\documentclass[letterpaper,12pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage[letterpaper,margin=1in,includehead=true]{geometry}
\usepackage{comment}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{enumerate}
\pagestyle{fancy}
\usepackage{color}
\usepackage{wrapfig}
%\usepackage{enumitem}
\usepackage{tikz, pgfplots}
%\usetikzlibrary{calc}
%\usepackage{titling}
%\setlength{\droptitle}{-1.5in}

%opening
\title{\vspace*{-1.5in}\textbf{\S7.5: The Logistic Equation}\\\normalsize (Thanks to Faan Tone Liu)}
\date{}
\author{}

\begin{document}

\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\chead{April 18, 2018}
\rhead{\S7.5: The Logistic Equation}
\vspace{-.5in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item Review: Often, population growth can be modeled by $P' = kP$. The solution is\\[6pt] \rule[0pt]{1.5in}{.5pt}, and this situation is called \rule[0pt]{2.1in}{.5pt}.\\

\item Exponential growth is not realistic in the long run because $\displaystyle\lim_{t \to \infty} P(t) = {}$\rule[0pt]{.55in}{.5pt}, so we modify it to get the \textbf{Logistic Equation}
\[
\frac{dP}{dt} = \phantom{kP(1-\frac{P}{M}) blahah},
\]
where $M$ is a constant that represents the carrying capacity of the population. The solution is \[\boxed{\phantom{this is the solution the solution.....\Bigg)}}\]\\

\item The slope field for the logistic growth equation is
\begin{center}
\includegraphics[trim={4.5in 0.6in 0in 8.6in}, clip]{2300SFp1.pdf}
\end{center}

\item Miscellaneous observations:
\begin{itemize}
\item If $P$ is small, then $\frac{dP}{dt} \approx {}$ \rule[-2pt]{.5in}{.5pt} (basically exponential growth).
\item If $P \approx M$, then $\frac{dP}{dt} \approx {}$ \rule[-2pt]{.5in}{.5pt} (growth slows to $0$).
\item Equilibrium (constant) solutions are: \rule[-1.5pt]{2in}{.5pt}.
\item If the population starts between $0$ and $M$: $\displaystyle \lim_{t\to \infty}P(t) = {}$ \rule{.5in}{.5pt}.
\item Using Calc I methods, we can show that $P(t)$ has an inflection point when \rule{.5in}{.5pt}.
\end{itemize}
\end{itemize}\newpage
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item (Review of exponential growth) Solve the differential equation $P' = kP$.\vfill
\item Solve the logistic differential equation $\displaystyle P' = kP\left(1-\frac{P}{M}\right)$.\vfill\vfill\vfill\vfill
\item Suppose a population grows according to the logistic model with an initial population of 1000 and a carrying capacity of 10,000. If the population grows to 2500 after one year, what is the population after three more years?\vfill
\end{enumerate}
\end{document}
