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\title{Mathematics 2A, Section E, Fall 1996}
\author{Marcel Oliver}
\date{Final Exam}

\maketitle

\begin{enumerate}

\item Compute the following limits, if they exist:
\begin{enumerate}
\item $\displaystyle \lim_{\theta \to 0}
  \frac{\sin a\theta}{\sin b\theta}$
\item $\displaystyle \lim_{t \to 4^-} \sqrt{4-t}$
\item $\displaystyle \lim_{z \to 4} \frac{z-4}{\sqrt{z} - 2}$
\item $\displaystyle \lim_{h \to 0} \frac{\sqrt{5+h}-\sqrt{5}}{h}$
\item $\displaystyle \lim_{x \to -\infty}
  \frac{x+1}{|x-1|}$
\end{enumerate}
\hfill (5+5+5+10+5)

\item The derivative of the exponential function is very simple, namely
\[ (e^x)' = e^x . \]
Use this fact to show that the second derivative of
\[ f(x) = e^{-x^2} \]
is
\[ f''(x) = (4 \, x^2 - 2) \, e^{-x^2} . \]
\hfill (20)

\item Find all points on the curve with equation
\[ x^2 -x\,y + y^2 = 3 \]
at which the tangent line is vertical.
\hfill (20)

\item A spherical tank of radius $r$ has many coats of old paint on
the inside.  The paint is sandblasted off  and the tank repainted,
with a net increase in the radius of $\tfrac1{10}\%$.  Find an
approximation to the percent increase in the volume of the tank.
\mbox{ }\hfill (20)

\item A rectangle is inscribed in a circle of radius $R$.  What is the
largest area the rectangle can have?  Use the first derivative test to show
that you have indeed found the maximum.
\hfill (20)


\item Consider the function
\[ f(x) = \frac{x^2-1}{x^2+1}. \]
% \[ f(x) = \frac{3\,x^2+1}{x^2+1}. \]
% \[ f(x) = \frac{1 - x^2}{1+x^2}. \]
% \[ f(x) = \frac{-1-3\,x^2}{1+x^2}. \]
Sketch the graph of $f$ \emph{into the coordinate system provided on the
next page}.  Your drawing must show clearly where the graph is
increasing, where decreasing, where concave upward, and where concave
downward.  Locate, identify, and label all extrema and inflection points.
Find and label all asymptotes.  Pay special attention to the behavior of
the function when $|x|$ is large and when $x$ is close to any discontinuity
of the function.  Find and label all intercepts.
\hfill (30)

\item Compute the following integrals:
\begin{enumerate}
\item $\displaystyle \int x^{n-1} \, dx $
\item $\displaystyle \int \sin^2 \alpha \, \cos \alpha \, d\alpha $
\item $\displaystyle \int_{-2}^2 4 z^3 \, dz $
\item $\displaystyle \int_{-2}^4 |x^2 - 1| \, dx $
\item $\displaystyle \int_{-2462 \pi}^{6468 \pi} \sin \phi \, d\phi  $
\end{enumerate}
\hfill (5+5+5+10+5)

\item Suppose you jump upward with initial velocity $v_0$, subject only to
the gravitational acceleration $g$.
\begin{enumerate}
\item Show, by twice solving an initial value problem, that your height
above the ground $h$ as a function of time $t$ is given by
\begin{equation}
  h(t) = v_0 \, t - \tfrac12 \, g \, t^2 . \tag{$*$}
\end{equation}
\item What is the maximum height you will reach?
\item If you were sent to the moon where the gravitational acceleration is
only $\tfrac16 g$, how many times higher could you jump?
\item For what values of $t$ does equation ($*$) make sense?  Explain your
answer. 
\end{enumerate}
\hfill (10+10+5+5)


\end{enumerate}

\end{document}
