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\usepackage{tikz}
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\begin{document}

\begin{questions}

\question

\begin{parts}

\part Suppose $0 < \theta < \frac{\pi}{2}$. Recall the formulas for the area of a triangle and a sector (pie slice) of a circle. Find the area of each shaded region below. Explain why the height of the triangle on the right is $\tan(\theta)$.

\begin{figure}[H]
\begin{center}
\includegraphics[width=4in]{esergel23.png}
\end{center}
\end{figure}

\part Order the areas you computed in part (a) from smallest to largest. Use this to argue that $\cos(\theta) \leq \frac{\sin(\theta)}{\theta} \leq 1$ for $0 < \theta < \frac{\pi}{2}$. [Actually, it is true for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$.]

\part Use the Squeeze Theorem to argue that $\displaystyle \lim_{x \to 0} \frac{\sin(\theta)}{\theta} = 1$. State specifically what $l(x)$, $f(x)$, $u(x)$, $c$ and $L$ are.

\part Consider the graph of $\sin(\theta)$. What is the slope of the secant line between the points $(0,0)$ and $(\theta, \sin(\theta))$? What is the slope of the tangent line through $(0,0)$?

\part Consider the graph of $\cos(\theta)$. What do you think the slope of the tangent line through $(0,1)$ is? How does this relate to the fact that $\displaystyle \lim_{x \to 0} \frac{1-\cos(\theta)}{\theta} = 0$?

\end{parts}


\end{questions}

\end{document}