Question

3. Use the definition of the Laplace Transform to find \( \mathscr{L}\{f(t)\} \) where \( f(t)=\left\{\begin{array}{ll}1, & 0 \leq t

A) 1-e⁻ˢ/s
B) 1-e⁻ˢ/s + 2e⁻ˢ-e⁻²ˢ/s
C) 1-e⁻ˢ/s + 2e⁻ˢ-e⁻²ˢ/s+e⁻²ˢ/s
D) 1-e⁻ˢ/s + 2e⁻ˢ-e⁻²ˢ/s+e⁻²ˢ/s

Answer 1

The problem requires finding the Laplace Transform of a piecewise function using its definition. Due to missing details of the function, a specific solution can't be provided. The general approach involves integrating each piece of the function over its respective interval, multiplied by
`e^(-st).`

Step-by-step explanation:

The student is asked to use the definition of the Laplace Transform to find
`\( \mathscr{L}\{f(t)\} \)` f(t) \).

To solve for
`\( \mathscr{L}\{f(t)\} \)`of the piecewise function multiplied by e^{-st} from the appropriate bounds. The bounds for integration are based on the intervals where the function changes its behavior. Here, since no explicit function is given, we're unable to provide the exact Laplace Transform solution. However, we could say that if the function follows a form like f(t) = 1 from t = 0 to t = A and f(t) = 2 for t > A, we'd perform two integrations: integrate
`1*e^(-st)`o A, and integrate
`2*e^(-st)` infinity.

The final answer would be combined from these two integrals. Unfortunately, there's a lack of clarity regarding the question's function details which prevents providing a definitive answer.