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\title{Exercise Week 12}
\author{COMP 150}
\date{Spring 2021}
\begin{document}
\maketitle
\begin{lstlisting}
# Your name here
\end{lstlisting}
\begin{lstlisting}
\end{lstlisting}

{{(1.) Define the functions ${\displaystyle f(x) = \tan\left(\frac{\pi x^2}{4}\right)}$ and $g(x) = \pi x + 1 - \pi$ in Sage for use throughout this exercise.}}


\begin{lstlisting}
\end{lstlisting}

{{(2.) Plot both functions on the same set of axes.  Choose a scaling that lets you see the key features of each graph.}



\begin{lstlisting}
\end{lstlisting}

{{(3.) Find an $x$-value where the two graphs intersect.  You may identify this value visually.  Make sure you also establish mathematically that you have found an $x$-value where the graphs intersect.}}



\begin{lstlisting}
\end{lstlisting}

{{(4.) Find the corresponding $y$-value where the intersect.  You may identify this value visually.  Make sure you also establish mathematically that you have found a point $(x, y)$ where the graphs intersect.}}



\begin{lstlisting}
\end{lstlisting}

{{(5.) One of the graphs is a line.  What is its slope?  In Sage, we can calculate the derivative of $f(x)$ using the following command: $\texttt{derivative(f(x), x)}$. Use Sage to find the derivative of $f(x)$. Use Sage to find the value of the derivative of $f(x)$ at $x = 1$.  Explain how this value compares with the slope of the line you identified in this question.  From a calculus point of view, what is the significance of how $f'(1)$ compares to the slope? Refer to your graph in your answer.}


\end{document}
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