With the assistance of Eq. [14] and showing appropriate intermediate steps, compute the

Identify all the complex frequencies present in these real functions: (a) \((2e^{?100t} + e^{?200t} )\ sin\ 2000t\) ; (b) \((2 ? e^{?10t})\ cos(4t + \phi)\); (c) \(e^{?10t}\ cos\ 10t\ sin\ 40t\).

Use real constants A, B, C, \(\phi\), and so forth, to construct the general form of the real function of time for a current having components at these frequencies: (a) 0, 10, \(?10\ s^{?1}\); (b) ?5, j8, \(?5 ? j8\ s^{?1}\); (c) ?20, 20, ?20 + j20, \(20 ? j20\ s^{?1}\).

Give the phasor current that is equivalent to the time-domain current: (a) \(24\ sin(90t + 60^{\circ})\ A\); (b) \(24e^{-10t}\ cos(90t + 60^{\circ})\ A\); (c) \(24e^{-10t}\ cos\ 60^{\circ}\ \times\ cos\ 90t\ A\). If \(\mathbf{V}=12\angle35^{\circ}\mathrm{\ V}\), find v(t) for s equal to (d) 0; (e) \(-20\ s^{-1}\); (f) \(-20 + j5\ s^{-1}\).

Let \(f(t) = ?6e^{?2t}\ [u(t + 3) ? u(t ? 2)]\). Find the (a) two-sided F(s); (b) one-sided F(s).

Determine V(s) if v(t) equals (a) \(4 \delta (t) ? 3u(t)\); (b) \(4 \delta (t ? 2) ? 3tu(t)\); (c) [u(t)] [u(t ? 2)].

Determine v(t) if V(s) equals (a) 10; (b) \(10 / s\); (c) \(10/s^2\); (d) 10/[s(s + 10)]; (e) 10s/(s + 10).

Given \(\mathbf{H}(\mathbf{s})=\frac{2}{\mathbf{s}}-\frac{4}{\mathbf{s}^{2}}+\frac{3.5}{(\mathbf{s}+10)(\mathbf{s}+10)}\), obtain h(t).

Given the function \(\mathbf{Q}(\mathbf{s})=\frac{3 \mathbf{s}^{2}-4}{\mathbf{s}^{2}}\), find q(t).

Given the function \(\mathbf{Q}\mathbf(s)=\frac{11 \mathbf{s}+30}{\mathbf{s}^{2}+3 \mathbf{s}}\), find q(t).

Determine g(t) if \(\mathbf{G}(\mathbf{s})=\frac{3}{\mathbf{s}^{3}+5 \mathbf{s}^{2}+8 \mathbf{s}+4}\).

Use Laplace transform methods to find i(t) in the circuit of Fig. 14.4.

Find v(t) at t = 800 ms for the circuit of Fig. 14.7.

Obtain the Laplace transform of the time function shown in Fig. 14.9.

Without finding f(t) first, determine \(f (0^+)\) and \(f(\infty)\) for each of the following transforms: (a) \(4e^{-2s}\ (s + 50) / s\); (b) \((s^2 + 6) / (s^2 + 7)\); (c) \((5s^2 + 10) / [2s(s^2 + 3s + 5)]\).

Determine the conjugate of each of the following: (a) \(8 ? j\) ; (b) \(8e^{?9t}\) ; (c) \(22.5\); (d) \(4e^{j9}\); (e) \(j2e^{?j11}\).

Obtain the time-domain expression which corresponds to the following s-domain functions: (a) \(\frac{\mathbf{s}}{(\mathbf{s}+2)^{3}}\); (b) \(\frac{4}{(s+1)^{4}(s+1)^{2}}\); (c) \(\frac{1}{\mathbf{s}^{2}(\mathbf{s}+4)^{2}(\mathbf{s}+6)^{3}}-\frac{2 \mathbf{s}^{2}}{\mathbf{s}}+9\). (d) Verify your solutions with MATLAB.

