Question Consider the function if 0 st< 8x sin(t – 8x) if 8x St. a. Use the graph of this function to write it in terms of the Heaviside function. Use h(t – a) for the Heaviside function shifted a units horizontally. s) = {am f() = help (formulas) b. Find the Laplace transform F(s) = L(S)). F(s) = L {f(1)} = help (formulas)

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Transcribed Image Text: Consider the function if 0 st< 8x sin(t – 8x) if 8x St. a. Use the graph of this function to write it in terms of the Heaviside function. Use h(t – a) for the Heaviside function shifted a units horizontally. s) = {am f() = help (formulas) b. Find the Laplace transform F(s) = L(S)). F(s) = L {f(1)} = help (formulas)

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Step 1Solution:- criven furction isf(t)=\left\{\begin{array}{ll}0 & \text { if } 0 \leq t<8 \pi \\\sin (t-8 \pi) & \text { if } 8 \pi \leq t\end{array}\right.Step 2a) Civen function can be wnitten in the form of heaviside function or unitstep function\begin{array}{l} f(t)= 0 \cdot(u(t-\theta)-u(t-\theta \pi)) \\+\sin (t-\theta \pi)(u(t-\theta \pi)) \\f(t)= \sin (t-\theta \pi) u(t-\theta \pi) \\\text { or } f(t)=\sin (t-\theta \pi) h(t-\theta \pi)\end{array}taking laplave transform of eqn (1) we get\begin{aligned}\mathcal{L}\{f(t)\} & =\mathcal{L}\{\sin (t-8 \pi) h(t-8 \pi)\} \\& =e^{-8 \pi s} \mathcal{L}\{\sin t\} \\& =e^{-8 \pi s} \frac{1}{s^{2}+1} \quad\left\{\begin{array}{l}\{\{f(t-a) u(t-a)\} \\=e^{-a s} F(s)\end{array}\right. \\\mathcal{L}\{f(t)\} & =\frac{e^{-\theta \pi s}}{s^{2}+1}\end{aligned}Af ...