what is x(t) = 0.5*t*u(t) when x(0) knowing that t>=0 x(t)=0.5*t and when t<0 x(t)=0?


Given the piecewise function definition of \( x(t) \), let's break it down step-by-step to clarify it:

1. **For \( t \geq 0 \)**: 
   The function \( x(t) = 0.5 \cdot t \cdot u(t) \).
   Here, \( u(t) \) represents the **unit step function**. The unit step function \( u(t) \) is defined as:

   \[
   u(t) = 
   \begin{cases}
   1, & \text{if } t \geq 0 \\
   0, & \text{if } t < 0
   \end{cases}
   \]

   Therefore, for \( t \geq 0 \), we have:

   \[
   x(t) = 0.5 \cdot t \cdot 1 = 0.5 \cdot t
   \]

2. **For \( t < 0 \)**:
   Since \( u(t) = 0 \) for \( t < 0 \), this implies:

   \[
   x(t) = 0.5 \cdot t \cdot 0 = 0
   \]

Thus, the piecewise function for \( x(t) \) is:

\[
x(t) = 
\begin{cases}
0.5 \cdot t, & \text{if } t \geq 0 \\
0, & \text{if } t < 0
\end{cases}
\]

### Finding \( x(0) \):
At \( t = 0 \):

\[
x(0) = 0.5 \cdot 0 = 0
\]

Thus, \( x(0) = 0 \).

### Summary:
The function \( x(t) \) is defined as:
\[
x(t) = 
\begin{cases}
0.5 \cdot t, & \text{if } t \geq 0 \\
0, & \text{if } t < 0
\end{cases}
\]
And \( x(0) = 0 \).

Would you like further details or explanations?

### Related questions:
1. How does the unit step function influence \( x(t) \) in this case?
2. What is the graphical representation of \( x(t) \) for this function?
3. How can \( x(t) \) be used in signal processing?
4. What happens to \( x(t) \) if we modify the scaling factor from 0.5 to another constant?
5. How would the function change if the unit step function \( u(t) \) had a delay, like \( u(t - T) \)?

**Tip**: The unit step function is commonly used to define signals that "turn on" at specific times, especially in control systems and signal processing.

what is x(t) = 0.5*t*u(t) when x(0) knowing that t>=0 x(t)=0.5*t and when t<0 x(t)=0?

Understand how to define the piecewise function x(t) = 0.5 * t * u(t) for t >= 0 and t < 0. This solution explores the unit step function and its role in signal processing. Suitable for students in undergraduate engineering or mathematics programs.

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