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Handout Lesson 35, Piecewise Continuous Input Functions
This lesson is based on Section 7.5 of your textbook by Edwards, Penney, and Calvis.
We can use
unit step functions for jump discontinuities and to help us model on/off switches. Recall:
Definition 240 .
The Heavyside function , denoted \(u(t)\text{,}\) is defined by
\begin{align*}
u(t)\defn \begin{cases} 1, \amp \text{if } t \geq 0 \\
0, \amp \text{if } t \lt 0 \end{cases}
\end{align*}
A unit step function is a horizontal shift of the Heavyside function. More specifically, the unit step function with shift \(a\text{,}\) denoted \(u_a(t)\text{,}\) is given by
\begin{align*}
u_a(t)\defn u(t-a)= \begin{cases} 1, \amp \text{if } t \geq a \\
0, \amp \text{if } t \lt a \end{cases}
\end{align*}
Example 241 . Graphs of fuctions that are multiplied by unit step functions.
Graph
\(s=u_2(t)t^2\text{.}\)
An axis system with horizontal
\(t\) -axis and vertical
\(s\) -axis.
Example 242 . Rewriting piecewise functions using unit step functions.
Rewrite
\begin{align*}
f(t)= \begin{cases} 0, \amp t\lt 1 \\ t+4, \amp 1 \leq t \lt 5 \\
7, \amp t \geq 5 \end{cases}
\end{align*}
using unit step functions.
Example 243 . Rewriting piecewise functions using unit step functions.
Rewrite
\begin{align*}
f(t)= \begin{cases} \sin(t), \amp 0 \leq t \lt 2\pi \\
0, \amp \text{otherwise} \end{cases}
\end{align*}
using unit step functions.
Laplace transforms and translations on the \(t\) -axis.
NOTE: In the last lesson, we looked at translations on the
\(s\) -axis.
Theorem 244 .
If \(\mathscr{L}\{f(t)\}=F(s)\) exists for \(s\gt c\text{,}\) then
\begin{gather*}
\mathscr{L}\{u(t-a)f(t-a)\}=\mathscr{L}\{u_a(t)f(t-a)\}=
e^{-as}F(s)
\end{gather*}
and
\begin{gather*}
\mathscr{L}^{-1}\{e^{-as}F(s)\}=u(t-a)f(t-a)=u_a(t)f(t-a)
\end{gather*}
Example 245 . \(\mathscr{L}\{u(t-a)f(t-a)\}\) .
Find \(\mathscr{L}\{h(t)\}\) if
\begin{align*}
h(t)=\begin{cases} 0, \amp t\lt 1 \\ t^2, \amp t \geq 1
\end{cases}
\end{align*}
In the last example, we converted a function of
\(t\) to a function of
\((t-c)=(t-1)\text{.}\) There has to be an easier way. There is an easier way, but it does not appear on your formula sheet.
Let’s redo the last example.
Find \(\mathscr{L}\{h(t)\}\) if
\begin{align*}
h(t)=\begin{cases} 0, \amp t\lt 1 \\ t^2, \amp t \geq 1
\end{cases}
\end{align*}
Example 246 . \(\mathscr{L}^{-1}\{e^{-as}F(s)\}\) . Find
\(\mathscr{L}^{-1}\{\frac{e^{-s}-e^{-3s}}{s^2}\}\text{.}\)
Example 247 . \(\mathscr{L}^{-1}\{e^{-as}F(s)\}\) . Find
\(\mathscr{L}^{-1}\{\frac{se^{-s}}{s^2+\pi^2}\}\text{.}\)
