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# The Heaviside function defined in Exercise 59 can also be used to define the ramp function $y = ctH(t)$ , which represents a gradual increase in voltage or current in a circuit.(a) Sketch the graph of the ramp function $y = tH(t)$.(b) Sketch the graph of the voltage $V(t)$ in a circuit if the switch is turned on at time $t = 0$ and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for $V(t)$ in terms of $H(t)$ for $t \le 60$.(c) Sketch the graph of the voltage $V(t)$ in a circuit if the switch is turned on at time $t = 7$ seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for $V(t)$ in terms of $H(t)$ for $t \le 32$.

## a) see step for solutionb) $V(t)=2 t H(t)$c) $V(t)=4(t-7) H(t-7)$

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Tuv D.

August 16, 2020

Part c), when t=7, V(7)=0, so the slope of the graph should be from 0 to 100 for [7,32]

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### Video Transcript

So for the given problem um we can use it to define the ramp function Y equals C. T. Hft. Um Which is going to represent a gradual increase in the voltage or current in a circuit. So we're going to sketch the graph of the ramp function. We see the voltage is going to be in this case C is going to be too so we'll have to t 3 50. And then given the ramp function we'll eventually end up getting um Since this is Changing turned on at seven seconds we'll have four Times T -7 yeah. of eight times at age 15 and seven. So that will be our final uh resulting function for the voltage.

California Baptist University

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