To solve this problem, let's address each part in turn.

Part (a): Representation of $f(t)$ using Heaviside step functions

The given function is:

$f(t) = 3 - 3h(t - 6) + (t - 6)h(t - 6) - (t - 6)h(t - 7) + 2h(t - 7)$

where $h(t - c)$ represents the Heaviside function shifted to start at $t = c$.

Each term here represents a piecewise function that activates at different points in $t$:

1. $3$: This is the base value of $f(t)$ before any Heaviside functions are applied.
2. $-3h(t - 6)$: This term subtracts 3 from $f(t)$ starting at $t = 6$.
3. $(t - 6)h(t - 6)$: This adds $t - 6$ starting from $t = 6$.
4. $-(t - 6)h(t - 7)$: This removes $t - 6$ starting from $t = 7$.
5. $2h(t - 7)$: This adds 2 starting from $t = 7$.

These terms collectively define $f(t)$ over different intervals of $t$, incorporating jumps and shifts at $t = 6$ and $t = 7$.

Part (b): Finding the Laplace Transform $F(s) = \mathcal{L}\{f(t)\}$

To find the Laplace transform, we will use the linearity of the Laplace transform and properties related to the Heaviside function. For a function $g(t)$ shifted by $h(t - c)$, the Laplace transform is:

$\mathcal{L}\{g(t)h(t - c)\} = e^{-cs} \mathcal{L}\{g(t + c)\}$

Applying this to each term in $f(t)$:

1. Transform of $3$: $\mathcal{L}\{3\} = \frac{3}{s}$
2. Transform of $-3h(t - 6)$: $-3 e^{-6s} \frac{1}{s} = \frac{-3 e^{-6s}}{s}$
3. Transform of $(t - 6)h(t - 6)$: Shift by 6, so $e^{-6s} \cdot \frac{1}{s^2}$ (as $\mathcal{L}\{t\} = \frac{1}{s^2}$).
4. Transform of $-(t - 6)h(t - 7)$: Shift by 7, so $-e^{-7s} \cdot \frac{1}{s^2}$.
5. Transform of $2h(t - 7)$: $2e^{-7s} \cdot \frac{1}{s}$.

Adding these transforms together:

$F(s) = \frac{3}{s} - \frac{3e^{-6s}}{s} + \frac{e^{-6s}}{s^2} - \frac{e^{-7s}}{s^2} + \frac{2e^{-7s}}{s}$

This is the Laplace transform of $f(t)$.

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

Related Questions:

1. How does the Heaviside function modify the Laplace transform for shifted terms?
2. What are other applications of the Heaviside step function in piecewise functions?
3. Can we use Laplace transforms to solve differential equations with piecewise inputs?
4. How does the Laplace transform handle discontinuous functions in general?
5. What other methods besides Heaviside functions can represent piecewise functions in Laplace transformations?

Tip: When using Heaviside functions, always consider the shift property in the Laplace transform to simplify calculations for shifted terms.