\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts}

\def\e{{\mathrm e}}

\begin{document}

\title{Applied Calculus}
\author{Homework 5}
\date{Due in class, November 3, 2015}
\maketitle

\begin{enumerate}

\item (From MLS, p.~405.)

The population of a flock of geese is modeled by
\[
  G(t) = 2000 + 150\, \sin \Bigl( \frac\pi6 \, t \Bigr)
         + 75 \, \sin \Bigl( \frac\pi3 \, t \Bigr) \,,
\]
where $t$ is measured in months from January 1.

\begin{enumerate}
\item Find any critical points and any inflection points of $G$.
State the location of any local maxima, minima, and inflection
points.  Use this information to sketch a graph of $G(t)$ over a
$2$-year period.
\item Interpret any local maxima, minima, and inflection points over
the first $2$ years in biological terms.
\item How fast is the flock growing on May 1?  On May 1, is the
\emph{rate} at which the flock is growing increasing or decreasing?
\end{enumerate}

\item (From MLS, p.~406.) During an epidemic, the number of infected
people $I(t)$ is approximated by
\[
  I(t) = \frac{a \, \ln(bt+1)}{bt+1} \,,
\]
where $t$ is measured in days since the initial outbreak.
\begin{enumerate}
\item Find an expression for the time when the number of infected
people starts to decline.
\item Suppose that you introduce a treatment for infected
individuals.  When treatment is added to the model, the value of $b$
is increased.  What effect does this have on the time at which the
number of infected people starts to decline?  Given your answer,
describe (in biological terms) the population-level effect of your
treatment. 
\item Consider your answers from (a) and (b).  Is there a
biological/physical limitation to how large the value of $b$ can be?
(\emph{Hint:} Try calculating the time at which the number of infected
people starts to decline for a \emph{very large} value of $b$.)
\end{enumerate}

\item (From MLS, p.~406.) For infants less than $9$ months old, the
relationship between the rate of growth $R$ (in pounds/month) and the
present weight $W$ (in pounds) is approximated by 
\[
  R = c \, W \, (21-W)
\]
for some constant $c$.  At what weight is the growth rate the largest?

\item (From MLS, p.~407.)  Imagine that you have just been given a
vacation home situated on an island $10$\,km off the coast.  The
nearest town is $60$\,km up the coast from the point of least distance
to the island.  Suppose you can travel at $100\,\mbox{km}/\mbox{h}$
along the highway and at $20\,\mbox{km}/\mbox{h}$ in water, at which
point should you land to minimize travel time from the island to town?

\item (From MLS, p.~406.) Find the rectangle with the largest area
that can be placed inside the region bounded between the curve
$y=4-x^2$ and the horizontal axis $y=0$.

% \begin{enumerate}
% \item Solve this problem by formulating it as a maximization problem
% for a function of one variable.

% \item Solve this problem by using a Lagrange multiplier.

% \end{enumerate}

% \item Use a Lagrange multiplier to find the point on the unit circle
% $x^2+y^2=1$ closest to $(1,2)$.

% \item Use Lagrange multipliers to find the maximum and minimum values
% of the function
% \[
%   f(x,y,z) = 3 \, x - y + 2\, z 
% \]
% subject to the constraints
% \[
%   x + y - z = 0  \qquad \text{and} \qquad
%   x^2 + 3 \, y^2 = 1 \,.
% \]


% \item Heron's formula states that the area of a triangle with side
% lengths $x$, $y$, and $z$ is
% \[
%   A = \sqrt{s \, (s-x) \, (s-y) \, (s-z)}
% \]
% where $s$ is the semiperimeter of the triangle, i.e.,
% \[
%   s = \frac{x+y+z}2 \,.
% \]
% Use a Lagrange multiplier to show that the triangle with maximum area
% for a given perimeter is equilateral.

\end{enumerate}




\end{document}
