A.8.1: Section 8.1 Answers
- Page ID
- 43783
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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.
- \(\frac{1}{s^{2}}\)
- \(\frac{1}{(s+1)^{2}}\)
- \(\frac{b}{s^{2}-b^{2}}\)
- \(\frac{-2s+5}{(s-1)(s-2)}\)
- \(\frac{2}{s^{3}}\)
2.
- \(\frac{s^{2}+2}{\left[(s-1)^{2}+1\right]\left[(s+1)^{2}+1\right]}\)
- \(\frac{2}{s(s^{2}+4)}\)
- \(\frac{s^{2}+8}{s(s^{2}+16)}\)
- \(\frac{s^{2}-2}{s(s^{2}-4)}\)
- \(\frac{4s}{(s^{2}-4)^{2}}\)
- \(\frac{1}{s^{2}+4}\)
- \(\frac{1}{\sqrt{2}}\frac{s+1}{s^{2}+1}\)
- \(\frac{5s}{(s^{2}+4)(s^{2}+9)}\)
- \(\frac{s^{3}+2s^{2}+4s+32}{(s^{2}+4)(s^{2}+16)}\)
4.
- \(f(3-)=-1,\: f(3)=f(3+)=1\)
- \(f(1-)=3,\: f(1)=4,\: f(1+)=1\)
- \(f\left(\frac{\pi }{2}-\right) =1,\: f\left(\frac{\pi }{2} \right) = f\left(\frac{\pi }{2}+ \right)=2,\: f(\pi -)=0,\: f(\pi )=f(\pi +)=-1\)
- \(f(1−) = 1,\: f(1) = 2,\: f(1+) = 1,\: f(2−) = 0,\: f(2) = 3,\: f(2+) = 6\)
5.
- \(\frac{1-e^{-(s+1)}}{s+1}+\frac{e^{-(s+2)}}{s+2}\)
- \(\frac{1}{s}+e^{-4s}\left(\frac{1}{s^{2}}+\frac{3}{s}\right)\)
- \(\frac{1-e^{-s}}{s^{2}}\)
- \(\frac{1-e^{-(s-1)}}{(s-1)^{2}}\)
7. \(\mathcal{L}(e^{\lambda t}\cos\omega t)=\frac{(s-\lambda )^{2}-\omega^{2}}{((s-\lambda )^{2}+\omega^{2})^{2}}\quad\mathcal{L}(e^{\lambda t}\sin\omega t)=\frac{2\omega (s-\lambda )}{((s-\lambda )^{2}+\omega ^{2})^{2}} \)
15.
- \(\tan ^{-1}\frac{\omega }{s},\quad s>0\)
- \(\frac{1}{2}\ln\frac{s^{2}}{s^{2}+\omega ^{2}},\quad s>0\)
- \(\ln\frac{s-b}{s-a},\quad s>\text{max}(a,b)\)
- \(\frac{1}{2}\ln\frac{s^{2}}{s^{2}-1},\quad s>1\)
- \(\frac{1}{4}\ln\frac{s^{2}}{s^{2}-4},\quad s>2\)
18.
- \(\frac{1}{s^{2}}\tanh\frac{s}{2}\)
- \(\frac{1}{s}\tanh\frac{s}{4}\)
- \(\frac{1}{s^{2}+1}\coth\frac{\pi s}{2}\)
- \(\frac{1}{(s^{2}+1)(1-e^{-\pi s})}\)