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\begin{center}
{\Large Warm-Up Problems and Lecture Problems

February 28, 2003}
\end{center}

\begin{enumerate}



\item  Solve the following equation for $P$:
	%
	\begin{equation} \label{logistic}
	\frac{M-P}{P} = Ae^{-kt}
	\end{equation}
	%

\item  Suppose you know that in equation~(\ref{logistic}) that $M$ and $k$ are constants given by $M=100$ and $k=2$.
	Suppose you also know that $P=20$ when $t=0$.  Find $A$.
	Using the previous problem, write $P$ as a function of $t$.

\newpage

\lecture

\item  Draw a slopefield and some possible trajectories for the logistic differential equation:
	\[
	\frac{dP}{dt} = 0.1P \left( 1 - \frac{P}{500} \right)
	\]

\item  Draw a slopefield and some possible trajectories for the differential equation:
	\[
	\frac{dP}{dt} = -0.1P \left( 1 - \frac{P}{500} \right)
	\]

\item  Draw a slopefield and some possible trajectories for the differential equation:
	\[
	\frac{dP}{dt} = 0.1P \left( 1 - \frac{P}{500} \right) \left( 1- \frac{P}{250} \right)
	\]






\end{enumerate}

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