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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) $V(t)=4(t-7) H(t-7)$

04:43

Clarissa N.

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 3

New Functions from Old Functions

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

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]

Leo L.

January 2, 2022

Looks like I am missing something basic. Both authors say c = 2t. Where is that 2 coming from? Could someone explain.

Nevermind. Got it 120 vols over 60 sec = slope of 2

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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.

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