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To find the Laplace transform \( F(s) = \mathcal{L}\{f(t)\} \) of the function \( f(t) = (7 - t) \left( u_4(t) - u_7(t) \right) \), let's break down the function and apply the Laplace transform.

### Step-by-Step Solution

1. **Interpret \( f(t) \):**
   \[
   f(t) = (7 - t) \left( u_4(t) - u_7(t) \right)
   \]
   where \( u_4(t) \) and \( u_7(t) \) are the Heaviside (unit step) functions shifted by 4 and 7, respectively. This means that \( f(t) \) is defined as \( 7 - t \) on the interval \( 4 \leq t < 7 \) and is zero otherwise.

2. **Rewrite \( f(t) \) using Heaviside functions:**
   We can express \( f(t) \) in terms of shifted Heaviside functions:
   \[
   f(t) = (7 - t) \, u_4(t) - (7 - t) \, u_7(t)
   \]
   
3. **Use the Laplace Transform of a function multiplied by a Heaviside function:**
   The Laplace transform of \( f(t - a) u_a(t) \) is \( e^{-as} \mathcal{L}\{f(t)\} \), where \( a \) is the shift and \( \mathcal{L}\{f(t)\} \) is the Laplace transform of \( f(t) \) without the shift.

   So, we need to compute:
   - The Laplace transform of \( (7 - t) u_4(t) \)
   - The Laplace transform of \( (7 - t) u_7(t) \)

4. **Transform \( (7 - t) u_4(t) \):**
   Let \( g(t) = 7 - t \). Then \( g(t - 4) = 7 - (t - 4) = 11 - t \).

   So,
   \[
   \mathcal{L} \{ (7 - t) u_4(t) \} = e^{-4s} \mathcal{L} \{ 11 - t \}
   \]
   Now, calculate \( \mathcal{L} \{ 11 - t \} \):
   - \( \mathcal{L} \{ 11 \} = \frac{11}{s} \)
   - \( \mathcal{L} \{ t \} = \frac{1}{s^2} \)

   Therefore,
   \[
   \mathcal{L} \{ 11 - t \} = \frac{11}{s} - \frac{1}{s^2}
   \]
   So,
   \[
   \mathcal{L} \{ (7 - t) u_4(t) \} = e^{-4s} \left( \frac{11}{s} - \frac{1}{s^2} \right)
   \]

5. **Transform \( (7 - t) u_7(t) \):**
   Similarly, for \( (7 - t) u_7(t) \), we get:
   \[
   \mathcal{L} \{ (7 - t) u_7(t) \} = e^{-7s} \mathcal{L} \{10 - t \} = e^{-7s} \left( \frac{10}{s} - \frac{1}{s^2} \right)
   \]

6. **Combine results:**
   \[
   F(s) = e^{-4s} \left( \frac{11}{s} - \frac{1}{s^2} \right) - e^{-7s} \left( \frac{10}{s} - \frac{1}{s^2} \right)
   \]

### Final Answer
\[
F(s) = e^{-4s} \left( \frac{11}{s} - \frac{1}{s^2} \right) - e^{-7s} \left( \frac{10}{s} - \frac{1}{s^2} \right)
\]

Would you like more details on any of these steps or have any specific questions?

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### Related Questions

1. How can we apply the Laplace transform to other piecewise functions with different intervals?
2. What is the Laplace transform of a function with a general Heaviside step function \( u_a(t) \)?
3. How would the result change if \( f(t) \) had a different linear form, like \( t - 3 \)?
4. What role does the exponential term \( e^{-as} \) play in the Laplace transform of shifted functions?
5. How can we use the Laplace transform for solving differential equations involving Heaviside functions?

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**Tip:** When dealing with Heaviside functions in Laplace transforms, always shift the variable appropriately to handle discontinuities in the function.

Find the Laplace transform \( F(s) = \mathcal{L}\{f(t)\} \) of the function \( f(t) = (7 - t) \left( u_4(t) - u_7(t) \right) \).

Learn how to find the Laplace transform of a piecewise function involving Heaviside step functions. This problem covers the use of Laplace transforms for shifted functions and provides a detailed step-by-step solution for \( f(t) = (7 - t)(u_4(t) - u_7(t)) \). Suitable for college-level students studying Laplace transforms.

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