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\begin{document}

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

\maketitle

\begin{enumerate}

\item Compute the following limits, if they exist:
\begin{enumerate}
\item $\displaystyle \lim_{x \to 0} \frac{\sin ax}{\sin bx}$
\item $\displaystyle \lim_{t \to 5^-} \frac{t-5}{|t-5|}$
\item $\displaystyle \lim_{t \to 5} \frac{t-5}{|t-5|}$
\item $\displaystyle \lim_{r \to 3} \frac{2\,r - 6}{r^2 - 4\, r + 3}$
\item $\displaystyle \lim_{\theta \to \infty}
  \frac{\sin \theta \cdot \cos \theta}{\theta^2}$
\end{enumerate}
\hfill (5+5+5+10+5)

\item The derivative of the logarithm is
\[ (\ln x)' = \frac1x . \]
Use this fact to show that $\ln x$ does not have a point of inflection.
\hfill (20)


\item Consider the equation
\begin{equation}
  x^3 + 2 x y + y^3 = 13 . \tag{$*$}
\end{equation}
\begin{enumerate}
\item Find $\tfrac{dy}{dx}$ by implicit differentiation.
\item Write an equation for the straight line which is tangent to the graph
of equation ($*$) at the point $(1,2)$.
\end{enumerate}
\hfill (10+10)

\item With what accuracy must the volume of a cube be measured to ensure an
error of at most $1\%$ in the calculated length of its sides? (Express your
answer in percent.)
\hfill (20)

\item A piece of string of length $L$ is shaped into two circles.  Where
should the string be cut to ensure that the combined area of the two
circles is a minimum (without any string being wasted)?  Use the first
derivative test to show that you have indeed found the minimum.
\hfill (20)


\item Consider the function
\[ f(x) = \frac{x^3-8}{x}. \]
% \[ f(x) = \frac{8-x^3}{x}. \]
% \[ f(x) = \frac{x^3+8}{x}. \]
% \[ f(x) = \frac{-x^3-8}{x}. \]
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 \sin (at) \, dt $
\item $\displaystyle \int \sin^2 \alpha \, \cos \alpha \, d\alpha $
\item $\displaystyle \int_0^\pi \cos \phi \, d\phi $
\item $\displaystyle \int_{-2}^4 |x^2 - 1| \, dx $
\item $\displaystyle \int_{-1}^1 \bigl(x + x^3 + x^5 + x^7 + x^9 + x^{11}
+ x^{13} + x^{15} + x^{17} + x^{19}\bigr) dx$
\end{enumerate}
\hfill (5+5+5+10+5)

\item A car is traveling at speed $v_0$ along a straight road.  At time
$t=0$ and position $x=0$ the brakes are applied, and the car decelerates at
a constant rate $a$.
\begin{enumerate}
\item Show, by twice solving an initial value problem, that
\begin{equation}
  x(t) = v_0 \, t - \tfrac12 \, a \, t^2 . \tag{$**$}
\end{equation}
\item How far does the car travel before coming to a stop?
\item If the car traveled at a speed of $2v_0$, the stopping distance would
increase by what factor?
\item For what values of $t$ does equation ($**$) make sense?  Explain your
answer. 
\end{enumerate}
\hfill (10+10+5+5)

\end{enumerate}

\end{document}
