Determine the Laplace Transform of the functionf(t)=−9δ(t−10)
Question
Determine the Laplace Transform of the functionf(t)=−9δ(t−10)
Solution
The Laplace Transform of a function f(t) is given by the integral from 0 to infinity of e^(-st)f(t) dt.
However, the function f(t) = -9δ(t-10) is a shifted delta function. The delta function δ(t) is a mathematical construct which is used to model an impulse. It is zero everywhere except at t = 0, where it is infinite. The area under the delta function is 1.
When the delta function is shifted by a, δ(t-a), it represents an impulse at t = a.
The Laplace Transform of a delta function δ(t-a) is e^(-as).
So, the Laplace Transform of -9δ(t-10) is -9e^(-10s).
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