Chapter 1.3, Problem 59E

The Heaviside function H is defined by

$H\left(t\right)=\{\begin{array}{l}0\text{ if }t<0\\ 1\text{ if }t\ge 0\end{array}$

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.

(b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t=0 and 120 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t).

(c) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t=5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (Note that starting at t = 5 corresponds to a translation.)