Exercise · ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-001 · ★☆☆ Basic
If \(f(t)=e^{3t}\), then \(F(s)=\dfrac1{s-3}\) and \(f(0)=1\). What is \(\mathcal L\{f'\}\)?
Choose one option.
Show answer
- A. \(\dfrac{3}{s-3}\)
Explanation
Use \(\mathcal L\{f'\}=sF(s)-f(0)=\frac{s}{s-3}-1=\frac3{s-3}\), which matches \(\mathcal L\{3e^{3t}\}\). Forgetting to subtract \(f(0)\) gives \(\frac{s}{s-3}\).
Review the concept: Laplace transform of a derivative
The Laplace transform \(\mathcal L\) of the derivative \(f'\) of a function \(f(t)\) with transform \(F(s)\) is an expression in \(F(s)\) and the initial value \(f(0)\).
\[\mathcal L\{f'\}(s)=sF(s)-f(0)\]
More practice problems: laplace transform of a derivative
- ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-002 Which is the correct formula for mathcal L\f''\?
- ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-003 For f(t)=sin t we have F(s)=1s^2+1 and f(0)=0. What is mathcal L\f'\?
- ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-004 Transforming y'+2y=0, y(0)=3, what is Y(s)=mathcal L\y\?
- ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-005 Transforming y'+y=1, y(0)=0, what Y(s) results?
- ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-006 Transforming y''+4y=0, y(0)=1, y'(0)=0, what Y(s) results?
- ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-007 Let f(t)=t^2. Using the derivative rule with F(s)=2/s^3, what is mathcal L\f'\?
- ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-008 Transforming y'+3y=e^-t, y(0)=1, what Y(s) results?
- ZW-LAPLACE-TRANSFORM-OF-A-DERIVATIVE-009 Transforming y''-3y'+2y=0, y(0)=1, y'(0)=0, what Y(s) results?