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Write the function in terms of unit step functions. Find the Laplace transform of the given function. f(t) = 5, 0 ≤ t < 7 −3, t ≥ 7

Respuesta :

LammettHash

Rewrite f in terms of the unit step function:

[tex]f(t)=\begin{cases}5&\text{for }0\le t<7\\-3&\text{for }t\ge7\end{cases}[/tex]

[tex]\implies f(t)=5(u(t)-u(t-7))-3u(t-7)=5u(t)-8u(t-7)[/tex]

where

[tex]u(t)=\begin{cases}1&\text{for }t\ge0\\0&\text{for }t<0\end{cases}[/tex]

Recall the time-shifting property of the Laplace transform:

[tex]L[u(t-c)f(t-c)]=e^{-cs}L[f(t)][/tex]

and the Laplace transform of a constant function,

[tex]L[k]=\dfrac ks[/tex]

So we have

[tex]L[f(t)]=L[5u(t)-8u(t-7)]=5L[1]-8e^{-7s}L[1]=\boxed{\dfrac{5-8e^{-7s}}s}[/tex]

lhmarianateixeira

In this exercise you have to find the laplace transform:

[tex]L[f(t)]=\frac{5-8e^{-7s}}{s}[/tex]

Rewrite f in terms of the unit step function:

[tex]f(t)=\left \{ {{5, for 0\leq t\leq 7} \atop {-3, for t\geq 7}} \right. \\f(t)= 5(u(t)-u(t-7)-3u(t-7)=5u(t)-8u(t-7)[/tex]

Where:

[tex]u(t)= \left \{ {{1, t\geq 0} \atop {0, t<0}} \right.[/tex]

Recall the time-shifting property of the Laplace transform:

[tex]L[u(t-c)f(t-c)]= e^{-cs}L[f(t)][/tex]

and the Laplace transform of a constant function,

[tex]L[k]=\frac{k}{s}[/tex]

So we have:

[tex]L[f(t)]= L[5u(t)-8u(t-7)]= 5L[1]-8e^{-7s}L[1]= \frac{5-8e^{-7s}}{s}[/tex]

See more about Laplace transform at : brainly.com/question/2088771

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