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\begin{document}
\begin{center}
\huge\textbf{Homework 1} \normalsize \\ Due Friday, June 30 2017
\end{center}
\problem{Problem 1 (10 points)}
(This is adapted from problem 5 from chapter 3 of the course textbook.)
A professor's daughter just started her freshman year at a private college that costs \$30,000 per year. When she was born (18 years ago) her grandparents put \$10,000 in a college fund for her. If the interest were compounded continuously, what annual percentage rate (APR) would the college fund need so that she has enough money today for the next four years of college tuition? What if the interest were compounded monthly? What if it were compounded yearly?
\problem{Problem 2 (10 points)}
(This is adapted from problem 7 from chapter 3 of the course textbook.)
You have just signed a contract that entitles you to receive \$1,000,000 twenty years from now, but you can't wait and want your money now. Assuming that the risk-free, inflation-adjusted APR is 3\% per year, compounded continuously, what is a fair price for your contract?
\problem{Problem 3 (5 points)}
Let $N_t$ be the number of rabbits in Australia at the end of year $t$ and let $M_t$ be the number of fox in Australia at the end of year $t$. Suppose that the populations interact in such a way that
\begin{align*} \label{eq:problem3}
N_{t+1} &= 6N_t - 4M_t, \numberthis \\
M_{t+1} &= 2N_t.
\end{align*}
If $N_0 = M_0 = 1$, find the populations $N_t$ and $M_t$ for all $t \geq 0$.
\emph{Extra credit (5 points)}: There is a very easy way to solve the preceding problem, but it only works because we chose convenient initial conditions. A slightly more systematic way to solve the problem is by assuming $N_t$ and $M_t$ are of the form
\begin{equation}
\begin{pmatrix} N_t \\ M_t \end{pmatrix} = \begin{pmatrix} u \\ v \end{pmatrix}\lambda^t
\end{equation}
and determining the values of $\lambda$, $u$ and $v$. Find a condition on $N_0$ and $M_0$ such that the solution will be of this form, and solve \eqref{eq:problem3} for any such initial conditions. Show that this agrees with your solution from above.
\problem{Problem 4 (15 points)}
(This is adapted from problem 4 from chapter 2 of the course textbook. Note that the text has a typo in part c -- it is missing a power of 1/4.)
In a growing organism, metabolism supplies energy to both maintain existing tissues and create new tissues by cell division. Let $Y_c$ be the metabolic rate of a single cell (that is, the amount of energy used by the cell per unit time) and let $E_c$ be the energy required to create a new cell and let $N_c(t)$ be the total number of cells at time $t$. The total mass of the organism is given by
\begin{equation} \label{eq:mass}
m(t) = m_cN_c(t),
\end{equation}
and the total metabolic rate $Y$ of the organism is given by
\begin{equation} \label{eq:metab}
Y(t) = Y_cN_c(t) + E_c\deriv{N_c}{t}.
\end{equation}
It has been argued (read http://science.sciencemag.org/content/276/5309/122 or chapter 2 of our textbook for more information) that $Y$ is related to $m$ by the equation
\begin{equation} \label{eq:metab-mass}
Y(t) = Y_0\cdot(m(t))^{3/4},
\end{equation}
where $Y_0$ is some constant.
\begin{enumerate}[(a)]
\item Show that equations \eqref{eq:mass}, \eqref{eq:metab} and \eqref{eq:metab-mass} can be combined to obtain
\begin{equation} \label{eq:problem1a}
\deriv{m}{t} = am^{3/4} - bm,
\end{equation}
where $a \equiv Y_0m_c/E_c$ and $b = Y_c/E_c$ are constants.
\item Suppose that when an organism matures, its mass stops changing. That is, at maturity $m(t) \equiv M$ is constant, so $m'(t) = 0$. Find $M$, and show that \eqref{eq:problem1a} can be rewritten as
\begin{equation} \label{eq:problem1b}
\deriv{m}{t} = am^{3/4}\left[1 - \left(\frac{m}{M}\right)^{1/4}\right].
\end{equation}
\item Let $r = (m/M)^{1/4}$ and $R = 1 - r$. Show that \eqref{eq:problem1b} becomes
\begin{equation}
\deriv{R}{t} = -\left(\frac{a}{4M^{1/4}}\right)R.
\end{equation}
Solve this differential equation for $R(t)$ with initial condition $R(0) = R_0$ and plot $\ln\left(R(t)/R_0\right)$ vs $\tau \equiv at/(4M^{1/4})$. This plot should be a straight line with slope $-1$, regardless of the values of any constants.
\end{enumerate}
\end{document}