0)\text{.}\) The direction of travel is that of increasing \(y\) value.
10.2.4.17.
Answer.
A flower-shaped curve, with 7 “petals”. Each petal is an arc that loops around and intersects itself before continuing to the next arc.
10.2.4.19.
10.2.4.19.a
Answer.
Traces the parabola \(y=x^2\text{,}\) moves from left to right.
10.2.4.19.b
Answer.
Traces the parabola \(y=x^2\text{,}\) but only from \(-1\leq x\leq 1\text{;}\) traces this portion back and forth infinitely.
10.2.4.19.c
Answer.
Traces the parabola \(y=x^2\text{,}\) but only for \(0\lt x\text{.}\) Moves left to right.
10.2.4.19.d
Answer.
Traces the parabola \(y=x^2\text{,}\) moves from right to left.
10.2.4.21.
Answer.
\(3x+2y = 17\)
10.2.4.25.
Answer.
\(y-2x = 3\)
10.2.4.35.
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.
\(1\)
10.2.4.37.
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.
\(1\)
10.2.4.39.
Answer 1.
\(-1, 1\)
Answer 2.
\(\left(3,-2\right)\)
10.2.4.51.
Answer.
\(3\cos\mathopen{}\left(2\pi t\right)+1;\,3\sin\mathopen{}\left(2\pi t\right)+1\)
10.3 Calculus and Parametric Equations
10.3.4 Exercises
Terms and Concepts
10.3.4.1.
Answer.
\(\text{False}\)
10.3.4.3.
Answer.
\(\text{False}\)
Problems
10.3.4.15.
Answer 1.
\(-0.5\)
Answer 2.
\(\left(0.75,-0.25\right)\)
10.3.4.21.
Answer 1.
\(0\)
Answer 2.
\(0\)
10.3.4.27.
Answer 1.
\(-\frac{4}{\left(2t-1\right)^{3}}\)
Answer 2.
\(\left(-\infty ,0.5\right]\)
Answer 3.
\(\left[0.5,\infty \right)\)
10.3.4.33.
Answer.
\(6\pi \)
10.3.4.35.
Answer.
\(2\sqrt{34}\)
10.4 Introduction to Polar Coordinates
10.4.4 Exercises
Terms and Concepts
10.4.4.1.
Answer.
Answers will vary.
10.4.4.3.
Answer.
\(\text{True}\)
Problems
10.4.4.5.
Answer.
On a polar grid, four points are plotted. The point \(A\) is at the intersection of the initial ray and the circle of radius 2. Points \(B\) and \(D\) are both on the circle of radius 1. The point \(B\) is on the same line as the initial ray, but in the opposite direction. The point \(D\) lies above the initial ray, making an angle of \(\pi/4\text{.}\) Finally, the point \(C\) is at the bottom of the circle of radius \(2\text{.}\)
10.4.4.7.
Answer.
\(A=P(2.5,\pi/4)\) and \(P(-2.5,5\pi/4)\text{;}\)
\(B=P(-1,5\pi/6)\) and \(P(1,11\pi/6)\text{;}\)
\(C=P(3,4\pi/3)\) and \(P(-3,\pi/3)\text{;}\)
\(D=P(1.5,2\pi/3)\) and \(P(-1.5,5\pi/3)\text{;}\)
10.4.4.9.
Answer 1.
\(\left(\sqrt{2},\sqrt{2}\right)\)
Answer 2.
\(\left(\sqrt{2},-\sqrt{2}\right)\)
Answer 3.
\(\left(\sqrt{5},\tan^{-1}\mathopen{}\left(\frac{-1}{2}\right)\right)\)
Answer 4.
\(\left(\sqrt{5},\pi +\tan^{-1}\mathopen{}\left(\frac{-1}{2}\right)\right)\)
10.4.4.11.
Answer.
An arc of the circle \(r=2\) is shown, for \(0\leq \theta\leq \pi/2\text{.}\) This is the quarter of a circle of radius 2, centered at the origin, that lies in the first quadrant.
10.4.4.13.
Answer.
The curve is a cardioid that is symmetric about the \(x\) axis. The cusp is at the origin, and the other \(x\) intercept is at \((-2,0)\text{.}\) (It is in the opposite direction of the example in the gallery of polar curves.)
10.4.4.15.
Answer.
The curve is a convex limaçon. This is the fourth type of limaçon in the gallery of polar curves. In this case, the limaçon is symmetric about the \(y\) axis, with the flattened part of the curve at the top.
10.4.4.17.
Answer.
The curve is a limaçon with an inner loop. It is symmetric about the \(y\) axis. The inner loop lies above the \(x\) axis, with \(y\) intercepts at \(y=0\) and \(y=1\text{.}\) The outer loop has its other \(y\) intercept at \(y=3\text{.}\)
10.4.4.19.
Answer.
A rose curve with three loops that all pass through the origin. One loop is along the negative \(y\) axis, with a \(y\) intercept at \((-1,0)\text{.}\) The other two loops lie in the first and second quadrants.
10.4.4.21.
Answer.
This is a more complicated curve. It passes several times through the origin, and has eight other points of self-intersection. The largest loops in the curve are similar to cardioids; there are four of these passing through the origin, with a second intercept at one of the four points \((\pm 1, 0)\text{,}\) \((0,\pm 1)\text{.}\) As these loops intersect each other, they create four other loops of intermediate size, and four smaller loops in the center.
10.4.4.23.
Answer.
A circle of radius \(3/2\) with its center at \((0,3/2)\text{.}\) It passes through the origin and the point \((0,3)\text{.}\)
10.4.4.25.
Answer.
The curve is a four-leafed rose that lies within the circle \(r=1/2\text{.}\) One leaf lies in each of the four quadrants.
10.4.4.27.
Answer.
