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\title{\vspace*{-1.5in}\textbf{Math 2300-007: Quiz 12}}
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%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\rhead{April 23, 2018}
\chead{Quiz 12}
\vspace{-.5in}
\noindent
{Name: \underline{\phantom{Aalsdkjflaskdjflaskjflksdjflkdsjf}}} \hfill{Score: \underline{\phantom{lksdjflkdsjf}}}\\
\noindent
\begin{enumerate}
\item (6 points) A population $P(t)$ grows according to a logistic model and satisfies the logistic differential equation
\[
\frac{dP}{dt} = \frac{4}{10}P\left(1 - \frac{P}{400}\right),\quad P(0) = 10,
\]
where $t$ is measured in years.
\begin{enumerate}[(a)]
\item What is the carrying capacity in this situation?\vfill
\item What is $P'(0)$?\vfill
\item Interpret the meaning of $P'(0)$. Mention the units in your answer.\vfill
\end{enumerate}
\newpage
\item (4 points) Suppose $P(t)$ represents the size of a population in millions $t$ years since $2000$ and we know that
\begin{itemize}
\item the birth rate is 0.05 births per person per year;
\item the death rate is 0.02 deaths per person per year;
\item 3 million immigrants join the population each year.
\end{itemize}
Write (\textbf{but do not solve}) a differential equation for $\frac{dP}{dt}$, the rate of change of the population at time $t$.
\end{enumerate}

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