Problem 1. For each of the following signals (represented via Laplace transforms X(s)),what can you say about their time-domain representation x(t) without solving for all the parameters of an explicit

analytical solution? (note: please show the general structure of the partial fraction expansion for each system, but DO NOT estimate coefficients or fully invert the transforms to obtain an analytical solution). Specifically, \bullet \text { What are } x(0) \text { and } x(\infty) ? \text { - Does } x(t) \text { as } t \rightarrow \infty \text { converge to a steady-state or diverge to infinity? } • Is x(t) smooth or oscillatory? For each case, draw a qualitative yet informative sketch of the time-domain response. In addition, apply the Initial Value Theorem to determine the slope of the response at t = 0,and make this information part of your sketch. a)X(s)=\frac{6(s+2)}{\left(s^{2}+9 s+20\right)(s+4)} b)X(s)=\frac{10 s^{2}-3}{\left(s^{2}-6 s+10\right)(s+2)} c)X(s)=\frac{16 s+5}{\left(s^{2}+9\right)(s+3)} d)\frac{d^{3} x}{d t^{3}}+2\frac{d^{2} x}{d t^{2}}+2\frac{d x}{d t}+x=3S(t) S(t) is the unit step change; all initial conditions at zero