The curve is a straight line with \(x\) intercept at \((-3,0)\) and \(y\) intercept \((0,3/5)\text{.}\)
10.4.4.29.
Answer.
The curve is the vertical line \(x=3\text{.}\)
10.4.4.31.
Answer.
\(\left(x-3\right)^{2}+y^{2} = 9\)
10.4.4.33.
Answer.
\(\left(x-0.5\right)^{2}+\left(y-0.5\right)^{2} = 0.5\)
10.4.4.35.
Answer.
\(x = 3\)
10.4.4.39.
Answer.
\(x^{2}+y^{2} = 4\)
10.4.4.41.
Answer.
\(\theta = \frac{\pi }{4}\)
10.4.4.43.
Answer.
\(r = 5\sec\mathopen{}\left(\theta\right)\)
10.4.4.45.
Answer.
\(r = \frac{\cos\mathopen{}\left(\theta\right)}{\sin^{2}\mathopen{}\left(\theta\right)}\)
10.4.4.47.
Answer.
\(r = \sqrt{7}\)
10.4.4.49.
Answer.
\(P\left(\frac{\sqrt{3}}{2},\frac{\pi }{6}\right), P\left(0,\frac{\pi }{2}\right), P\left(\frac{-\sqrt{3}}{2},\frac{5\pi }{6}\right)\)
10.4.4.51.
Answer.
\(P\left(0,0\right), P\left(\sqrt{2},\frac{\pi }{4}\right)\)
10.5 Calculus and Polar Functions
10.5.5 Exercises
Problems
10.5.5.3.
Answer 1.
\(-\cot\mathopen{}\left(\theta\right)\)
Answer 2.
\(y = -\left(x-\frac{\sqrt{2}}{2}\right)+\frac{\sqrt{2}}{2}\)
Answer 3.
\(y = x\)
10.5.5.7.
Answer 1.
\(\frac{\theta\cos\mathopen{}\left(\theta\right)+\sin\mathopen{}\left(\theta\right)}{\cos\mathopen{}\left(\theta\right)-\theta\sin\mathopen{}\left(\theta\right)}\)
Answer 2.
\(y = \frac{-2}{\pi }x+\frac{\pi }{2}\)
Answer 3.
\(y = \frac{\pi }{2}x+\frac{\pi }{2}\)
10.5.5.9.
Answer 1.
\(\frac{4\sin\mathopen{}\left(\theta\right)\cos\mathopen{}\left(4\theta\right)+\sin\mathopen{}\left(4\theta\right)\cos\mathopen{}\left(\theta\right)}{4\cos\mathopen{}\left(\theta\right)\cos\mathopen{}\left(4\theta\right)-\sin\mathopen{}\left(\theta\right)\sin\mathopen{}\left(4\theta\right)}\)
Answer 2.
\(y = 5\sqrt{3}\mathopen{}\left(x+\frac{\sqrt{3}}{4}\right)-\frac{3}{4}\)
Answer 3.
\(y = \frac{-1}{5\sqrt{3}}\mathopen{}\left(x+\frac{\sqrt{3}}{4}\right)-\frac{3}{4}\)
10.5.5.19.
Answer.
\(\frac{\pi }{12}\)
10.5.5.21.
Answer.
\(\frac{3\pi }{2}\)
10.5.5.23.
Answer.
\(2\pi +\frac{3\cdot 1.73205}{2}\)
10.5.5.25.
Answer.
\(1\)
10.5.5.29.
Answer.
\(4\pi \)
10.5.5.31.
Answer.
\(\sqrt{2}\pi\)
10.5.5.33.
Answer.
\(2.2592\hbox{ or }2.22748\)
11 Vectors
11.1 Introduction to Cartesian Coordinates in Space
11.1.7 Exercises
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.
\(3\)
11.1.7.15.
Answer.
11.1.7.17.
Answer.
11.1.7.19.
Answer.
\(x^{2}+z^{2} = \left(\frac{1}{1+y^{2}}\right)^{2}\)
11.1.7.21.
Answer.
\(x^{2}+y^{2} = z\)
11.1.7.23.
Answer.
(a)\(\ds x=y^2+\frac{z^2}{9}\)
11.1.7.25.
Answer.
(b) \(\ds x^2+\frac{y^2}9+\frac{z^2}4=1\)
11.1.7.27.
Answer.
11.1.7.29.
Answer.
11.1.7.31.
Answer.
11.2 An Introduction to Vectors
Exercises
Problems
11.2.7.
Answer.
\(\left<1,6\right>\)
Answer.
\(\boldsymbol{i}+6\boldsymbol{j}\)
11.2.9.
Answer.
\(\left<6,-1,6\right>\)
Answer.
\(6\boldsymbol{i}-\boldsymbol{j}+6\boldsymbol{k}\)
11.2.11.
11.2.11.a
Answer.
\(\vec u+\vec v = \la 2,-1\ra\text{;}\) \(\vec u -\vec v = \la0,-3\ra\text{;}\) \(2\vec u-3\vec v = \la -1,-7\ra\text{.}\)
11.2.11.c
Answer.
\(\vec x = \la 1/2,2\ra\text{.}\)
11.2.17.
Answer 1.
\(\sqrt{5}\)
Answer 2.
\(\sqrt{13}\)
Answer 3.
\(\sqrt{26}\)
Answer 4.
\(\sqrt{10}\)
11.2.19.
Answer 1.
\(\sqrt{5}\)
Answer 2.
\(3\sqrt{5}\)
Answer 3.
\(2\sqrt{5}\)
Answer 4.
\(4\sqrt{5}\)
11.2.23.
Answer.
\(\left<0.6,0.8\right>\)
11.2.25.
Answer.
\(\left<\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right>\)
11.2.27.
Answer.
