Exercise 1.3.64

Question

The Ramp Function The Heaviside function defined in Exercise 63 can also be used to define the ramp function $y = ctH(t)$, which represents a gradual increase in voltage or current in a circuit.
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of the ramp function $y = tH(t)$.
\item 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 $t \leq 60.$
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item $$R(t)=t H(t)= \begin{cases}0, & t<0 \ t, & t \geqslant 0\end{cases}$$
    \begin{figure}[H]\centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-3.2:0, samples=2, mark=none, color=violet, ultra thick] {0};
         \addplot[domain=0:3.2, samples=2, mark=none, color=violet, ultra thick] {x};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    \item $$V(t)=2 t H(t)= \begin{cases}0, & 	ext { if } t<0 \ 2 t, & 	ext { if } t \geqslant 0 .\end{cases}$$
    \begin{figure}[H]\centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-3.2:0, samples=2, mark=none, color=violet, ultra thick] {0};
         \addplot[domain=0:3.2, samples=2, mark=none, color=violet, ultra thick] {2*x};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    \item $$
V(t)=4 t H(t-7)=\left\{\begin{array}{l}
0, 	ext { if } t<7 \
4 t, 	ext { if } t \geqslant 7
\end{array}ight.
$$
    \begin{figure}[H]\centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-1.2:7, samples=2, mark=none, color=violet, ultra thick] {0};
         \addplot[domain=7:12.2, samples=2, mark=none, color=violet, ultra thick] {4*x};
         \draw[dashed] (axis cs:7, 0) -- (axis cs:7, 28);
         \addplot[mark=*, mark options={scale=1, fill=white}] coordinates {(7,0)};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
\end{enumerate}

Exercise 1.3.64

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Exercise 1.3.64

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