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To solve this problem, we need to find the Laplace transform \( F(s) = \mathcal{L}\{f(t)\} \) of the given function \( f(t) \) by splitting the integral according to the intervals specified for \( f(t) \). Here’s the approach in steps.

### Step 1: Understand the Piecewise Definition of \( f(t) \)
The function \( f(t) \) is defined as:
\[
f(t) = \begin{cases} 
-5t & \text{for } 0 \le t < 4, \\
5 + 3t & \text{for } 4 \le t < 7, \\
4 & \text{for } t \ge 7.
\end{cases}
\]

### Step 2: Set Up the Laplace Transform with Splitting
The Laplace transform \( F(s) \) for a piecewise function \( f(t) \) can be written as a sum of transforms over each interval:
\[
F(s) = \int_{0}^{4} (-5t)e^{-st} \, dt + \int_{4}^{7} (5 + 3t)e^{-st} \, dt + \int_{7}^{\infty} 4e^{-st} \, dt.
\]

### Step 3: Compute Each Integral Separately
Let's evaluate each part.

#### Part 1: \( \int_{0}^{4} (-5t)e^{-st} \, dt \)
This integral represents the Laplace transform of \( -5t \) over the interval \( [0, 4] \).

#### Part 2: \( \int_{4}^{7} (5 + 3t)e^{-st} \, dt \)
This integral represents the Laplace transform of \( 5 + 3t \) over the interval \( [4, 7] \).

#### Part 3: \( \int_{7}^{\infty} 4e^{-st} \, dt \)
This integral represents the Laplace transform of the constant \( 4 \) starting from \( t = 7 \).

### Step 4: Solve Each Integral
1. **First Integral**:
   \[
   \int_{0}^{4} (-5t)e^{-st} \, dt
   \]
   Calculate this integral by using integration by parts or a Laplace table if you have one.

2. **Second Integral**:
   \[
   \int_{4}^{7} (5 + 3t)e^{-st} \, dt
   \]
   Similarly, use integration by parts or a table for this function over the interval.

3. **Third Integral**:
   \[
   \int_{7}^{\infty} 4e^{-st} \, dt = 4 \int_{7}^{\infty} e^{-st} \, dt
   \]
   This is a simpler integral since it involves a constant multiplied by the exponential term, which can be directly evaluated.

Would you like me to proceed with detailed calculations for each integral, or do you have any specific questions about the setup? 

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### Practice Questions
1. How would you evaluate the Laplace transform of a function with more than three piecewise segments?
2. What is the Laplace transform of \( t^n \) for any integer \( n \)?
3. How does the Laplace transform handle exponential growth or decay functions?
4. What happens to the Laplace transform of a function as \( s \to \infty \)?
5. How can we use Laplace transforms to solve differential equations involving piecewise functions?

### Tip
When handling piecewise functions in Laplace transforms, make sure to carefully evaluate each segment independently and add the results, considering the domain for each piece.

Find the Laplace transform F(s) = L{f(t)} of the function: f(t) = -5t for 0 ≤ t < 4, 5 + 3t for 4 ≤ t < 7, and 4 for t ≥ 7 by splitting the integral into three pieces. Enter your answers in order of increasing domain.

Calculate the Laplace transform of the function f(t) with three distinct intervals. Learn how to split the integral and find the Laplace transform for each interval. This example involves piecewise functions and integration.

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