\(\left<\frac{-1}{2},\frac{\sqrt{3}}{2}\right>\)
11.3 The Dot Product
11.3.2 Exercises
Terms and Concepts
11.3.2.1.
Answer.
Scalar
Problems
11.3.2.5.
Answer.
\(-22\)
11.3.2.7.
Answer.
\(3\)
11.3.2.9.
Answer.
not defined
11.3.2.11.
Answer.
Answers will vary.
11.3.2.13.
Answer.
\(\cos^{-1}\mathopen{}\left(\frac{3}{\sqrt{10}}\right)\)
11.3.2.15.
Answer.
\(\frac{\pi }{4}\)
11.3.2.17.
Answer 1.
\(\left<-7,4\right>\)
Answer 2.
\(\left<4,7\right>\)
11.3.2.19.
Answer 1.
\(\left<1,0,-1\right>\)
Answer 2.
\(\left<1,1,1\right>\)
11.3.2.21.
Answer.
\(\left<\frac{-5}{10},\frac{15}{10}\right>\)
11.3.2.23.
Answer.
\(\left<\frac{-1}{2},\frac{-1}{2}\right>\)
11.3.2.25.
Answer.
\(\left<\frac{14}{14},\frac{28}{14},\frac{42}{14}\right>\)
11.3.2.27.
Answer 1.
\(\left<\frac{-5}{10},\frac{15}{10}\right>\)
Answer 2.
\(\left<\frac{15}{10},\frac{5}{10}\right>\)
11.3.2.29.
Answer 1.
\(\left<\frac{-1}{2},\frac{-1}{2}\right>\)
Answer 2.
\(\left<\frac{-5}{2},\frac{5}{2}\right>\)
11.3.2.31.
Answer 1.
\(\left<\frac{14}{14},\frac{28}{14},\frac{42}{14}\right>\)
Answer 2.
\(\left<\frac{0}{14},\frac{42}{14},\frac{-28}{14}\right>\)
11.3.2.33.
Answer.
1.96lb
11.3.2.35.
Answer.
\(141.42\)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
Terms and Concepts
11.4.3.1.
Answer.
vector
11.4.3.3.
Answer.
“Perpendicular” is one answer.
11.4.3.5.
Answer.
Torque
Problems
11.4.3.7.
Answer.
\(\left<12,-15,3\right>\)
11.4.3.9.
Answer.
\(\left<-5,-31,27\right>\)
11.4.3.11.
Answer.
\(\left<0,-2,0\right>\)
11.4.3.13.
Answer.
\(\vec u\times \vec v = \langle 0,0,ad-bc\rangle\)
11.4.3.15.
Answer.
\(-\boldsymbol{j}\)
11.4.3.17.
Answer.
Answers will vary.
11.4.3.19.
Answer.
\(5\)
11.4.3.21.
Answer.
\(0\)
11.4.3.23.
Answer.
\(\sqrt{14}\)
11.4.3.25.
Answer.
\(3\)
11.4.3.27.
Answer.
\(\frac{5\sqrt{2}}{2}\)
11.4.3.29.
Answer.
\(1\)
11.4.3.31.
Answer.
\(7\)
11.4.3.33.
Answer.
\(2\)
11.4.3.35.
Answer.
\(\left<0.408248,0.408248,-0.816497\right>\hbox{ or }\left<-0.408248,-0.408248,0.816497\right>\)
11.4.3.37.
Answer.
\(\left<0,1,0\right>\hbox{ or }\left<0,-1,0\right>\)
11.4.3.39.
Answer.
\(87.5\)ft–lb
11.4.3.41.
Answer.
\(200/3\approx 66.67\)ft–lb
11.5 Lines
11.5.4 Exercises
Terms and Concepts
11.5.4.1.
Answer.
A point on the line and the direction of the line.
11.5.4.3.
Answer.
parallel, skew
Problems
11.5.4.11.
Answer 1.
\(\left(7,2,-1\right)+t\mathopen{}\left<1,-1,2\right>\)
Answer 2.
\(x = 7+t, y = 2-t, z = -1+2t\)
Answer 3.
\(x-7 = 2-y = \frac{z+1}{2}\)
11.5.4.15.
Answer.
\(\text{parallel}\)
11.5.4.19.
Answer.
\(\text{skew}\)
11.5.4.23.
Answer.
\(\sqrt{41}/3\)
11.5.4.25.
Answer.
\(5\sqrt{2}/2\)
11.5.4.27.
Answer.
\(3/\sqrt{2}\)
11.6 Planes
11.6.2 Exercises
Terms and Concepts
11.6.2.1.
Answer.
A point in the plane and a normal vector (i.e., a direction orthogonal to the plane).
Problems
11.6.2.3.
Answer.
Answers will vary.
11.6.2.5.
Answer.
Answers will vary.
11.6.2.17.
Answer 1.
\(x-5+y-7+z-3 = 0\)
Answer 2.
\(x+y+z = 15\)
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.
\(3x+8y-10z = 24\)
11.6.2.27.
Answer.
\(\sqrt{5/7}\)
11.6.2.29.
Answer.
\(1/\sqrt{3}\)
12 Vector Valued Functions
12.1 Vector-Valued Functions
12.1.4 Exercises
Terms and Concepts
12.1.4.1.
Answer.
parametric equations
12.1.4.3.
Answer.
displacement
Problems
12.1.4.15.
Answer.
Graph of the function \(\vec r(t) = \la \cos(t) , \sin(t) ,\sin(t) \ra\) on \([0,2\pi]\text{.}\) The graph of the function is an oval lying in the plane coming from rotating the \(xy\) plane \(45\) degrees towards the \(z\)-axis. The oval lying in this plane has a horizontal width of \(\sqrt{2}\) and a height of \(1\text{.}\) Ignoring the \(z\) coordinate, the curve is a unit circle in the \(xy\) plane. Similarly ignoring the \(y\) coordinate, the curve is a unit circle in the \(xz\) plane. If we now ignore the \(x\) coordinate, the resulting curve is a diagonal line given by \(z=y\) in the \(yz\) plane. This line turns back on itself, which can be seen in the image of the oval when considering all three coordinate axes.
