Math Exam

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\usepackage{amsmath}

\ExamClass{Sample Class}
\ExamName{Sample Exam}
\ExamHead{\today}

\let\ds\displaystyle

\begin{document}
\ExamInstrBox{
Please show \textbf{all} your work! Answers without supporting work will not be given credit. Write answers in spaces provided. You have 1 hour and 50 minutes to complete this exam.}
\ExamNameLine
\begin{enumerate}
   \item Calculate the following limits.  If a limit is $\infty$ or $-\infty$,
      please say so.  Make sure you show all your work and justify all your
      answers.  
      \begin{enumerate}
	 \item $\ds{\lim_{x\rightarrow3}\frac{\sqrt{x+1} - 2}{x-3}}$\answer
	 \item $\ds{\lim_{x\rightarrow0}\frac{\sin(4x)}{8x}}$\answer
      \end{enumerate}
   \item Use the $\varepsilon$-$\delta$ definition of limit to prove that 
      \[\lim_{x\rightarrow 2} x^2 - 3x + 2 = 0\]\noanswer[2.5in]
      \newpage
   \item If $h(x) = \sqrt{x^2 + 2} - 1$, find a \textbf{non-trivial} decomposition of $h$ into $f$ and $g$ such that $h = f\circ g$.
      \answer*{$f(x)=$}\addanswer*{$g(x)=$}
   \item Find the first two derivatives of the function $f(x) = x^2\cos(x)$.  Simplify
      your answers as much as possible.  Show all your work.
      \answer*{$f'(x)=$}\answer*{$f''(x)=$}
      \newpage
   \item Find the derivative of the function $\ds{f(x) = \int_{x^2}^2
      \frac{\cos(t)}{t} \,dt}$.\answer[1in plus 1fill]
   \item Set up, but do not evaluate, the integral for the volume of the solid obtained by rotating the area between the curves $y = x$ and $y = \sqrt{x}$ about the $x$-axis.\noanswer
\end{enumerate}
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