Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 58

Assume that the Heaviside function defined by

$H(t)$ = $\left\{\begin{array}{cccc} 0 & if & t < 0 \\ 1 & if & t\geq 0 \end{array} \right.$

can also be used to define the ramp function $y= tH(t)$, which represents a gradual increase in voltage or current in a circuit.


a.) Draw the graph of the ramp function $y = tH(t)$.










b.) Find a formula for $V(t)$ in terms of $H(t)$ for $t \leq 60$ and 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.









slope = $\frac{120 - 0}{60 - 0} = 2$

c.) Find a formula for $V(t)$ in terms of $H(t)$ for $t \leq 32$ and 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.












slope = $\frac{100 -0}{32-7} = 4$