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To find the Laplace transform of the function 
\[
f(t) = (7 - t) \left( u_4(t) - u_7(t) \right),
\]
we'll follow these steps:

1. **Interpret the function**: 
   - \( u_4(t) \) and \( u_7(t) \) are unit step functions. Specifically, \( u_4(t) \) activates the function at \( t = 4 \) and \( u_7(t) \) deactivates it at \( t = 7 \).
   - Therefore, \( f(t) \) is defined as \( 7 - t \) for \( t \) in the interval \( [4, 7] \) and \( 0 \) elsewhere.

2. **Rewrite using shifted unit step functions**:
   \[
   f(t) = (7 - t) \left( u_4(t) - u_7(t) \right) = (7 - t) u_4(t) - (7 - t) u_7(t).
   \]

3. **Express each part using shifts**:
   - For the part \( (7 - t) u_4(t) \), we rewrite \( t \) in terms of \( (t - 4) \):
     \[
     (7 - t) u_4(t) = \left(7 - (t - 4 + 4)\right) u_4(t) = (3 - (t - 4)) u_4(t).
     \]
   - Similarly, for \( (7 - t) u_7(t) \), we rewrite \( t \) in terms of \( (t - 7) \):
     \[
     (7 - t) u_7(t) = (7 - (t - 7 + 7)) u_7(t) = (- (t - 7)) u_7(t).
     \]

4. **Take the Laplace Transform**:
  

Find the Laplace transform F(s) = L{f(t)} of the function f(t) = (7 - t)(u4(t) - u7(t))

Find the Laplace transform of f(t) = (7 - t)(u4(t) - u7(t)), a function defined by unit step functions. This problem involves understanding Laplace transforms of shifted functions and piecewise functions. Ideal for undergraduate students studying engineering or applied mathematics.

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