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\title{Applied Calculus}
\author{Homework 1}
\date{Due in class, September 15, 2015}
\maketitle

\emph{Note}: You may express your answer in terms of unevaluated
exponential or logarithmic functions so long as everything that can be
simplified by hand is simplified.  There is no need to use a
calculator unless explicitly requested.

\begin{enumerate}

\item
\begin{enumerate}

\item Find the equation of the line through the points $(4,5)$ and
$(8,7)$.

\item Find $x$ and $y$ intercepts for the line with equation $y =
2x-5$.

\item Find the point of intersection of the lines with equations
\begin{gather*}
  y = -x+7 \qquad \text{and} \qquad y = 3x-9 \,.
\end{gather*}

\item Find an equation for the line parallel to the line with equation
$y=10x+5$ through the point $(0,2)$.

\end{enumerate}

\item The cost of a taxi ride is the sum of a flat call out fee and a
charge for every kilometer traveled.  In some city, a ride of
$6\,\text{km}$ length costs $\text{EUR}\,12$ while a ride of
$10\,\text{km}$ length costs $\text{EUR}\,18$.  How much are the call
out fee and the per-kilometer charge?  How much does a trip of
$15\,\text{km}$ cost?

\item Find the equation for a quadratic through the points $(1,15)$,
$(3,31)$, and $(-2,6)$.


\item (From MLS, p.~67.) Solve each of the exponential equations for
$t$.
\begin{enumerate}
\item $4 \, (3^{2t}) = 12$
\item $6 \, (0.6)^t = 10$
\item $4 \, \e^{0.05t} = 8$
\item $3^{t^2-t}=9$
\end{enumerate}

\item (From MLS, p.~67.) Solve each of the following logarithmic
equations for $x$.

\begin{enumerate}
\item $\ln(2x+3) = 5$
\item $\log_2(\log_3 2x) = 4$
\end{enumerate}


\item (From MLS, p.~67.) Salmonella bacteria grow rapidly at room
temperature.  If some bacteria are left on the cutting board when a
chicken with salmonella is cut up and they get into a salad, the
population of bacteria begins growing.  Suppose that the number
present in the salad after $t$ hours is given by
\[
  f(t) = 200 \cdot \e^{rt} 
\]
where $r=2 \, \text{h}^{-1}$.

\begin{enumerate}
\item If the salad is left at room temperature, how many bacteria are
present $1$ hour later?
\item How many bacteria were initially present in the salad?
\item What is the doubling time for the number of bacteria?
\end{enumerate}

\item (From MLS, p.~68.) Suppose that the island area $A$ and the
number of species $S$ (of a certain taxa) on that island are given by
\[
  S = \alpha\, A^\beta \,,
\]
where $\alpha$ and $\beta$ are constants.
\begin{enumerate}
\item What kind of function is this?
\item Suppose you are studying land and freshwater birds among a small
cluster of islands.  The first island has an area of $50$ square
miles, and you count $13$ species of birds.  the second island has an
area of $125$ square miles, and you count $17$ species of birds.
Given this information, find the values of $\alpha$ and $\beta$ so
that you have a description of how bird species count depends on
island size.

\item If you count $24$ species of birds on a nearby island, estimate
the area of this island.
\end{enumerate}

\item (From MLS, p.~68.) Suppose that for a certain data set, the
semilog (base-$10$) graph is a line through the points $(2,-2)$ and
$(3,-4)$.  Give an equation for $y$ as a function of $x$.


\item \emph{Note:} The following problem requires the use of
\emph{Scientific Python}.  It should be attempted this week, but is
due only one week later to allow ironing out potential technical
problems.  An Ipython noteboook from the class session September 3 is
available on my web page, it already contains the solution for part
(a) below.

(From MLS, p.~45.) Use the data from the Example shown in class.
\begin{enumerate}
\item Find the equation for the least-squares fit of the $y$ data in
terms of the $x$ data.

\item Find the equation for the least-squares fit of the $x$ data in
terms of the $y$ data.

\item Solve the equation in (b) for $y$ in terms of $x$.  Is this the
same equation that you obtained in (a)?  Explain why you think the
answers were the same or different.

\item Compute the $\rho^2$ value in each case and interpret the
results. 
\end{enumerate}


\end{enumerate}


\end{document}
