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

March 10, 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 $t$.


\item  (Review problem)
	Suppose you use Simpson's Rule for the following integral:  $\int_1^5 e^{3x} \; dx$.
	\begin{enumerate}
	\item  What is the error formula for Simpson's rule (and what does everything mean)?
	\item  Find a good value for $K$ for out integral.
	\item  If $n=100$, then what is the worst the error can be?
	\item  Find $n$ so that the error is less then $0.01$.
	\end{enumerate}

\newpage

\lecture

\item  Consider the curve $y=x^2+1$.
	\begin{enumerate}
	\item  Draw this curve and rotate it around the $y$-axis.
	
	\item  Find the area function $A(y)$ (the area of a $y$-cross section).

	\item  Set up an integral representing the volume of the solid of revolution from $y=2$ to $y=4$.  (What does this part of the solid look like?)

	\end{enumerate}


\item  Repeat the previous exercise, but rotate the curve around the $x$-axis and find the volume from $x=-1$ to $x=3$.



\end{enumerate}

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