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Problem 59 Hard Difficulty

The Heaviside function $ H $ is defined by
$ H(t) = \left\{
\begin{array}{ll}
0 & \mbox{if $ t < 0 $}\\
1 & \mbox{if $ t \ge 0 $}
\end{array} \right.$
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.)

Answer

(a) $$H(t)=\left\{\begin{array}{ll} 0 & \text { if } t<0 \\ 1 & \text { if } t \geq 0\end{array}\right.$$
(b) $V(t)=\left\{\begin{array}{ll}0 & \text { if } t<0 \\ 120 & \text { if } t \geq 0\end{array} \quad \text { so } V(t)=120 H(t)\right.$
(c) Starting with the formula in part (b), we replace 120 with 240 to reflect the
different voltage. Also, because we are starting 5 units to the right of $t=0$
we replace $t$ with $t-5 .$ Thus, the formula is $V(t)=240 H(t-5)$.

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Video Transcript

have you said function? Each is defined by this thing. It is used in the study of like, electric circuits to represent the Senate surge of electric current or voltage when a switches instantaneously turned on or a tells us sketch a graph of the heavyset function. So it looks like when time is ah less than zero Here this thing is turned off. But then right when time zero this scene turns onto one. Until let me just show you what this looks like here. Okay, so here is each in here is t OK And so we have each equal zero fourteen is ah less than zero. And so we'll have that there and then when t equals zero, this thing is turned on sketch. The graph of the voltage in a circuit of the switch is turned on at time equals zero and a twenty volt are applied instantaneously. Teo circuit breakdown A formula for beauty in terms of ht. Okay, support me here. So what time equals zero where we switch on one hundred twenty holes. Okay, so formula will say Susan Green here, B fifty iss twenty Time's agent T for H is Oh, no tea is less than zero. And then, of course, it zero otherwise Oh, uh, sorry, There's my this looks here under twenty h of tea when tea is greater than or equal to zero and zero. Otherwise, okay, in the now sketched a graph of the voltage dft in the circuit, if the switch is turned on at a time T equals five seconds and two hundred forty volts or plain instantaneously to the circuit. So same kind of question here we'LL draw this Rath or C H and tea. So it looks like we're turned off until time equals five seconds. I know what you're switched on, Teo Teo, forty volts. So this would be, ah, equation. Your tea cools, uh, two hundred forty times h of t minus five. Since there's a horizontal shift. And, uh, teen, this is applied when he created an equal to five and zero. Otherwise,