12.1.4.17.
Answer.
\(\left|t\right|\sqrt{1+t^{2}}\)
12.1.4.19.
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
\(\vec r(t) = \la 3t+1, 3t+2, 3t+3\ra\text{.}\)
12.1.4.29.
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
12.2.5.1.
Answer.
component
Problems
12.2.5.5.
Answer.
\(\left<11,74,\sin\mathopen{}\left(5\right)\right>\)
12.2.5.7.
Answer.
\(\left<1,e\right>\)
12.2.5.9.
Answer.
\(\left(-\infty ,0\right)\cup \left(0,\infty \right)\)
12.2.5.11.
Answer.
\(\left<-\sin\mathopen{}\left(t\right),e^{t},\frac{1}{t}\right>\)
12.2.5.13.
Answer.
\(\left<2t\sin\mathopen{}\left(t\right)+t^{2}\cos\mathopen{}\left(t\right),6t^{2}+10t\right>\)
12.2.5.15.
Answer.
\(\left<-1,\cos\mathopen{}\left(t\right)-2t,6t^{2}+10t+2+\cos\mathopen{}\left(t\right)-\sin\mathopen{}\left(t\right)-t\cos\mathopen{}\left(t\right)\right>\)
12.2.5.21.
Answer.
\(\left<2+3t,t\right>\)
12.2.5.23.
Answer.
\(\ell(t) = \la -3,0,\pi\ra + t\la0,-3,1\ra\)
12.2.5.33.
Answer.
\(\la \frac14t^4,\sin(t) ,te^t-e^t\ra + \vec C\)
12.2.5.35.
Answer.
\(\left<-2,0\right>\)
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.
\(2\cdot 3.60555\pi \)
12.2.5.43.
Answer.
\(\frac{1}{54}\mathopen{}\left(22^{\frac{3}{2}}-8\right)\)
12.3 The Calculus of Motion
12.3.3 Exercises
Problems
12.3.3.7.
Answer.
\(\vvt = \la 2,5,0\ra\text{,}\) \(\vat = \la 0,0,0\ra\)
12.3.3.19.
Answer 1.
\(\left|\sec\mathopen{}\left(t\right)\right|\sqrt{\tan^{2}\mathopen{}\left(t\right)+\sec^{2}\mathopen{}\left(t\right)}\)
Answer 2.
\(0\)
Answer 3.
\(\frac{\pi }{4}\)
12.3.3.39.
Answer.
\(0.013\ {\rm radians}\)
Answer.
\(11.7\ {\rm ft}\)
12.4 Unit Tangent and Normal Vectors
12.4.4 Exercises
Terms and Concepts
12.4.4.1.
Answer.
\(1\)
12.4.4.3.
Answer.
\(\unittangent(t)\) and \(\unitnormal(t)\text{.}\)
Problems
12.4.4.5.
Answer.
\(\unittangent(t) = \la\frac{4 t}{\sqrt{20 t^2-4t+1}},\frac{2 t-1}{\sqrt{20 t^2-4t+1}}\ra\text{;}\) \(\unittangent(1) = \la 4/\sqrt{17},1/\sqrt{17}\ra\)
12.4.4.9.
Answer.
\(\left(2,0\right)+t\mathopen{}\left<\frac{4}{\sqrt{17}},\frac{1}{\sqrt{17}}\right>\)
12.4.4.13.
Answer.
\(\unittangent(t) = \la -\sin(t) ,\cos(t) \ra\text{;}\) \(\unitnormal(t) = \la -\cos(t) ,-\sin(t) \ra\)
12.4.4.15.
Answer.
\(\unittangent(t) = \la -\frac{\sin(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }},\frac{2\cos(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }}\ra\text{;}\) \(\unitnormal(t) = \la -\frac{2\cos(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }},-\frac{\sin(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }}\ra\)
12.5 The Arc Length Parameter and Curvature
12.5.4 Exercises
Terms and Concepts
12.5.4.1.
Answer.
time and/or distance
12.5.4.3.
Answer.
Answers may include lines, circles, helixes
12.5.4.5.
Answer.
\(\kappa\)
Problems
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.
\(8\)
12.5.4.23.
Answer.
\(\frac{\sqrt{2}}{\sqrt[4]{5}}, \frac{-\sqrt{2}}{\sqrt[4]{5}}\)
12.5.4.25.
Answer.
\(\frac{1}{4}\)
13 Functions of Several Variables
13.2 Limits and Continuity of Multivariable Functions
13.2.5 Exercises
Problems
13.2.5.7.
Answer.
- Answers will vary. interior point: \((1,3)\) boundary point: \((3,3)\)
- \(S\) is a closed set
- \(S\) is bounded
13.2.5.11.
Answer.
- \(D = \left\{(x,y)\, |\, 9-x^2-y^2\geq 0\right\}\text{.}\)
- \(D\) is a closed set.
- \(D\) is bounded.
13.2.5.13.
Answer.
- \(D = \left\{(x,y)\, |\, y \gt x^2\right\}\text{.}\)
- \(D\) is an open set.
- \(D\) is unbounded.
13.3 Partial Derivatives
13.3.7 Exercises
Terms and Concepts
13.3.7.3.
Answer.
\({\verb!f_x!}\)
Problems
13.3.7.19.
Answer 1.
\(\frac{2y^{2}}{\sqrt{4xy^{2}+1}}\)
Answer 2.
\(\frac{4xy}{\sqrt{4xy^{2}+1}}\)
Answer 3.
\(\frac{-4y^{4}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}\)
Answer 4.
