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.