A.8.4: Section 8.4 Answers
- Page ID
- 43786
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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. \(1+u(t-4)(t-1);\quad\frac{1}{s}+e^{-4s}\left(\frac{1}{s^{2}}+\frac{3}{s}\right)\)
2. \(t+u(t-1)(1-t);\quad\frac{1-e^{-s}}{s^{2}}\)
3. \(2t-1-u(t-2)(t-1);\quad\left(\frac{2}{s^{2}}-\frac{1}{s}\right)-e^{-2s}\left(\frac{1}{s^{2}}+\frac{1}{s}\right)\)
4. \(1+u(t-1)(t+1);\quad\frac{1}{s}+e^{-s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)
5. \(t-1+u(t-2)(5-t);\quad\frac{1}{s^{2}}-\frac{1}{s}-e^{-2s}\left(\frac{1}{s^{2}}-\frac{3}{s}\right)\)
6. \(t^{2}(1-u(t-1));\quad\frac{2}{s^{3}}-e^{-s}\left(\frac{2}{s^{3}}+\frac{2}{s^{2}}+\frac{1}{s}\right)\)
7. \(u(t-2)(t^{2}+3t);\quad e^{-2s}\left(\frac{2}{s^{3}}+\frac{7}{s^{2}}+\frac{10}{s}\right)\)
8. \(t^{2}+2+u(t-1)(t-t^{2}-2);\quad\frac{2}{s^{3}}+\frac{2}{s}-e^{-s}\left(\frac{2}{s^{3}}+\frac{1}{s^{2}}+\frac{2}{s}\right)\)
9. \(te^{t}+u(t-1)(e^{t}-te^{t});\quad\frac{1-e^{-(s-1)}}{(s-1)^{2}}\)
10. \(e^{-t}+u(t-1)(e^{-2t}-e^{-t});\quad\frac{1-e^{-(s+1)}}{s+1}+\frac{e^{-(s+2)}}{s+2}\)
11. \(-t+2u(t-2)(t-2)-u(t-3)(t-5);\quad-\frac{1}{s^{2}}+\frac{2e^{-2s}}{s^{2}}+e^{-3s}\left(\frac{2}{s}-\frac{1}{s^{2}}\right)\)
12. \(\left[u(t-1)-u(t-2)\right] t;\quad e^{-s}\left(\frac{1}{s^{2}}+\frac{1}{s}\right)-e^{-2s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)
13. \(t+u(t-1)(t^{2}-t)-u(t-2)t^{2};\quad\frac{1}{s^{2}}+e^{-s}\left(\frac{2}{s^{3}}+\frac{1}{s^{2}}\right)-e^{-2s}\left(\frac{2}{s^{3}}+\frac{4}{s^{2}}+\frac{4}{s}\right)\)
14. \(t+u(t-1)(2-2t)+u(t-2)(4+t);\quad\frac{1}{s^{2}}-2\frac{e^{-s}}{s^{2}}+e^{-2s}\left(\frac{1}{s^{2}}+\frac{6}{s}\right)\)
15. \(\sin t+u(t-\pi /2)\sin t+u(t-\pi )(\cos t-2\sin t);\quad \frac{1+e^{-\frac{\pi }{2}s}s-e^{-\pi s}(s-2)}{s^{2}+1}\)
16. \(2-2u(t-1)t+u(t-3)(5t-2);\quad\frac{2}{s}-e^{-s}\left(\frac{2}{s^{2}}+\frac{2}{s}\right)+e^{-3s}\left(\frac{5}{s^{2}}+\frac{13}{s}\right)\)
17. \(3+u(t-2)(3t-1)+u(t-4)(t-2);\quad\frac{3}{s}+e^{-2s}\left(\frac{3}{s^{2}}+\frac{5}{s}\right)+e^{-4s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)