Compute the complex conjugate of each of the following expressions: (a) ?1; (b) \(\frac{-j}{5 \angle 20^{\circ}}\); (c) \(5e^{-j5} + 2e^{j3}\); (d) \((2+j)\left(8 \angle 30^{\circ}\right) e^{j 2 t}\).

Employ the initial-value theorem to determine the initial value of each of the following time-domain functions: (a) \(2u(t)\); (b) \(2e^{?t}\ u(t)\); (c) \(u(t ? 6)\); (d) \(cos\ 5t\ u(t)\).

Several real voltages are written down on a piece of paper, but coffee spills across half of each one. Complete the voltage expression if the legible part is (a) \(5e^{-j50t}\); (b) \((2 + j)e^{j9t}\); (c) \((1 - j)e^{j78t}\); (d) \(-je^{-5t}\). Assume the units of each voltage are volts (V).

Determine which of the following functions are appropriate for the final-value theorem: (a) \(\frac{1}{(s-1)}\); (b) \(\frac{10}{\mathbf{s}^{2}-4 s+4}\); (c) \(\frac{13}{\mathbf{s}^{3}-5 \mathbf{s}^{2}+8 \mathbf{s}-6}\); (d) \(\frac{3}{2 s^{3}-10 s^{2}+16 s-12}\).

State the complex frequency or frequencies associated with each function: (a) \(f(t) = sin\ 100t\); (b) \(f(t) = 10\); (c) \(g(t) = 5e^{-7t}\ cos\ 80t\); (d) \(f(t) = 5e^{8t}\); (e) \(g(t) = (4e^{-2t} - e^{-t})\ cos(4t - 95^{\circ})\).

The voltage \(v(t) = 8e^{?2t}\ u(t)\ V\) is applied to an unlabeled two-terminal device. Your assistant misunderstands you and only records the s-domain current which results. Determine what type of element it is and its value if I(s) is equal to (a) \(\frac{1}{s+2} A\); (b) \(\frac{4}{\mathbf{s}(\mathbf{s}+2)} \mathrm{A}\).

For each of the following functions, determine both the complex frequency s as well as \(\mathbf{s}^{*}\): (a) \(7e^{-9t}\ sin\ (100t + 9^{\circ})\); (b) \(cos\ 9t\); (c) \(2\ sin\ 45t\); (d) \(e^{7t}\ cos\ 7t\).

(a) Create an s-domain function F(s) which corresponds to an initial value \(f (0^- ) = 16\) yet has an indeterminate final value. (b) Obtain an expression for f(t). (c) If this waveform represents the voltage across a 2 F capacitor, determine the current flowing through the device (assume the passive sign convention).

Use real constants A, B, \(\theta\), \(\phi\), etc. to construct the general form of a real time function characterized by the following frequency components: (a) \(10 - j3\ s^{-1}\); (b) \(0.25\ s^{-1}\); (c) 0, 1, -j, 1 + j (all \(s^{-1}\)).

For the circuit of Fig. 14.19, let \(i_s (t) = 5u(t)\ A\) and \(v_s (t) = e^{?4t}\ u(t + 1)\ V\). Working initially in the s-domain, obtain an expression for \(i_C(t)\) valid for t > 0.

The following voltage sources \(Ae^{Bt}\ cos(Ct + \theta)\) are connected (one at a time) to a \(280\ \Omega\) resistor. Calculate the resulting current at t = 0, 0.1, and 0.5 s, assuming the passive sign convention: (a) A = 1 V, B = 0.2 Hz, C = 0, \(\theta = 45^{\circ}\); (b) A = 285 mV, B = -1 Hz, C = 2 rad/s, \(\theta = -45^{\circ}\).

Referring to the circuit depicted in Fig. 14.19 and working in the s-domain to develop an expression for \(\mathbf{I}_{C}(\mathbf{s})\), determine \(i_C(t)\) for t > 0 if \(i_s (t) = 2u(t + 2)\ A\) and \(v_s (t)\) is equal to (a) \(2u(t)\ V\); (b) \(te^{-t}\ u(t)\ V\).

