https://file.aiquickdraw.com/mathbot/file/1731071212027w0a5hs5l.jpg

Let's tackle this step-by-step.

### Part (a): Represent \( f(t) \) using Heaviside step functions

The function \( f(t) \) is defined in pieces, so we need to rewrite it using the Heaviside step function \( h(t - c) \), which "activates" the function at \( t = c \).

Given:
\[
f(t) = \begin{cases} 
2, & t < 2, \\
t - 2, & 2 \leq t < 7, \\
4, & t \geq 7.
\end{cases}
\]

We can represent each piece with a Heaviside function that "turns on" the appropriate parts of \( f(t) \) at each boundary.

1. For \( t < 2 \): \( f(t) = 2 \).
2. For \( 2 \leq t < 7 \): \( f(t) = t - 2 \).
3. For \( t \geq 7 \): \( f(t) = 4 \).

The Heaviside function representation of \( f(t) \) is:
\[
f(t) = 2 + (t - 2 - 2) h(t - 2) + (4 - (t - 2)) h(t - 7)
\]

Simplifying this, we get:
\[
f(t) = 2 + (t - 4) h(t - 2) +

Consider the function:

\[ f(t) = \begin{cases} 2, & t < 2, \\ t - 2, & 2 \leq t < 7, \\ 4, & t \geq 7. \end{cases} \]

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

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

Learn to represent a piecewise function using Heaviside step functions and find its Laplace transform. This solution covers key topics including the Heaviside function, properties of Laplace transforms, and representation of piecewise functions. Ideal for undergraduate students studying calculus and differential equations.

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