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Let's work through each part of this problem step-by-step.

### Part (a): Represent \( f(t) \) using the Heaviside step function
The function \( f(t) \) is defined as:
\[
f(t) = 
\begin{cases} 
t^2 & \text{for } 0 \leq t < 2, \\
7 & \text{for } t \geq 2.
\end{cases}
\]

To represent \( f(t) \) using the Heaviside step function \( h(t - c) \), we can rewrite \( f(t) \) in a piecewise form that transitions from \( t^2 \) to 7 at \( t = 2 \).

The Heaviside step function \( h(t - 2) \) "activates" at \( t = 2 \). We can use it to modify \( f(t) \) as follows:
\[
f(t) = t^2 + (7 - t^2) \cdot h(t - 2)
\]
Here's how this expression works:
- For \( 0 \leq t < 2 \), \( h(t - 2) = 0 \), so \( f(t) = t^2 \).
- For \( t \geq 2 \), \( h(t - 2) = 1 \), making \( f(t) = t^2 + (7 - t^2) = 7 \).

So, the answer to part (a) is:
\[
f(t) = t^2 + (7 - t^2) \cdot h(t - 2).
\]

### Part (b): Find the Laplace transform \( F(s) = \mathcal{L}\{f(t)\} \)

Now, let's find the Laplace transform of \( f(t) \) as defined in part (a).

#### Step-by-Step Laplace Transform Calculation
1. **Laplace transform of \( t^2 \)**:
   \[
   \mathcal{L}\{t^2\} = \frac{2}{s^3}.
   \]

2. **Laplace transform of \( (7 - t^2) \cdot h(t - 2) \)**:
   Using the **second shifting theorem** (i.e., \( \mathcal{L}\{f(t - a) h(t - a)\} = e^{-as} \mathcal{L}\{f(t)\} \) where \( f(t) = 7 - t^2 \) and \( a = 2 \)), we get:
   \[
   \mathcal{L}\{(7 - t^2) h(t - 2)\} = e^{-2s} \cdot \mathcal{L}\{7 - (t + 2)^2\}.
   \]

Let's break this down:
1. \( 7 - (t + 2)^2 = 7 - (t^2 + 4t + 4) = -t^2 - 4t + 3

Consider the function: 
\( f(t) = \begin{cases} t^2, & 0 \leq t < 2 \\ 7, & t \geq 2 \end{cases} 
\)

(a) Represent \( f(t) \) using a combination of Heaviside step functions. Use \( h(t - c) \) for the Heaviside function.

(b) Find the Laplace transform \( F(s) = \mathcal{L} \{ f(t) \} \).

Learn how to represent a piecewise function using Heaviside step functions and calculate its Laplace transform. This problem covers piecewise function representation, the Heaviside step function, and applying the second shifting theorem for Laplace transforms. Suitable for undergraduate mathematics students.

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