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\begin{document}

\title{Applied Calculus}
\author{Midterm Exam I}
\date{September 24, 2015}
\maketitle

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% \noindent
% Useful identities:
% \begin{gather*}
%   \cos \theta = \frac{\e^{\i\theta}+\e^{-\i\theta}}{2} \qquad
%   \sin \theta = \frac{\e^{\i\theta}-\e^{-\i\theta}}{2i} \\
%   \cos^2 \theta = \tfrac12 \, (1 + \cos 2\theta) \qquad
%   \sin^2 \theta = \tfrac12 \, (1 - \cos 2\theta) \vphantom\int \\
%   \cos \theta \, \sin \theta = \tfrac12 \, \sin 2 \theta \vphantom\int \\
%   \tilde f(\xi)
%   = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty \e^{-\i\xi x} \, f(x) \,
%   dx \qquad 
%   f(x)
%   = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty \e^{\i\xi x} \,
%   \tilde f(\xi) \, d\xi \\
%   \delta(x)
%   = \frac1{2\pi} \int_{-\infty}^\infty \e^{\i\xi x} \, d\xi  \\
%   c_k = \frac1L \int_{-L/2}^{L/2} \e^{-2 \pi \i k x/L} \, f(x) \, \d x
%   \qquad
%   f(x) = \sum_{k=-\infty}^\infty c_k \, \e^{2 \pi \i k x/L}
% \end{gather*}

 

\begin{enumerate}
\item\begin{enumerate}

\item Find the equation of the line through the points $(3,-2)$ and
$(-1,1)$. 

\item Find the equation of the quadratic polynomial $y = a \, x^2 + b
\, x + c$ through the points $(0,0)$, $(3,0)$, and $(1,1)$.
\end{enumerate}
\hfill(5+10)
 
\item Solve the following equations for $x$.
\begin{enumerate}
\item $\displaystyle 2^x=4$
\item $\displaystyle \log_{10}(6 \log_2 x) = 3$
\item $\displaystyle \e^{\tfrac{2 \ln x^s}s} = 4$
\end{enumerate}
\hfill(5+5+5)
 
\item Suppose that for a certain data set, the doubly logarithmic
(base-$10$) graph is a line through points $(1,3)$ and $(2,-4)$.  Give
an equation for $y$ as a function of $x$.
\hfill(10)
 
\item The number of bacteria in milk grows exponentially, at least for
some time.  At bottling time, it is known that there are $10^6$
bacteria per bottle, the next day, there are three times as many.  The
milk can be consumed with up to $10^9$ bacteria per bottle.

\begin{enumerate}
\item Determine the shelf-life of bottled milk under these
assumptions. 

\item You are investigating the growth of bacteria in a sample of food
and want to plot the number of bacteria vs.\ time.  Which scaling
function will you use on each of the coordinate axes, and why?
\end{enumerate}
\hfill(10+5)
 

\item Compute the following limits.
\begin{enumerate}
\item $\displaystyle \lim_{r \to 10} \frac{\log_{10} r}r$
\item $\displaystyle \lim_{r \to 2} \frac{r^2-4}{r-2}$
\item $\displaystyle \lim_{r \to \infty} \frac{\e^r+r}{\e^r}$
\end{enumerate}
\hfill(5+5+5)
 
\item Determine whether $g(x)$ is continuous.  If $g(x)$ has a
discontinuity, state the type of discontinuity (removable
discontinuity, jump discontinuity, vertical asymptote, or other).

\begin{enumerate}
\item $\displaystyle g(x) = \frac1x$
\item $\displaystyle g(x) = x \, \ln x^2$
\item $\displaystyle g(x) =
\begin{cases}
0 & \text{for } x \leq 0 \\
x^2 & \text{for } x > 0
\end{cases}$

\end{enumerate}
\hfill(5+5+5)
 
\item An elevator is driven by a motor which is either off or moves it
with the constant speed of $1\,\text{m}/\text{s}$ up or down.  

The elevator is initially at the bottom of a building.  At time
$t=20\,\text{s}$, it visits the third floor at height
$h=12\,\text{m}$, then at $t=60\,\text{s}$ the first floor at height
$h=4\,\text{m}$. 
\begin{enumerate}
\item Is the height function $h(t)$ continuous?  Why or why not?
\item Draw a possible height function $h(t)$ onto the graph paper
provided.  Label the coordinate axes carefully.

(Note: The answer is not unique!)
\end{enumerate}
\hfill(5+10)



\end{enumerate}
\end{document}