Find the Laplace Transform of  f(t)=0     0≤t<3
                                                                 = 4      t≥3

The Laplace Transform \( \mathcal{L}\{f(t)\} \) of a piecewise function is found by taking the transform of each piece of the function within its respective interval. The given function is:

\[
f(t) = 
\begin{cases} 
0, & 0 \leq t < 3 \\
4, & t \geq 3 
\end{cases}
\]

### Step 1: Express the function using the Heaviside (Unit Step) function
To handle piecewise functions, we can rewrite \( f(t) \) using the Heaviside step function \( u(t) \). The function \( f(t) \) can be expressed as:

\[
f(t) = 4 \cdot u(t - 3)
\]

Where \( u(t - 3) \) shifts the function so that it becomes active (non-zero) at \( t = 3 \).

### Step 2: Apply the Laplace Transform
The Laplace Transform of \( u(t - a) \cdot f(t - a) \) is given by:

\[
\mathcal{L}\{u(t - a) \cdot f(t - a)\} = e^{-as} \cdot \mathcal{L}\{f(t)\}
\]

In this case, \( f(t - 3) = 4 \), and the Laplace Transform of a constant \( 4 \) is:

\[
\mathcal{L}\{4\} = \frac{4}{s}
\]

Thus, the Laplace Transform of \( f(t) \) is:

\[
\mathcal{L}\{f(t)\} = 4 \cdot \frac{e^{-3s}}{s}
\]

### Final Answer:
The Laplace Transform of the given function is:

\[
\mathcal{L}\{f(t)\} = \frac{4e^{-3s}}{s}
\]

Do you have any questions or would you like further details?

Here are 5 related questions:
1. How do you find the Laplace Transform of other step functions?
2. How does the Laplace Transform handle discontinuous functions in general?
3. What are the properties of the Heaviside function in Laplace Transforms?
4. How can Laplace Transforms be used to solve differential equations?
5. What is the inverse Laplace Transform of \( \frac{4e^{-3s}}{s} \)?

**Tip:** Remember that the Heaviside function simplifies the analysis of piecewise functions in Laplace Transforms.

Find the Laplace Transform of f(t)=0 for 0≤t<3 and f(t)=4 for t≥3

This solution finds the Laplace Transform of a piecewise function where f(t)=0 for 0≤t<3 and f(t)=4 for t≥3. The Heaviside step function is used to express the function, and the final Laplace Transform is determined as \( \frac{4e^{-3s}}{s} \). Ideal for undergraduate students studying calculus and differential equations.

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