Your neighbor’s cell phone interferes with your laptop speaker system whenever the phone is connecting to the local network. Connecting an oscilloscope to the output jack of your computer, you observe a voltage waveform that can be described by a complex frequency \(s = -1 + j200 \pi\ s^{-1}\). (a) What can you deduce about your neighbor’s movements? (b) The imaginary part of the complex frequency starts to decrease suddenly. Alter your deduction as appropriate.

For the circuit of Fig. 14.20, \(\mathbf{I}(\mathrm{s})=5\frac{\mathrm{s}+1}{(\mathrm{~s}+1)^2+10^4}\mathrm{\ A}\). (a) Determine the initial value of the inductor current. (b) Determine the final value of the inductor voltage, assuming it is defined consistent with the passive sign convention.

Compute the real part of each of the following complex functions: (a) \(v(t) = 9e^{-j4t}\ V\); (b) \(v(t) = 12 - j9\ V\); (c) \(5\ cos\ 100t - j43\ sin\ 100t\ V\); (d) \((2+j)e^{j3t}\mathrm{\ V}\).

Evaluate the following: (a) \(\int _{-\infty} ^{+\infty}\ e^{-100} \delta(t-\frac{1}{5})\ dt\); (b) \(\int _{-\infty} ^{+\infty}\ 4t \delta(t - 2)\ dt\); (c) \(\int _{-\infty} ^{+\infty}\ 4t^2 \delta(t-1.5)\ dt\); (d) \(\frac{\int _{-\infty} ^{+\infty}\ (4 - t) \delta (t - 1)\ dt}{\int _{-\infty} ^{+\infty}\ (4 - t) \delta (t + 1)\ dt}\).

Your new assistant has measured the signal coming from a piece of test equipment, writing \(v(t) = V_x e^{(-2 + j60)t}\), where \(V_x = 8 - j100\ V\). (a) There is a missing term. What is it, and how can you tell it’s missing? (b) What is the complex frequency of the signal? (c) What is the significance of the fact that \(Im\{V_x\}\ >\ Re\{V_x\}\)? (d) What is the significance of the fact that \(|Re\{s\}|\ <\ |Im\{s\}|\)?

Determine the inverse transform of F(s) equal to (a) \(5 + \frac{5}{s^2} - \frac{5}{(s + 1)}\); (b) \(\frac{1}{s} + \frac{5}{(0.1s + 4}-3\); (c) \(-\frac{1}{2s} + \frac{1}{(0.5s)^2} + \frac{4}{(s + 5) (s + 5)} + 2\); (d) \(\frac{4}{(s + 5)(s + 5)} + \frac{2}{s+1} + \frac{1}{s+3}\).

State the time-domain voltage v(t) which corresponds to the voltage \(\mathbf{V}=19\angle84^{\circ}\mathrm{\ V}\) if s is equal to (a) \(5\ s^{-1}\); (b) 0; (c) \(-4 + j\ s^{-1}\).

Obtain an expression for g(t) if G(s) is given by (a) \(\frac{3(s+1)}{(s+1)^2} + \frac{2s}{s^2} - \frac{1}{(s + 2)^2}\) (b) \(-\frac{10}{(s+3)^3}\); (c) \(19-\frac{8}{(s+3)^2}+\frac{18}{s^2 + 6s +9}\).

For the circuit of Fig. 14.10, the voltage source is chosen such that it can be represented by the complex frequency domain function \(\mathbf{V} e^{\mathrm{st}}\), with \(\mathrm{V}=2.5 \angle -20^{\circ}\mathrm{\ V}\) and \(s = -1 + j100\ s^{-1}\). Calculate (a) \(s^*\); (b) v(t), the time- domain representation of the voltage source; (c) the current i(t).

Reconstruct the time-domain function if its transform is (a) \(\frac{s}{s(s+2)}\); (b) 1; (c) \(3 \frac{s+2}{(s^2 + 2s +4)}\); (d) \(4 \frac{s}{2s+3}\).

