3 years ago

< The map below is being reduced by a scale factor of 2/3 and printed at

Mathematics

1 answer:

Orlov [11]3 years ago

3 0

Answer:

Step-by-step explanation:

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Use Definition 7.1.1. DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t ≥ 0. Then the integral ℒ{f(t)} = [inf

Alla [95]

$f(t)=\begin{cases}-1&\text{for }0\le t$

Write $f(t)$ in terms of the unit step function $u(t)$ :

$f(t)=-1(u(t)-u(t-1))+1u(t-1)=2u(t-1)-u(t)$

where

$u(t)=\begin{cases}1&\text{for }t\ge0\\0&\text{for }t$

Then the Laplace transform of $f(t)$ , denoted $F(s)$ , is

$F(s)=\displaystyle\int_0^\infty f(t)e^{-st}\,\mathrm dt$

$F(s)=\displaystyle2\int_1^\infty e^{-st}\,\mathrm dt-\int_0^\infty e^{-st}\,\mathrm dt$

$\implies F(s)=\dfrac{2e^{-s}}s-\dfrac1s=\dfrac{2e^{-s}-1}s$

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3 years ago