Chapter 1.3, Problem 57E

(a)

To determine

To sketch: The graph of the Heaviside function.

Explanation of Solution

The Heaviside function is, $H\left(t\right)=\{\begin{array}{l}0\text{ if }t<0\\ 1\text{if }t\ge 0\end{array}$.

Use online graphing calculator and draw the graph of the Heaviside function as shown below in Figure1.

From Figure1, it is observed the Heaviside function is a piece-wise continuous function.

(b)

To determine

To sketch: The graph of $V\left(t\right)$ in a circuit if the switch is turned on at time t = 0 and 120 volts are applied to the circuit; write the formula for $V\left(t\right)$ in terms of $H\left(t\right)$.

Answer to Problem 57E

The formula for $V\left(t\right)$ in terms of $H\left(t\right)$ is $V\left(t\right)=120H\left(t\right)$.

Explanation of Solution

Observe that when $t<0$, the voltage is 0 and $\text{when }t\ge 0$, the voltage is 120.

Thus, the formula for $V\left(t\right)$ can be written in form of $H\left(t\right)$ as, $V\left(t\right)=\{\begin{array}{l}0\text{ if }t<0\\ 120\text{if }t\ge 0\end{array}$.

Use online graphing calculator and draw the graph of $V\left(t\right)=\{\begin{array}{l}0\text{ if }t<0\\ 120\text{if }t\ge 0\end{array}$ as shown below in Figure 2.

From Figure2, it is observed that the voltage is 0 when t = 0.

(c)

To determine

To sketch: The graph of the voltage in a circuit if the switch is turned on at t = 5 seconds and 240 volts are applied to the circuit; write the formula for $V\left(t\right)$ in terms of $H\left(t\right)$.

Explanation of Solution

Given that the time is t = 5 seconds.

Express the time as, $t-5=0$.

Thus, the formula for $V\left(t\right)$ after $t=5\text{s}$ is $V\left(t\right)=240H\left(t-5\right)$.

Use online graph calculator and draw the graph of the voltage $V\left(t\right)=240H\left(t-5\right)$ as shown below in Figure 3.

From Figure 3, notice that the number 5 is not included in the domain as the switch is turned on at t =  5 seconds.