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\title{Applied Calculus}
\author{Homework 4}
\date{Due in class, October 13, 2015}
\maketitle

\begin{enumerate}

\item (From MLS, p.~322.) Suppose that the population of a town can be
approximated by 
\[
  P(t) = 16\,250 \cdot (0.87)^t \,,
\]
where $P$ is the population of the town $t$ years after 1985 (starting
January 1, 1985).
\begin{enumerate}
\item Find the rate of change of the population on January 1, 2000.
Is the population increasing or decreasing?
\item Find the rate of change of the population on January 1, 2010.
Is the rate of change faster or slower than in was in 2000?
\item In this model, what happens for very long times (mathematically
in the limit as $t\to\infty$?)
\end{enumerate}

\item (From MLS, p.~322.) A laboratory study investigating the
relationship between diet and weight in adult humans found that the
weight of a subject, $W$ (in pounds), was a function $W=f(c)$, of the
average number of calories per day, $c$, consumed by the subject.

\begin{enumerate}
\item Interpret the statements $f(1800)=155$, $f'(2000)=0$ each in
terms of diet and weight.
\item What are the units of $f'(c)$?
\end{enumerate}

\item You are recording the lap times of a runner each of the four
$400\,\text{m}$ laps of a $1\,600\,\text{m}$ race.  Is the average
velocity of the runner over the entire race equal to the arithmetic
mean of the average lap velocities computed from the lap times?
Explain why or why not.

\item Find the equation of the tangent line at $x=1$ for the following
functions:
\begin{enumerate}
\item $\displaystyle f(x) = \frac{x \, \ln x}{x+1}$
\item $\displaystyle f(x) = \sqrt{1+\e^x}$
\end{enumerate}

\item For each function $f$: Find the vertical and horizontal
asymptotes (if any), find and classify all critical points, determine
where the function is concave up or concave down, find all points of
inflection, and sketch the graph.

\begin{enumerate}
\item $\displaystyle f(x) = x^5-2 \, x^3 + x - 2$
\item $\displaystyle f(x) = \frac{x+1}{x-1}$
\item $\displaystyle f(x) = \ln(x^2+1)$
\item $\displaystyle f(x) = x \, \e^{-x}$
\end{enumerate}
\hfill (5 Points each)

\item \emph{Note:} The solution to this homework may be turned in
\emph{after} the midterm exam on Tuesday, October 20, in class.

In \emph{Scientific Python}, after setting up your environment with
\verb+%pylab inline+, generate a data set as follows
\begin{verbatim}
  x = linspace(0,10,30)
  y = 0.5*x + 0.1*normal(size=30)
\end{verbatim}

\begin{enumerate}
\item Explain what these commands do and what you expect from this data
set.

\item Compute the local rate of change in between each of the data
points.

\emph{Hint:} You might find the partial data vectors \verb+y[1:]+ and
\verb+y[:-1]+ useful.  Find out what they are, and how to use them in
the context of this question.

\item Compute the average rate of change using two different
methods: (i) by doing a least-square fit of a straight line and
looking at its slope, (ii) by computing the average rate of change
from the local rates of change in part (b).  Do the results coincide?
Which computation is better?  Explain.

\end{enumerate}


\end{enumerate}




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