Exercise · ZW-UNIT-STEP-FUNCTION-003 · ★☆☆ Basic
If \(F(s)=\mathcal L\{f\}\) and \(c\ge0\), what is \(\mathcal L\{u_c(t)\,f(t-c)\}\)?
Choose one option.
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- B. \(e^{-cs}F(s)\)
Explanation
Delaying the function by \(c\) multiplies its transform by \(e^{-cs}\). Do not confuse it with \(F(s-c)\), which corresponds to multiplying by \(e^{ct}\).
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
More practice problems: unit step function
- ZW-UNIT-STEP-FUNCTION-001 What is mathcal L\u_3(t)\?
- ZW-UNIT-STEP-FUNCTION-002 What is mathcal L\u_2(t) (t-2)\?
- ZW-UNIT-STEP-FUNCTION-004 Write, using steps, the function equal to 2 for t<4 and 5 for t≥4.
- ZW-UNIT-STEP-FUNCTION-005 What is mathcal L\u_1(t)-u_3(t)\?
- ZW-UNIT-STEP-FUNCTION-006 What is mathcal L\u_1(t)sin (t-1)\?
- ZW-UNIT-STEP-FUNCTION-007 What is mathcal L^-1\(e^-4s)/(s+1)\?
- ZW-UNIT-STEP-FUNCTION-008 What is mathcal L\t u_2(t)\? (Careful: the factor is t, not t-2.)
- ZW-UNIT-STEP-FUNCTION-009 The solution of y'+y=u_1(t), y(0)=0, is y=u_1(t)bigl(1-e^-(t-1)bigr). What is y(2)?