With regard to the circuit depicted in Fig. 14.10, determine the time-domain voltage v(t) which corresponds to a frequency-domain current \(\mathbf{i}(t)=5\angle30^{\circ}\mathrm{\ A}\) for a complex frequency of (a) \(s = -2 + j2\ s^{-1}\); (b) \(s = -3 + j\ s^{-1}\).

Determine the inverse transform of V(s) equal to (a) \(\frac{s^2 +2}{s} +1\); (b) \(\frac{s+8}{s}+\frac{2}{s^2}\); (c) \(\frac{s+1}{s(s+2)} + \frac{2s^2 -1}{s^2}\); (d) \(\frac{s^2 +4s + 4}{s}\).

For the circuit depicted in Fig. 14.11, take \(\mathbf{s}=-200+j150\mathrm{\ s}^{-1}\). Determine the ratio of the frequency-domain voltages \(\mathbf{V}_{2}\) and \(\mathbf{V}_{1}\), which correspond to \(v_2(t)\) and \(v_1(t)\), respectively.

Obtain the time-domain expression which corresponds to each of the following s-domain functions: (a) \(2\frac{3\mathbf{s}+\frac{1}{2}}{\mathbf{s}^2+3\mathbf{s}}\); (b) \(7-\frac{\mathbf{s}+\frac{1}{s}}{\mathbf{s}^{2}+3 \mathbf{s}+1}\); (c) \(\frac{2}{\mathbf{s}^{2}}+\frac{1}{\mathbf{s}}+\frac{\mathbf{s}+2}{\left(\frac{s}{2}\right)^{2}+4 s+6}\); (d) \(\frac{2}{(\mathbf{s}+1)(\mathbf{s}+1)}\); (e) \(\frac{14}{(s+1)^{2}(s+4)(s+5)}\).

If the complex frequency describing the circuit of Fig. 14.11 is \(s = ?150 + j100\ s^{?1}\), determine the time-domain voltage which corresponds to a frequency-domain voltage \(\mathbf{V}_2=5\angle{-25^{\circ}}\mathrm{\ V}\).

Find the inverse Laplace transform of the following: (a) \(\frac{1}{s^{2}+9 s+20}\); (b) \(\frac{4}{\mathbf{s}^{3}+18 \mathbf{s}^{2}+17 \mathbf{s}}+\frac{1}{\mathbf{s}}\); (c) \((0.25) \frac{1}{\left(\frac{s}{2}\right)^{2}+1.75 s+2.5}\); (d) \(\frac{3}{\mathbf{s}(\mathbf{s}+1)(\mathbf{s}+4)(\mathbf{s}+5)(\mathbf{s}+2)}\); (e) Verify your answers with MATLAB.

Calculate the time-domain voltage v in the circuit of Fig. 14.12 if the frequency-domain representation of the current source is \(2.3 \angle 5^{\circ} \mathrm{A}\) at a complex frequency of \(s = -1 + j2\ s^{-1}\).

Determine the inverse Laplace transform of each of the following s-domain expressions: (a) \(\frac{1}{(s+2)^{2}(s+1)}\); (b) \(\frac{\mathbf{s}}{\left(\mathbf{s}^{2}+4 \mathbf{s}+4\right)(\mathbf{s}+2)}\); (c) \(\frac{8}{\mathbf{s}^{3}+8 \mathbf{s}^{2}+21 \mathbf{s}+18}\) (d) Verify your answers with MATLAB.

The circuit of Fig. 14.12 is operated for an extended period of time without interruption. The frequency-domain voltage which develops across the three elements can be represented as \(1.8\angle75^{\circ}\mathrm{\ V}\) at a complex frequency of \(s = -2 + j1.5\ s^{-1}\). Determine the time-domain current \(i_s\).

Given the following expressions in the s-domain, determine the corresponding time-domain functions: (a) \(\frac{1}{3 \mathbf{s}}-\frac{1}{2 \mathbf{s}+1}+\frac{3}{\mathbf{s}^{3}+8 \mathbf{s}^{2}+16 \mathbf{s}}-1\); (b) \(\frac{1}{3 \mathbf{s}+5}+\frac{3}{\mathbf{s}^{3} / 8+0.25 \mathbf{s}^{2}}\); (c) \(\frac{2 \mathbf{s}}{(\mathbf{s}+a)^{2}}\).

