\documentclass[11pt]{article}

\usepackage{pandora,euler,amsmath}

\begin{document}

\title{Math 2B, Section A, Winter 1997}
\author{Marcel Oliver}
\date{Midterm Exam I}

\maketitle

\noindent Some trigonometric identities:
\begin{align*}
  {} &  \sin^2 \theta + \cos^2 \theta = 1
     && \sin 2x = 2 \, \sin x \, \cos x \\
     &  1 + \tan^2 \theta = \sec^2 \theta
     && \cos 2x = 2 \, \cos^2 x - 1 = 1 - 2 \, \sin^2 x \\
     &  1 + \cot^2 \theta = \csc^2 \theta
     && \tan 2x = \frac{2 \, \tan x}{1 - \tan^2 x} \\
     &  \sin^2 x = \frac{1 - \cos 2 x}{2}
     && \cos^2 x = \frac{1 + \cos 2 x}{2} 
\end{align*}
\bigskip

\begin{enumerate}

\item Compute the area bounded by the $x$-axis, the line $x=e$, and
the graph of the function
\begin{equation*}
  y = \ln x.
\end{equation*}
Hint: Remember that $\ln 1 = 0$ and $\ln e = 1$.
\hfill (15)

\item Compute the following integrals.  Be sure to show all your work.
\begin{enumerate}
\item $\displaystyle \int \frac{x^3}{(x^2+1)^2} \, dx$
\item $\displaystyle \int \sin \sqrt{x} \, dx$
\item $\displaystyle \int \frac{1}{1 + e^x} \, dx$
\item $\displaystyle \int x \, e^{-x} \, dx$
\end{enumerate}
\hfill (10+10+10+10)

\item Solve the initial value problem
\begin{equation*}
  \frac{dy}{dt} = \alpha \, y^2, \qquad y(0) = 1,
\end{equation*}
where $\alpha$ is a constant.  Show that the solution ``blows up'' at
time $t = 1/\alpha$.
\hfill (15)

\item The hyperbolic sine and cosine functions are defined as follows:
\begin{gather*}
  \sinh x  = \tfrac12 \bigl(e^x - e^{-x} \bigr) \\
  \cosh x  = \tfrac12 \bigl(e^x + e^{-x} \bigr)
\end{gather*}
\begin{enumerate}
\item Verify the following three identities:
\begin{gather*}
  \frac{d}{dx} \sinh x = \cosh x, \\
  \frac{d}{dx} \cosh x = \sinh x, \\
  \cosh^2 x - \sinh^2 x = 1.
\end{gather*}
\item Use part (a) to show that 
\begin{equation*}
  \int \frac{1}{\sqrt{a^2 + x^2}} \, dx = \sinh^{-1} \frac{x}{a} + C.
\end{equation*}
\end{enumerate}
\hfill (5+10)

\item Find the area of the ellipse
\begin{equation*}
  x^2 + 9 \, y^2 = 1
\end{equation*}
by integration.  Show all steps in the evaluation of the integral.
\hfill (15)

  
\end{enumerate}

\end{document}
