Exercise · ZW-UNIT-STEP-FUNCTION-008 · ★★★ Advanced

What is \(\mathcal L\{t\,u_2(t)\}\)? (Careful: the factor is \(t\), not \(t-2\).)

Choose one option.

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  • B. \(e^{-2s}\left(\dfrac1{s^2}+\dfrac2s\right)\)

Explanation

Rewrite \(t=(t-2)+2\): \(t\,u_2=(t-2)u_2+2u_2\), whose transform is \(e^{-2s}/s^2+2e^{-2s}/s\).

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Review the concept: Unit step function

The unit step function \(u_c(t)\), with \(c\ge0\), equals 0 while \(t<c\) and equals 1 from \(t=c\) on; it is a switch that turns on at time \(c\).

\[u_c(t)=\begin{cases}0,&t

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