\(\frac{-8xy^{3}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4y}{\sqrt{4xy^{2}+1}}\)
Answer 5.
\(\frac{-8xy^{3}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4y}{\sqrt{4xy^{2}+1}}\)
Answer 6.
\(\frac{-16x^{2}y^{2}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4x}{\sqrt{4xy^{2}+1}}\)
13.5 The Multivariable Chain Rule
13.5.3 Exercises
Terms and Concepts
13.5.3.5.
Answer.
F
Problems
13.5.3.7.
Answer.
- \(\frac{dz}{dt} = 3(2t)+4(2) = 6t+8\text{.}\)
- At \(t=1\text{,}\) \(\frac{dz}{dt} = 14\text{.}\)
13.5.3.9.
Answer.
- \(\displaystyle \frac{dz}{dt} = 5(-2\sin(t) )+2(\cos(t) ) = -10\sin(t) +2\cos(t)\)
- At \(t=\pi/4\text{,}\) \(\frac{dz}{dt} = -4\sqrt{2}\text{.}\)
13.5.3.11.
Answer.
- \(\ds\frac{dz}{dt} = 2x(\cos(t) ) + 4y(3\cos(t) )\text{.}\)
- At \(t=\pi/4\text{,}\) \(x=\sqrt{2}/2\text{,}\) \(y=3\sqrt{2}/2\text{,}\) and \(\frac{dz}{dt} = 19\text{.}\)
13.5.3.21.
Answer 1.
\(2x\cos\mathopen{}\left(t\right)+2y\sin\mathopen{}\left(t\right)\)
Answer 2.
\(-2xs\sin\mathopen{}\left(t\right)+2ys\cos\mathopen{}\left(t\right)\)
Answer 3.
\(4\)
Answer 4.
\(0\)
13.6 Directional Derivatives
13.6.3 Exercises
Terms and Concepts
13.6.3.3.
Answer.
\(\boldsymbol{j}\)
Problems
13.6.3.13.
13.6.3.13.a
Answer.
\(2/5\)
13.6.3.13.b
Answer.
\(-2/\sqrt{5}\)
13.6.3.15.
13.6.3.15.a
Answer.
\(0\)
13.6.3.15.b
Answer.
\(2\sqrt{2}/9\)
13.6.3.17.
13.6.3.17.a
Answer.
\(0\)
13.6.3.17.b
Answer.
\(0\)
13.6.3.19.
13.6.3.19.a
Answer.
\(\nabla f(2,1) = \la -2,2\ra\)
13.6.3.19.b
Answer.
\(\sqrt{8}\)
13.6.3.19.c
Answer.
\(\la 2, -2\ra\)
13.6.3.19.d
Answer.
\(\vec u = \la 1/\sqrt{2},1/\sqrt{2}\ra\)
13.6.3.21.
13.6.3.21.a
Answer.
\(\nabla f(1,1) = \la -2/9,-2/9\ra\)
13.6.3.21.b
Answer.
\(2\sqrt{2}/9\)
13.6.3.21.c
Answer.
\(\la 2/9,2/9\ra\)
13.6.3.21.d
Answer.
\(\vec u = \la 1/\sqrt{2},-1/\sqrt{2}\ra\)
13.6.3.23.
13.6.3.23.a
Answer.
No such direction
13.6.3.23.b
Answer.
\(0\)
13.6.3.23.c
Answer.
No such direction
13.6.3.23.d
Answer.
All directions
13.6.3.25.
13.6.3.25.a
Answer.
\(\nabla F(x,y,z) = \la 6xz^3+4y, 4x, 9x^2z^2-6z\ra\)
13.6.3.25.b
Answer.
\(113/\sqrt{3}\)
13.6.3.27.
13.6.3.27.a
Answer.
\(\nabla F(x,y,z) = \la 2xy^2, 2y(x^2-z^2), -2y^2z\ra\)
13.6.3.27.b
Answer.
\(0\)
13.7 Tangent Lines, Normal Lines, and Tangent Planes
13.7.5 Exercises
Terms and Concepts
13.7.5.3.
Answer.
\(\text{True}\)
13.8 Extreme Values
13.8.3 Exercises
Terms and Concepts
13.8.3.1.
Answer.
\(\text{False}\)
13.8.3.3.
Answer.
\(\text{True}\)
Problems
13.8.3.15.
Answer 1.
\(3\)
Answer 2.
\(\left(0,1\right)\)
Answer 3.
\(\frac{3}{4}\)
Answer 4.
\(\left(0,\frac{-1}{2}\right)\)
14 Multiple Integration
14.1 Iterated Integrals and Area
14.1.4 Exercises
Terms and Concepts
14.1.4.3.
Answer.
curve to curve, then from point to point
Problems
14.1.4.5.
14.1.4.5.a
Answer.
\(18x^2+42x-117\)
14.1.4.5.b
Answer.
\(-108\)
14.1.4.7.
14.1.4.7.a
Answer.
\(x^4/2-x^2+2x-3/2\)
14.1.4.7.b
Answer.
\(23/15\)
14.1.4.9.
14.1.4.9.a
Answer.
\(\sin^2(y)\)
14.1.4.9.b
Answer.
\(\pi/2\)
14.3 Double Integration with Polar Coordinates
Exercises
Problems
14.3.3.
Answer.
\(4\pi \)
14.3.5.
Answer.
\(16\pi \)
14.5 Surface Area
Exercises
Problems
14.5.7.
Answer.
\(\ds SA = \int_0^{2\pi}\int_0^{2\pi} \sqrt{1+ \cos^2(x) \cos^2(y) +\sin^2(x) \sin^2(y) }\, dx\, dy\)
14.5.9.
Answer.