The circuit of Fig. 14.13 is driven by \(v_S(t) = 10\ cos\ 5t\ V\). (a) Determine the complex frequency of the source. (b) Determine the frequency-domain representation of the source: (c) Compute the frequency-domain representation of \(i_x\). (d) Obtain the time-domain expression for \(i_x\).

Compute \(\mathscr{L}^{-1}\{\mathbf{G}(\mathbf{s})\}\) if G(s) is given by (a) \(\frac{3 \mathbf{s}}{(\mathbf{s} / 2+2)^{2}(\mathbf{s}+2)}\); (b) \(3-3 \frac{\mathbf{s}}{\left(2 \mathbf{s}^{2}+24 \mathbf{s}+70\right)(\mathbf{s}+5)}\); (c) \(2-\frac{1}{\mathbf{s}+100}+\frac{\mathbf{s}}{\mathbf{s}^{2}+100}\); (d) \(\mathscr{L}\{t u(2 t)\}\).

The frequency-domain current \(\mathbf{I}_{x}\) which flows through the \(2.2\ \Omega\) resistor of Fig. 14.13 can be represented as \(2\angle10^{\circ}\mathrm{\ A}\) at a complex frequency of \(s = -1 + j0.5\ s^{-1}\). Determine the time-domain voltage \(v_s\).

Take the Laplace transform of the following equations: (a) \(5\ di/dt - 7\ d^2 i / dt^2 + 9i = 4\); (b) \(m \frac{d^{2} p}{d t^{2}}+\mu_{f} \frac{d p}{d t}+k p(t)=0\), the equation that describes the “force-free’’ response of a simple shock absorber system; (c) \(\frac{d \Delta n_{p}}{d t}=-\frac{\Delta n_{p}}{\tau}+G_{L}\), with \(\tau = constant\), which describes the recombination rate of excess electrons \((\Delta n_p )\) in p-type silicon under optical illumination (\(G_L\) is a constant proportional to the light intensity).

Let \(i_{s1} = 20e^{-3t}\ cos\ 4t\) and \(i_{s2}=30e^{-3t}\sin4t\mathrm{\ A}\) in the circuit of Fig. 14.14. (a) Work in the frequency domain to find \(\mathbf{V}_{x}\). (b) Find \(v_x (t)\).

With regard to the circuit depicted in Fig. 14.15, the initial voltage across the capacitor is \(v(0^-) = 1.5\ V\) and the current source is \(\mathbf{i}_s=700u(t)\mathrm{\ mA}\). (a)Write the differential equation which arises from KCL, in terms of the nodal voltage v(t). (b) Take the Laplace transform of the differential equation. (c) Determine the frequency-domain representation of the nodal voltage. (d) Solve for the time-domain voltage v(t).

Calculate, with the assistance of Eq. [14] (and showing intermediate steps), the Laplace transform of the following: (a) \(2.1\ u(t)\); (b) \(2u(t ? 1)\); (c) \(5u(t ? 2) ? 2u(t)\); (d) \(3u(t ? b)\), where \(b\ >\ 0\).

For the circuit of Fig. 14.15, if \(I_s = \frac{2}{s+1}\ mA\), (a) write the time-domain nodal equation in terms of v(t); (b) solve for V(s); (c) determine the time-domain voltage v(t).

Employ the one-sided Laplace transform integral (with intermediate steps explicitly included) to compute the s-domain expressions which correspond to the following: (a) \(5u(t ? 6)\); (b) \(2e^{?t}\ u(t)\); (c) \(2e^{?t}\ u(t ? 1)\); (d) \(e^{?2t}\ sin\ 5tu(t)\).

The voltage source in the circuit of Fig. 14.4 is replaced with one whose s-domain equivalent is \(\frac{2}{s} - \frac{1}{s+1}\ V\). The initial condition is unchanged. (a) Write the s-domain KVL equation in terms of I(s). (b) Solve for i(t).

