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To find the Laplace transform \( F(s) = \mathcal{L} \{ f(t) \} \) of the given piecewise function \( f(t) \), we can split the integral into three parts according to the intervals provided:

Given:
\[
f(t) = \begin{cases} 
-3t & 0 \leq t < 3 \\
2 - 4t & 3 \leq t < 4 \\
4 & t \geq 4 
\end{cases}
\]

### Step 1: Laplace Transform Setup
The Laplace transform \( \mathcal{L} \{ f(t) \} \) is calculated as:
\[
F(s) = \int_0^{\infty} f(t) e^{-st} \, dt
\]
We break it up based on the intervals:
\[
F(s) = \int_0^3 (-3t)e^{-st} \, dt + \int_3^4 (2 - 4t)e^{-st} \, dt + \int_4^{\infty} 4e^{-st} \, dt
\]

### Step 2: Calculate Each Integral Separately

#### Integral 1: \( \int_0^3 (-3t)e^{-st} \, dt \)
1. **Set up the integral**:
   \[
   \int_0^3 (-3t)e^{-st} \, dt = -3 \int_0^3 t e^{-st} \, dt
   \]
2. **Use integration by parts** with \( u = t \) and \( dv = e^{-st} dt \).
3. Solve to find this term.

#### Integral 2: \( \int_3^4 (2 - 4t)e^{-st} \, dt \)
1. **Set up the integral**:
   \[
   \int_3^4 (2 - 4t)e^{-st} \, dt = 2 \int_3^4 e^{-st} \, dt - 4 \int_3^4 t e^{-st} \, dt
   \]
2. Separate the terms and solve each using appropriate techniques.

#### Integral 3: \( \int_4^{\infty} 4e^{-st} \, dt \)
1. **Set up the integral**:
   \[
   \int_4^{\infty} 4e^{-st} \, dt = 4 \int_4^{\infty} e^{-st} \, dt
   \]
2. This is a standard Laplace transform of a constant starting at \( t = 4 \), and can be solved accordingly.

### Solution
After evaluating each integral, enter the expressions in the form:
\[
F(s) = \text{(Expression for 1st integral)} + \text{(Expression for 2nd integral)} + \text{(Expression for 3rd integral)}
\]

Would you like further step-by-step calculations on any specific interval?

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1. How do we apply Laplace transforms to piecewise functions?
2. What is the role of the Heaviside function in transforming piecewise functions?
3. How does integration by parts help in solving Laplace transforms?
4. What are common Laplace transforms for functions involving constants and linear terms?
5. Why do we split the integral by intervals in this piecewise function?

**Tip:** When dealing with piecewise functions, remember to split each interval and apply transformations independently to simplify the overall process.

Find the Laplace transform F(s) = L{f(t)} of the function f(t) by splitting the integral into three pieces, where f(t) = {-3t for 0 <= t < 3, 2 - 4t for 3 <= t < 4, and 4 for t >= 4}.

This page provides a step-by-step solution for finding the Laplace transform of a piecewise function by splitting the integral into three parts based on intervals. Learn how to apply integration by parts and transform techniques for each segment of f(t). Suitable for undergraduate calculus students.

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