18. \((t+1)^{2}+u(t-1)(2t+3);\quad\frac{2}{s^{3}}+\frac{2}{s^{2}}+\frac{1}{s}+e^{-s}\left(\frac{2}{s^{2}}+\frac{5}{s}\right)\)
19. \(u(t-2)e^{2(t-2)}=\left\{\begin{array}{cc}{0,}&{0\leq t<2,}\\[4pt]{e^{2(t-2)},}&{t\geq 2}\end{array} \right.\)
20. \(u(t-1)\left(1-e^{-(t-1)}\right)=\left\{\begin{array}{cc}{0,}&{0\leq t<1,}\\[4pt]{1-e^{-(t-1)},}&{t\geq 1}\end{array} \right.\)
21. \(u(t-1)\frac{(t-1)^{2}}{2}+u(t-2)(t-2)=\left\{\begin{array}{cc}{0,}&{0\leq t<1,}\\[4pt]{\frac{(t-1)^{2}}{2},}&{1\leq t<2,}\\[4pt]{\frac{t^{2}-3}{2},}&{t\geq 2}\end{array} \right.\)
22. \(2+t+u(t-1)(4-t)+u(t-3)(t-2)=\left\{\begin{array}{cc}{2+t,}&{0\leq t<1,}\\[4pt]{6,}&{1\leq t<3,}\\[4pt]{t+4,}&{t\geq 3}\end{array} \right.\)
23. \(5-t+u(t-3)(7t-15)+\frac{3}{2}u(t-6)(t-6)^{2}=\left\{\begin{array}{cc}{5-t,}&{0\leq t<3,}\\[4pt]{6t-10,}&{3\leq t<6,}\\[4pt]{44-12t+\frac{3}{2}t^{2},}&{t\geq 6}\end{array} \right.\)
24. \(u(t-\pi )e^{-2(t-\pi )}(2\cos t-5\sin t)=\left\{\begin{array}{cc}{0,}&{0\leq t<\pi ,}\\[4pt]{e^{-2(t-\pi )}(2\cos t-5\sin t)}&{t\geq\pi }\end{array} \right.\)
25. \(1-\cos t+u(t-\pi /2)(3\sin t+\cos t)=\left\{\begin{array}{cc}{1-\cos t,}&{0\leq t<\frac{\pi }{2},}\\[4pt]{1+3\sin t,}&{t\geq\frac{\pi }{2}}\end{array} \right.\)
26. \(u(t-2)(4e^{-(t-2)}-4e^{2(t-2)}+2e^{(t-2)}=\left\{\begin{array}{cc}{0,}&{0\leq t<2,}\\[4pt]{4e^{-(t-2)}-4e^{2(t-2)}+2e^{(t-2)},}&{t\geq 2}\end{array} \right.\)
27. \(1+t+u(t-1)(2t+1)+u(t-3)(3t-5)=\left\{\begin{array}{cc}{t+1,}&{0\leq t<1,}\\[4pt]{3t+2,}&{1\leq t<3,}\\[4pt]{6t-3,}&{t\geq 3}\end{array} \right.\)
28. \(1-t^{2}+u(t-2)\left(-\frac{t^{2}}{2}+2t+1\right)+u(t-4)(t-4)=\left\{\begin{array}{cc}{1-t^{2},}&{0\leq t<2,}\\[4pt]{-\frac{3t^{2}}{2}+2t+2,}&{2\leq t<4,}\\[4pt]{-\frac{3t^{2}}{2}+3t-2,}&{t\geq 4}\end{array} \right.\)
29. \(\frac{e^{-\tau s}}{s}\)
30. For each \(t\) only finitely many terms are nonzero.
33. \(1+\sum_{m=1}^{\infty}u(t-m);\quad\frac{1}{s(1-e^{-s})}\)
34. \(1+2\sum_{m=1}^{\infty}(-1)^{m}u(t-m);\quad\frac{1}{s};\quad\frac{1-e^{-s}}{1+e^{-s}}\)
35. \(1+\sum_{m=1}^{\infty}(2m+1)u(t-m);\quad\frac{e^{-s}(1+e^{-s})}{s(1-e^{-s})^{2}}\)
36. \(\sum_{m=1}^{\infty}(-1)^{m}(2m-1)u(t-m);\quad\frac{1}{s}\frac{(1-e^{s})}{(1+e^{s})^{2}}\)