With the assistance of Eq. [14] and showing appropriate intermediate steps, compute the one-sided Laplace transform of the following: (a) \((t ? 1)u(t ? 1)\); (b) \(t2u(t)\); (c) \(sin\ 2tu(t)\); (d) \(cos\ 100t\ u(t)\).

For the circuit of Fig. 14.16, \(v_s(t) = 2u(t)\ V\) and the capacitor initially stores zero energy. (a) Write the time-domain loop equation in terms of the current i(t). (b) Obtain the s-domain representation of this integral equation. (c) Solve for i(t).

The Laplace transform of tf(t), assuming \(\mathscr{L}\{f(t)\}=\mathbf{F}(\mathbf{s})\), is given by \(-\frac{d}{d \mathbf{s}} \mathbf{F}(\mathbf{s})\). Test this by comparing the predicted result to what is found by directly employing Eq. [14] for (a) \(tu(t)\); (b) \(t^2\ u(t)\); (c) \(t^3u(t)\); (d) \(te^{?t}\ u(t)\).

The s-domain representation of the voltage source in Fig. 14.16 is \(\mathbf{V}_s(\mathbf{s})=\frac{2}{\mathbf{s}+1}\mathrm{\ V}\). The initial voltage across the capacitor, defined using the passive sign convention in terms of the current i , is 4.5 V. (a) Write the time domain integral equation that arises from application of KVL. (b) By first solving for I(s), determine the time-domain current i(t).

For the following functions, specify the range of \(\sigma_{0}\) for which the one-sided Laplace transform exists: (a) \(t + 4\); (b) \((t + 1)\ (t ? 2)\); (c) \(e^{?t/2}\ u(t)\); (d) \(sin\ 10t\ u(t + 1)\).

If the current source of Fig. 14.17 is given by 450u(t) mA, and \(i_x (0) = 150\ mA\), work initially in the s-domain to obtain an expression for v(t) valid for t > 0.

Show, with the assistance of Eq. [14], that \(\mathscr{L}\{f(t)+g(t)+h(t)\}=\mathscr{L}\{f(t)\}+\mathscr{L}\{g(t)\}+\mathscr{L}\{h(t)\}\).

Obtain, through purely legitimate means, an s-domain expression which corresponds to the time-domain waveform plotted in Fig. 14.18.

Determine F(s) if f(t) is equal to (a) \(3u(t ? 2)\); (b) \(3e^{?2t}\ u(t) + 5u(t)\); (c) \(\delta (t) + u(t) - t u(t)\); (d) \(5 \delta (t)\).

Apply the Routh test to the following system functions, and state whether the system is stable or unstable: (a) \(\mathbf{H}(\mathbf{s})=\frac{\mathbf{s}-500}{\mathbf{s}^{3}+13 \mathbf{s}^{2}+47 \mathbf{s}+35}\); (b) \(\mathbf{H}(\mathbf{s})=\frac{\mathbf{s}-500}{\mathbf{s}^{3}+13 \mathbf{s}^{2}+\mathbf{s}+35}\).

Obtain an expression for G(s) if g(t) is given by (a) \([5u(t)]^2 ? u(t)\); (b) \(2u(t) ? 2u(t ? 2)\); (c) \(t u(2 t)\); (d) \(2e^{?t}\ u(t) + 3u(t)\).

Apply the Routh test to the following system functions, and state whether the system is stable or unstable; then factor each denominator to identify the poles of H(s) and verify the accuracy of the Routh test for these functions: (a) \(\mathbf{H}(\mathbf{s})=\frac{4 \mathbf{s}}{\mathbf{s}^{2}+3 \mathbf{s}+8}\); (b) \(\mathbf{H}(\mathbf{s})=\frac{\mathbf{s}-9}{\mathbf{s}^{2}+2 \mathbf{s}+1}\).

Without recourse to Eq. [15], obtain an expression for f(t) if F(s) is given by (a) \(\frac{1}{s}\); (b) \(1.55 - \frac{2}{s}\); (c) \(\frac{1}{s+1.5}\); (d) \(\frac{5}{s^2} + \frac{5}{s} + 5\). (Provide some brief explanation of how you arrived at your solution.)

