Section The Basic Unit Step Function
In differential equations, we often need a simple mathematical βswitchβ that can turn a function on or off at a specific time. The simplest of these is the unit step function, which jumps from \(0\) to \(1\) at a chosen point.
β11β
Also called the Heaviside function.
Checkpoint 239. πβ Function Output.
π Definition 240. Unit Step Function.
Think of the unit step function as an ONβOFF switch. Before \(t = 0\text{,}\) the switch is \(0\) (OFF); after \(t = 0\text{,}\) it is \(1\) (ON). When you multiply another function by \(u(t)\text{,}\) that function is either turned OFF (multiplied by 0) or turned ON (multiplied by 1).
Checkpoint 241. πβ Unit Step Function Outputs.
The unit step function \(u(t)\) accepts any real number as input. Which of the following best describes its possible outputs?
- Only 0
- Only 1
- Only 0 and 1
- Any positive number
- Any real number
π Example 242. Switching ON a Parabola at \(t = 0\).
Consider the function
\begin{equation*}
g(t) = \frac{1}{5}t^2 - 1\text{.}
\end{equation*}
Describe the difference between \(g(t)\) and \(g(t)\cdot u(t)\) and give a graph of both.
Solution.
Because \(u(t)=1\) when \(t \ge 0\) and \(0\) otherwise, multiplying by \(u(t)\) βswitches offβ the parabola for \(t \lt 0\) and βswitches it onβ at \(t=0\text{.}\) In piecewise form:
\begin{equation*}
g(t)\cdot u(t) =
\left\{
\begin{array}{ll}
0, \amp t \lt 0 \\
\ds\frac{1}{5}t^2 - 1, \amp t \ge 0
\end{array}
\right.
\end{equation*}
and here are the graphs:
This example shows how multiplying by \(u(t)\) acts like a switch: the parabola is OFF for \(t \lt 0\text{,}\) then instantly turns ON at \(t=0\text{.}\)
Checkpoint 244. πβ Output of \(f(t) \cdot u_c(t)\).
π 245. Saying that a Function is ON or OFF.
Throughout this chapter, weβll often refer to a function as βONβ or βOFFβ. Hereβs what we mean:
-
The unit step function itself is ON when it equals \(1\) and OFF when it equals \(0\text{.}\)
-
A function multiplied by the unit step is ON when multiplied by \(1\) and OFF when multiplied by \(0\text{.}\)
The unit step function is the foundation for all step function work in this chapter. Next, weβll see how to shift the step so it can turn things on at times other than \(t=0\text{.}\)
Subsection π€ Wrap-Up
ποΈ \(\textbf{Key Takeaways...}\)
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The unit step function \(u(t)\) jumps from \(0\) to \(1\) at \(t=0\text{.}\)
-
Multiplying by \(u(t)\) βturns offβ a function before \(t=0\) and βturns it onβ after.
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