The Heaviside function H is defined by
H(t) =

0	    if t < 0
1	    if t ≥ 0.
It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on.
(a)
Sketch the graph of the Heaviside function.

To sketch the graph of the Heaviside function \( H(t) \), you need to consider its definition:

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

### Steps to Sketch the Graph:

1. **For \( t < 0 \)**: The function \( H(t) = 0 \). This means that for all negative values of \( t \), the graph will be a horizontal line along the x-axis (at \( y = 0 \)).

2. **For \( t \geq 0 \)**: The function \( H(t) = 1 \). This implies that for \( t = 0 \) and all positive values of \( t \), the graph will be a horizontal line at \( y = 1 \).

3. **At \( t = 0 \)**: The value of the function jumps from 0 to 1, so there will be a discontinuity at \( t = 0 \). This is represented by an open circle at \( (0, 0) \) and a closed circle at \( (0, 1) \) on the graph.

### Description of the Graph:

- **For \( t < 0 \)**: The graph is a straight line along the x-axis at \( y = 0 \).
- **At \( t = 0 \)**: There is a jump from \( y = 0 \) to \( y = 1 \).
- **For \( t > 0 \)**: The graph is a straight line parallel to the x-axis at \( y = 1 \).

This is how the graph looks:

- A horizontal line along the x-axis from \( t = -\infty \) to \( t = 0 \) (not including 0).
- A jump to \( y = 1 \) at \( t = 0 \).
- A horizontal line at \( y = 1 \) for \( t > 0 \).

If you'd like, I can provide a visual sketch of the graph.

Would you like to see the sketch or have any questions? Here are some related topics that might interest you:

1. How is the Heaviside function used in solving differential equations?
2. What is the relationship between the Heaviside function and the Dirac delta function?
3. How does the Heaviside function apply to Laplace transforms?
4. What are step functions, and how are they related to the Heaviside function?
5. How does the Heaviside function model real-world scenarios like electric circuits?

**Tip:** The Heaviside function is commonly used in control theory and signal processing to model sudden changes or "steps" in a system.

Explore how to graph the Heaviside function H(t), used in electric circuits to represent sudden surges of current or voltage. Understand its definition and graphing techniques, including handling discontinuities and interpreting step functions. Suitable for advanced high school students and anyone interested in mathematical functions.

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