Chapter 1.3, Problem 60E

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8th Edition

James Stewart

ISBN: 9781305270343

Textbook Problem

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* ≤ 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* ≤ 32.

(a)

To determine

**To sketch:** The graph of the ramp function

Explanation

Heaviside function is,

So, the ramp function is,

(b)

To determine

**To sketch:** The graph of circuit if the switch is turned at t = 0 and the voltage gradually increased to 120 volts in 60 seconds; write the formula for

(c)

To determine

**To sketch:** The graph of the voltage in a circuit if the switch is turned at *t* = 7 seconds and gradually increased t0 100 volts over a period of 25 seconds; write the formula for

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