\(\ds SA = \int_{-1}^{1}\int_{-1}^{1} \sqrt{1+ 4x^2+4y^2}\, dx\, dy\)
14.6 Volume Between Surfaces and Triple Integration
14.6.4 Exercises
Problems
14.6.4.9.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^3\int_0^{1-x/3}\int_0^{2-2x/3-2y}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_0^1\int_0^{3-3y}\int_0^{2-2x/3-2y}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^3\int_0^{2-2x/3}\int_0^{1-x/3-z/2}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^2\int_0^{3-3z/2}\int_0^{1-x/3-z/2}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_0^1\int_0^{2-2y}\int_0^{3-3y-3z/2}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^2\int_0^{1-z/2}\int_0^{3-3y-3z/2}\, dx\, dy\, dz\)
\(\ds V = \int_0^3\int_0^{1-x/3}\int_0^{2-2x/3-2y}\, dz\, dy\, dx =1\text{.}\)
14.6.4.11.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^2\int_{-2}^{0}\int_{y^2/2}^{-y}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{-2}^0\int_0^{2}\int_{y^2/2}^{-y}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{-2}^0\int_{y^2/2}^{-y}\int_0^{2}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^2\int_{-\sqrt{2z}}^{-z}\int_0^{2}\, dx\, dy\, dz\) \(\ds V = \int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dz\, dx =4/3\text{.}\)
14.6.4.13.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^2\int_{1-x/2}^{1}\int_{0}^{2x+4y-4}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^1\int_{2-2y}^{2}\int_{0}^{2x+4y-4}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^2\int_0^{2x}\int_{z/4-x/2+1}^{1}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^4\int_{z/2}^{2}\int_{z/4-x/2+1}^{1}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{4y}\int_{z/2-2y+2}^2\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^4\int_{z/4}^{1}\int_{z/2-2y+2}^2\, dx\, dy\, dz\) \(\ds V = \int_0^4\int_{z/4}^{1}\int_{z/2-2y-2}^2\, dx\, dy\, dz = 4/3\text{.}\)
14.6.4.15.
Answer.
\(dz\, dy\, dx\text{:}\) \(\ds\int_{0}^1\int_{0}^{1-x^2}\int_{0}^{\sqrt{1-y}}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{\sqrt{1-y}}\int_{0}^{\sqrt{1-y}}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_{0}^1\int_0^{x}\int_{0}^{1-x^2}\, dy\, dz\, dx + \int_{0}^1\int_x^{1}\int_{0}^{1-z^2}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^1\int_{0}^{z}\int_{0}^{1-z^2}\, dy\, dx\, dz+\int_0^1\int_{z}^{1}\int_{0}^{1-x^2}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{\sqrt{1-y}}\int_{0}^{\sqrt{1-y}}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^1\int_{0}^{1-z^2}\int_{0}^{\sqrt{1-y}}\, dx\, dy\, dz\) Answers will vary. Neither order is particularly “hard.” The order \(dz\, dy\, dx\) requires integrating a square root, so powers can be messy; the order \(dy\, dz\, dx\) requires two triple integrals, but each uses only polynomials.
14.7 Triple Integration with Cylindrical and Spherical Coordinates
14.7.3 Exercises
Problems
14.7.3.11.
Answer.
\(\ds\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}\int_{z_1}^{z_2}h(r,\theta,z)r\, dz\, dr\, d\theta\)
14.7.3.19.
Answer.
Describes the portion of the unit ball that resides in the first octant.
15 Vector Analysis
15.1 Introduction to Line Integrals
15.1.4 Exercises
Terms and Concepts
15.1.4.1.
Answer.
When \(C\) is a curve in the plane and \(f\) is a function defined over \(C\text{,}\) then \(\int_C f(s)\, ds\) describes the area under the spatial curve that lies on \(f\text{,}\) over \(C\text{.}\)
15.1.4.3.
Answer.
The variable \(s\) denotes the arc-length parameter, which is generally difficult to use. The Key Idea allows one to parametrize a curve using another, ideally easier-to-use, parameter.
Problems
15.1.4.5.
Answer.
\(12\sqrt{2}\)
15.1.4.7.
Answer.
\(40\pi\)
15.1.4.9.
Answer.
Over the first subcurve of \(C\text{,}\) the line integral has a value of \(3/2\text{;}\) over the second subcurve, the line integral has a value of \(4/3\text{.}\) The total value of the line integral is thus \(17/6\text{.}\)
15.1.4.11.
Answer.
\(\int_0^1(5t^2+_2t+2)\sqrt{(4t+1)^2+1}\, dt \approx 17.071\)
15.1.4.13.
Answer.
\(\oint_0^{2\pi} \big(10-4\cos^2t-\sin^2t\big)\sqrt{\cos^2t+4\sin^2t}\, dt \approx 74.986\)
15.1.4.15.
Answer.
\(7\sqrt{26}/3\)
15.1.4.17.
Answer.
\(8\pi^3\)
15.1.4.19.
Answer.
\(M=8\sqrt{2}\pi^2\text{;}\) center of mass is \((0,-1/(2\pi), 8\pi/3)\text{.}\)
15.2 Vector Fields
15.2.3 Exercises
Terms and Concepts
15.2.3.1.
Answer.
Answers will vary. Appropriate answers include velocities of moving particles (air, water, etc.); gravitational or electromagnetic forces.
15.2.3.3.
Answer.
Specific answers will vary, though should relate to the idea that the vector field is spinning clockwise at that point.
Problems
15.2.3.5.
Answer.
Correct answers should look similar to
15.2.3.7.
Answer.
Correct answers should look similar to
15.2.3.9.
Answer.
\(\divv \vec F = 1+2y\)
\(\curl \vec F = 0\)
15.2.3.11.
Answer.
\(\divv \vec F = x\cos(xy)-y\sin(xy)\)
\(\curl \vec F = y\cos(xy)+x\sin(xy)\)
15.2.3.13.