Employ the initial-value theorem to determine the initial value of each of the following time-domain functions: (a) \(u(t ? 3)\); (b) \(2e^{?(t?2)}\ u(t ? 2)\); (c) \(\frac{u(t-2)+[u(t)]^{2}}{2}\); (d) \(sin\ 5t\ e^{-2t}\ u(t)\).

Obtain an expression for g(t) without employing the inverse Laplace transform integral, if G(s) is known to be (a) \(\frac{1.5}{(s+9)^{2}}\); (b) \(\frac{2}{s}-0\); (c) \(\pi\); (d) \(\frac{a}{(\mathrm{s}+1)^2}-a,\ a>0\). (Provide some brief explanation of your solution process for each.)

Make use of the final-value theorem (if appropriate) to ascertain \(f (\infty)\) for (a) \(\frac{1}{\mathbf{s}+2}-\frac{2}{\mathbf{s}}\); (b) \(\frac{2 \mathbf{s}}{(\mathbf{s}+2)(\mathbf{s}+1)}\); (c) \(\frac{1}{(\mathbf{s}+2)(\mathbf{s}+4)}+\frac{2}{\mathbf{s}}\); (d) \(\text { (d) } \frac{1}{\left(s^{2}+s-6\right)(s+9)}\).

Evaluate the following: (a) \(\delta (t)\) at \(t = 1\); (b) \(5 \delta (t + 1) + u(t + 1)\) at \(t = 0\); (c) \(\int _{-1} ^{2}\ \delta(t)\ dt\); (d) \(3 - \int _{-1} ^{2}\ 2 \delta (t)\ dt\).

Without recourse to f(t), determine \(f(0^+)\) and \(f (\infty)\) (or show they do not exist) for each of the following s-domain expressions: (a) \(\frac{1}{s+18}\); (b) \(10\left(\frac{1}{\mathbf{s}^{2}}+\frac{3}{\mathbf{s}}\right)\); (c) \(\frac{\mathbf{s}^{2}-4}{\mathbf{s}^{3}+8 \mathbf{s}^{2}+4 \mathbf{s}}\); (d) \(\frac{\mathbf{s}^{2}+2}{\mathbf{s}^{3}+3 \mathbf{s}^{2}+5 \mathbf{s}}\).

Evaluate the following: (a) \([\delta (2t)]^2\) at \(t = 1\); (b) \(2 \delta (t ? 1) + u(?t + 1)\) at \(t = 0\); (c) \(\frac{1}{3} \int_{-0.001} ^{0.003}\ \delta(t)\ d(t)\); (d) \(\frac{1}{[\frac{1}{2} \int_0 ^2\ \delta (t - 1)\ dt| ]^2}\).

Apply the initial- or final-value theorems as appropriate to determine \(f(0^+)\) and \(f (\infty)\) for the following functions: (a) \(\frac{\mathbf{s}+2}{\mathbf{s}^{2}+8 \mathbf{s}+4}\); (b) \(\frac{1}{\mathbf{s}^{2}(\mathbf{s}+4)^{2}(\mathbf{s}+6)^{3}}-\frac{2 \mathbf{s}^{2}}{\mathbf{s}}+9\); (c) \(\frac{4 s^{2}+1}{(s+1)^{2}(s+2)^{2}}\).

Evaluate the following expressions at t = 0: (a) \(+\int _{-\infty} ^{\infty}\ 2 \delta(t-1)\ dt\); (b) \(\frac{\int _{-\infty} ^{+\infty}\ \delta (t + 1)\ dt}{u(t + 1)}\); (c) \(\frac{\sqrt{3\ \int _{-\infty} ^{+\infty}\ \delta (t - 2)\ dt}}{[u(1 - t)]^3} - \sqrt{u(t + 2)}\); (d) \([\frac{\int _{-\infty} ^{+\infty}\ \delta (t - 1)\ dt}{\int _{-\infty} ^{+\infty}\ \delta (t + 1)\ dt}|]^2\).