4: Section 2.1 Answers
- Page ID
- 103619
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. \(y=2\pm\sqrt{2(x^{3}+x^{2}+x+c)}\)
2. \(\ln (|\sin y|)=\cos x+c; \quad y \equiv k\pi,\quad k=\text{integer}\)
3. \(-{1\over 5}e^{-5y}={1\over 2}e^{2x}+c\)
4. \(y=\frac{c}{x-c}; \quad y≡-1\)
5. \({y\over y-1}=ce^x; \quad y≡1\)
6. \(\frac{(\ln y)^{2}}{2}=-\frac{x^{3}}{3}+c\)
7. \(y^{3}+3\sin y+\ln |y|+\ln (1+x^{2})+\tan ^{-1}x=c;\quad y≡0\)
8. \((y^2-1)^{-1/2}={1\over x}+c;\quad y≡\pm 1\)
9. \(y=\tan \left(\frac{x^{3}}{3}+c \right)\)
10. \(y=\frac{c}{\sqrt{1+x^{2}}}\)
11. \({y-2\over y-1}=ce^{(x^2-2x)/2};\quad y≡1\)
12. \(y=1+(3x^{2}+9x+c)^{1/3}\)
13. \(y=2+\sqrt{\frac{2}{3}x^{3}+3x^{2}+4x-\frac{11}{3}}\)
14. \(y=\frac{e^{-(x^{2}-4)/2}}{2-e^{-(x^{2}-4)/2}}\)
15. \(xy=e^{-(1+1/x)}\)
16. \(y^{3}+2y^{2}+x^{2}+\sin x=3\)
17. \((y+1)(y-1)^{-3}(y-2)^{2}=-256(x+1)^{-6}\)
18. \(y=-1+3e^{-x^{2}}\)
19. \({y^2\over y^2+1}={1\over 2}e^{2x^2}\)
20. \(y≡-1;\quad (-\infty,\infty)\)
21. \({y-2\over y-1}={1\over 2}e^{-x^2};\quad (-\infty,\infty)\)
22. \(y=\frac{-1+\sqrt{4x^{2}-15}}{2};\quad ({\sqrt {15}\over 2},\infty)\)
23. \(y=\frac{2}{1+e^{-2x}}; \quad (-\infty ,\infty )\)
24. \(y=-\sqrt{25-x^{2}};\quad (-5,5)\)
25. \(y≡2;\quad (-\infty ,\infty )\)
26. \(y=3\left(\frac{x+1}{2x-4} \right)^{1/3};\quad (-\infty ,2)\)
27. \(y=-\sqrt{25-x^2}; \quad (-5,5)\)
28. \(y=\frac{x+c}{1-cx}\)
29. \(y=-x\cos c+\sqrt{1-x^{2}}\sin c;\quad y≡1; \:y≡-1\)
30. \(y=-x+3\pi /2\)
31. a. \(y={1\over 1-{1\over 2}e^x}; \quad (-\infty,\ln 2)\)
b. \(y≡0;\quad (-\infty ,\infty )\)
32. a. \(y={x\over x+1}; \quad (-1,\infty)\)
b. \(y≡1;\quad (-\infty ,\infty )\)
33. \(y=1+(x^2+4)^{3/2}; \quad (-\infty,\infty)\)
34. \(y=\left\{\begin{array}{cc}{1,}&{0<x\leq \sqrt{5}}\\{1-(x^2-5)^{3/2},}&{\sqrt{5}<x<\infty}\end{array} \right.\)