Laplace Transform Quiz

Laplace Transform of Exponential Function

• The Laplace transform of the function ( f(t) = e^{at} ) is given by ( \mathcal{L}{e^{at}} = \frac{1}{s - a} ) for ( s > a ).

Theorems on Time Approaching Infinity

• The theorem used to evaluate the behavior of functions as time approaches infinity is the Final Value Theorem.
• The Final Value Theorem states that if the limit exists, ( \lim_{t \to \infty} f(t) = \lim_{s \to 0} s F(s) ).

Systems Represented by Linear ODEs

• Linear ordinary differential equations (ODEs) represent linear time-invariant (LTI) systems, which are systems where the principle of superposition applies.
• Such systems can describe various physical phenomena, including electrical circuits and mechanical systems.

Laplace Transform of Functions' Integrals

• The Laplace transform of an integral ( \mathcal{L}\left{\int_0^t f(\tau) d\tau\right} ) is given by ( \frac{F(s)}{s} ), where ( F(s) ) is the Laplace transform of ( f(t) ).

Time Delay Property of Laplace Transform

• The property of Laplace transform used for a time delay or translation in time is known as the Second Shifting Theorem.
• If ( f(t) ) is delayed by ( a ) time units, then ( \mathcal{L}{f(t - a)u(t - a)} = e^{-as}F(s) ), where ( u(t) ) is the unit step function.