Answer.
\(\divv \vec F = 3\)
\(\curl \vec F = \la -1,-1,-1\ra\)
15.2.3.15.
Answer.
\(\divv \vec F = 1+2y\)
\(\curl\vec F = 0\)
15.2.3.17.
Answer.
\(\divv \vec F = 2y-\sin z\)
\(\curl\vec F = \vec 0\)
15.3 Line Integrals over Vector Fields
15.3.4 Exercises
Terms and Concepts
15.3.4.1.
Answer.
False. It is true for line integrals over scalar fields, though.
15.3.4.3.
Answer.
True.
15.3.4.5.
Answer.
We can conclude that \(\vec F\) is conservative.
Problems
15.3.4.7.
Answer.
\(11/6\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle 3t,t\rangle\) on \(0\leq t\leq 1\text{.}\))
15.3.4.9.
Answer.
\(0\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle \cos t,\sin t\rangle\) on \(0\leq t\leq \pi\text{.}\))
15.3.4.11.
Answer.
\(12\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle 1,2,3\rangle+t\langle 3,1,-1\rangle\) on \(0\leq t\leq 1\text{.}\))
15.3.4.13.
Answer.
\(5/6\) joules. (One parametrization for \(C\) is \(\vec r(t) = \langle t,t\rangle\) on \(0\leq t\leq 1\text{.}\))
15.3.4.15.
Answer.
\(24\) ft-lbs.
15.3.4.17.
Answer.
- \(\displaystyle f(x,y) = xy+x\)
- \(\curl \vec F = 0\text{.}\)
- \(1\text{.}\) (One parametrization for \(C\) is \(\vec r(t) = \langle t,t-1\rangle\) on \(0\leq t\leq 1\text{.}\))
- \(1\) (with \(A = (0,1)\) and \(B = (1,0)\text{,}\) \(f(B) - f(A) = 1\text{.}\))
15.3.4.19.
Answer.
- \(\displaystyle f(x,y) = x^2yz\)
- \(\curl \vec F = \vec 0\text{.}\)
- \(250\text{.}\)
- \(250\) (with \(A = (1,-1,0)\) and \(B = (5,5,2)\text{,}\) \(f(B) - f(A) = 250\text{.}\))
15.4 Flow, Flux, Green’s Theorem and the Divergence Theorem
15.4.4 Exercises
Terms and Concepts
15.4.4.1.
Answer.
along, across
15.4.4.3.
Answer.
the curl of \(\vec F\text{,}\) or \(\curl \vec F\)
15.4.4.5.
Answer.
\(\curl \vec F\)
Problems
15.4.4.7.
Answer.
\(12\)
15.4.4.9.
Answer.
\(-2/3\)
15.4.4.11.
Answer.
\(1/2\)
15.4.4.13.
Answer.
The line integral \(\oint_C\vec F\cdot d\vec r\text{,}\) over the parabola, is \(38/3\text{;}\) over the line, it is \(-10\text{.}\) The total line integral is thus \(38/3-10 = 8/3\text{.}\) The double integral of \(\curl \vec F = 2\) over \(R\) also has value \(8/3\text{.}\)
15.4.4.15.
Answer.
Three line integrals need to be computed to compute \(\oint_C \vec F\cdot d\vec r\text{.}\) It does not matter which corner one starts from first, but be sure to proceed around the triangle in a counterclockwise fashion.
From \((0,0)\) to \((2,0)\text{,}\) the line integral has a value of 0. From \((2,0)\) to \((1,1)\) the integral has a value of \(7/3\text{.}\) From \((1,1)\) to \((0,0)\) the line integral has a value of \(-1/3\text{.}\) Total value is 2.
The double integral of \(\curl\vec F\) over \(R\) also has value 2.
15.4.4.17.
Answer.
Any choice of \(\vec F\) is appropriate as long as \(\curl \vec F = 1\text{.}\) When \(\vec F = \langle -y/2,x/2\rangle\text{,}\) the integrand of the line integral is simply 6. The area of \(R\) is \(12\pi\text{.}\)
15.4.4.19.
Answer.
Any choice of \(\vec F\) is appropriate as long as \(\curl \vec F = 1\text{.}\) The choices of \(\vec F = \langle -y,0\rangle\text{,}\) \(\langle 0,x\rangle\) and \(\langle -y/2,x/2\rangle\) each lead to reasonable integrands. The area of \(R\) is \(16/15\text{.}\)
15.4.4.21.
Answer.
The line integral \(\oint_C\vec F\cdot \vec n\, ds\text{,}\) over the parabola, is \(-22/3\text{;}\) over the line, it is \(10\text{.}\) The total line integral is thus \(-22/3+10 = 8/3\text{.}\) The double integral of \(\divv \vec F = 2\) over \(R\) also has value \(8/3\text{.}\)
15.4.4.23.
Answer.
Three line integrals need to be computed to compute \(\oint_C \vec F\cdot \vec n\, ds\text{.}\) It does not matter which corner one starts from first, but be sure to proceed around the triangle in a counterclockwise fashion.
From \((0,0)\) to \((2,0)\text{,}\) the line integral has a value of 0. From \((2,0)\) to \((1,1)\) the integral has a value of \(1/3\text{.}\) From \((1,1)\) to \((0,0)\) the line integral has a value of \(1/3\text{.}\) Total value is \(2/3\text{.}\)
The double integral of \(\divv\vec F\) over \(R\) also has value \(2/3\text{.}\)
15.5 Parametrized Surfaces and Surface Area
15.5.3 Exercises
Terms and Concepts
15.5.3.1.
Answer.
Answers will vary, though generally should meaningfully include terms like “two sided”.
Problems
15.5.3.3.
Answer.
