show steps to take the laplace transform of a piecewise function that is u(t)=t from 0 to 1 u(t)=1 from 1 to 2 and u(t)=3-t from 2 to 3 and u(t) = 0 otherwise

To take the Laplace transform of a piecewise function, you need to express the function using Heaviside step functions (also called unit step functions), and then apply the Laplace transform to each term individually.

Let \( u(t) \) be the given piecewise function:

\[
u(t) =
\begin{cases} 
t, & 0 \leq t < 1 \\
1, & 1 \leq t < 2 \\
3 - t, & 2 \leq t < 3 \\
0, & t \geq 3
\end{cases}
\]

### Steps to take the Laplace transform of the piecewise function:

#### 1. **Rewrite the function using the Heaviside function:**
The Heaviside function \( u_c(t) \) is defined as:
\[
u_c(t) =
\begin{cases}
0, & t < c \\
1, & t \geq c
\end{cases}
\]
Using Heaviside functions, you can express the piecewise function as:
\[
u(t) = t u_0(t) - t u_1(t) + u_1(t) - u_2(t) + (3 - t) u_2(t) - (3 - t) u_3(t)
\]
Simplify the expression by grouping terms:
\[
u(t) = t u_0(t) + (1 - t) u_1(t) + (3 - t) u_2(t)
\]

#### 2. **Break the function into intervals using the Heaviside function:**
This can now be written in terms of simpler Laplace transforms:
\[
u(t) = 
\begin{cases}
t, & 0 \leq t < 1 \\
1, & 1 \leq t < 2 \\
3 - t, & 2 \leq t < 3
\end{cases}
\]



Learn how to compute the Laplace transform of a piecewise function defined by different expressions over various intervals. This detailed step-by-step guide uses Heaviside step functions to simplify the process. Suitable for students in advanced calculus or engineering mathematics.

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