- \(\vec r(u,v) = \langle u, v, 3u^2v\rangle\) on \(-1\leq u\leq 1\text{,}\) \(0\leq v\leq 2\text{.}\)
- \(\vec r(u,v) = \langle 3v\cos u+1, 3v\sin u+2, 3(3v\cos u+1)^2(3v\sin u+2)\rangle\text{,}\) on \(0\leq u\leq 2\pi\text{,}\) \(0\leq v\leq 1\text{.}\)
- \(\vec r(u,v) = \langle u, v(2-2u), 3u^2v(2-2u)\rangle\) on \(0\leq u, v\leq 1\text{.}\)
- \(\vec r(u,v) = \langle u, v(1-u^2), 3u^2v(1-u^2)\rangle\) on \(-1\leq u\leq 1\text{,}\) \(0\leq v\leq 1\text{.}\)
15.5.3.5.
Answer.
\(\vec r(u,v) = \langle 0, u, v\rangle\) with \(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\text{.}\)
15.5.3.7.
Answer.
\(\vec r(u,v) = \langle 3\sin u\cos v, 2\sin u\sin v, 4\cos u\rangle\) with \(0\leq u\leq \pi\text{,}\) \(0\leq v\leq 2\pi\text{.}\)
15.5.3.9.
Answer.
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 \(1\leq u\leq 3\text{,}\) \(0\leq v\leq 2\)
15.5.3.11.
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 \(0\leq v\leq 1\text{.}\)
15.5.3.13.
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 \(0\leq v\leq 1\text{.}\)
15.5.3.15.
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 \(-1\leq v\leq 2\text{.}\)
15.5.3.17.
Answer.
\(S = 2\sqrt{14}\text{.}\)
15.5.3.19.
Answer.
\(S = 4\sqrt{3}\pi\text{.}\)
15.5.3.21.
Answer.
\(S =\int_0^3\int_0^{2\pi}\sqrt{v^2+4v^4}\, du\, dv= (37\sqrt{37}-1)\pi/6 \approx 117.319\text{.}\)
15.5.3.23.
Answer.
\(S =\int_0^1\int_{-1}^{1}\sqrt{(5u^2-5)^2+2(1-u^2)^2}\, du\, dv = 4\sqrt{3}\approx 6.9282\text{.}\)
15.6 Surface Integrals
15.6.3 Exercises
Terms and Concepts
15.6.3.1.
Answer.
curve; surface
15.6.3.3.
Answer.
outside
Problems
15.6.3.5.
Answer.
\(240\sqrt{3}\)
15.6.3.7.
Answer.
\(24\)
15.6.3.9.
Answer.
\(0\)
15.6.3.11.
Answer.
\(-1/2\)
15.6.3.13.
Answer.
\(0\text{;}\) the flux over \(\surfaceS_1\) is \(-45\pi\) and the flux over \(\surfaceS_2\) is \(45\pi\text{.}\)
15.7 The Divergence Theorem and Stokes’ Theorem
15.7.4 Exercises
Terms and Concepts
15.7.4.1.
Answer.
Answers will vary; in Section 15.4, the Divergence Theorem connects outward flux over a closed curve in the plane to the divergence of the vector field, whereas in this section the Divergence Theorem connects outward flux over a closed surface in space to the divergence of the vector field.
15.7.4.3.
Answer.
Curl.
Problems
15.7.4.5.
Answer.
Outward flux across the plane \(z=2-x/2-2y/3\) is 14; across the plane \(z=0\) the outward flux is \(-8\text{;}\) across the planes \(x=0\) and \(y=0\) the outward flux is 0.
Total outward flux: \(14\text{.}\)
\(\iint_D\divv\vec F\, dV = \int_0^{4}\int_0^{3-3x/4}\int_0^{2-x/2-2y/3}(2x+2y)\, dz\, dy\, dx = 14\text{.}\)
15.7.4.7.
Answer.
Outward flux across the surface \(z=xy(3-x)(3-y)\) is 252; across the plane \(z=0\) the outward flux is \(-9\text{.}\)
Total outward flux: \(243\text{.}\)
\(\iint_D\divv\vec F\, dV = \int_0^{3}\int_0^{3}\int_{0}^{xy(3-x)(3-y)}12\, dz\, dy\, dx = 243\text{.}\)
15.7.4.9.
Answer.
Circulation on \(C\text{:}\) \(\oint_C \vec F\cdot d\vec r = \pi\)
\(\iint_\surfaceS\big(\curl \vec F\big)\cdot\vec n\, dS = \pi\text{.}\)
15.7.4.11.
Answer.
Circulation on \(C\text{:}\) The flow along the line from \((0,0,2)\) to \((4,0,0)\) is 0; from \((4,0,0)\) to \((0,3,0)\) it is \(-6\text{,}\) and from \((0,3,0)\) to \((0,0,2)\) it is 6. The total circulation is \(0+(-6)+6=0\text{.}\)
\(\iint_\surfaceS\big(\curl \vec F\big)\cdot\vec n\, dS = \iint_\surfaceS 0 \, dS = 0\text{.}\)
15.7.4.13.
Answer.
\(128/225\)
15.7.4.15.
Answer.
\(8192/105\approx 78.019\)
15.7.4.17.
Answer.
\(5/3\)
15.7.4.19.
Answer.
\(23\pi\)
15.7.4.21.
Answer.
Each field has a divergence of 1; by the Divergence Theorem, the total outward flux across \(\surfaceS\) is \(\iint_D 1\, dS\) for each field.
15.7.4.23.
Answer.
Answers will vary. Often the closed surface \(\surfaceS\) is composed of several smooth surfaces. To measure total outward flux, this may require evaluating multiple double integrals. Each double integral requires the parametrization of a surface and the computation of the cross product of partial derivatives. One triple integral may require less work, especially as the divergence of a vector field is generally easy to compute.
