# Basic Classes of Mathematical Models

Following David Hestenes on page 6 of Modeling Instruction for STEM Education Reform, I wanted to create a poster like in my previous post of graphical methods and linearizing graphs but this time about the basic classes of mathematical models that Hestenes lists.  I’m not sure that I like all the equation gobbledegook, but I think students need something to which to aspire, so I just made it less prominent.  I’d also like a better presentation of some of the equations.

Update: You can now find source code for this and other posters in my GitHub repository.

\documentclass[final]{beamer} % beamer 3.10: do NOT use option hyperref={pdfpagelabels=false} !
%\documentclass[final,hyperref={pdfpagelabels=false}]{beamer} % beamer 3.07: get rid of beamer warnings
\mode<presentation> {  %% check http://www-i6.informatik.rwth-aachen.de/~dreuw/latexbeamerposter.php for examples
\definecolor{royalblue}{rgb}{0,0.13725490196078433,0.4}
\definecolor{royalblueweb}{rgb}{0.25490196078431371,0.41176470588235292,0.88235294117647056}
\definecolor{burntorange}{rgb}{0.8,0.3333333333333333,0}
\setbeamercolor{frametitle}{fg=blue!80!black}
\setbeamertemplate{frametitle} {
\begin{center}
\vspace{-1.2cm}\textbf{\insertframetitle} \par
\normalsize\textbf{\insertframesubtitle}
\end{center}
}
}
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{amsmath,amsthm, amssymb, latexsym}
%\usepackage{times}\usefonttheme{professionalfonts}  % times is obsolete
\usefonttheme[onlymath]{serif}
\boldmath
%\usepackage[orientation=portrait,size=a0,scale=1.4,debug]{beamerposter}                       % e.g. for DIN-A0 poster
%\usepackage[orientation=portrait,size=a1,scale=1.4,grid,debug]{beamerposter}                  % e.g. for DIN-A1 poster, with optional grid and debug output
\usepackage[size=custom,width=45.72,height=60.96,scale=1.8,debug]{beamerposter}                     % e.g. for custom size poster (18in x 24in w/ printable 17in x 23in)
%\usepackage[orientation=portrait,size=a0,scale=1.0,printer=rwth-glossy-uv.df]{beamerposter}   % e.g. for DIN-A0 poster with rwth-glossy-uv printer check
% ...
%
\geometry{margin=.5in}
\usepackage{array}
\usepackage{booktabs}
\newcolumntype{P}[1]{>{\raggedright\large}p{#1}}
\def\imagetop#1{\vtop{\vspace{-1.5cm}\null\hbox{#1}\vspace{-1.5cm}}}
\usepackage{tikz}

\newcommand{\xx}{\textcolor{variable}{x}}
\newcommand{\yy}{\textcolor{variable}{y}}
\newcommand{\versus}{vs\ }
\newcommand{\plotscale}{2.0}
\newcommand{\plotline}{6pt}
\newcommand{\formatmm}[1]{\textcolor{royalblue}{\textbf{#1}}}
\colorlet{plot}{burntorange}
\colorlet{variable}{blue!80!black}

% From Hestenes' list of 4 basic mathematical models
\title[Mathematical Models]{Basic Classes of Mathematical Models}
\author[Vancil]{Brian Vancil}
\institute[Sumner]{Sumner Academy of Arts & Sciences}
\date{2012-04-07}

\begin{document}
\begin{frame}{Basic Classes of Mathematical Models}
\framesubtitle{with sample equations}
\vspace{-2cm}
\begin{center}
\normalsize Mathematical model & \normalsize Kind of change & \normalsize Graph shape  \\ \midrule[.1em] \addlinespace

\formatmm{Linear model}
\par \normalsize $\yy=A\xx+B$
\par $\dfrac{d\yy}{d\xx}=A$ &
Rate of change is constant. &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth]
\draw[color=plot] plot (\x,.6*\x+1);
\draw[<->] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}

\par \normalsize $\yy=A\xx^{2}+B\xx+C$
\par $\dfrac{d^{2}\yy}{d\xx^{2}}=A$ &
Rate of change of rate of change is constant. &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
\draw[color=plot] plot (\x,{4-0.7*(\x-2)*(\x-2)});
\draw[<->] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}

\formatmm{Exponential model}
\par \normalsize $\yy=Ab^{\xx}$ or $\yy=Ae^{\frac{\xx}{\xi}}$
\par $\dfrac{d\yy}{d\xx}=\ln b\cdot\yy$ or $\dfrac{d\yy}{d\xx}=\frac{\yy}{\xi}$ &
Rate of change is proportional to amount. &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
\draw[color=plot] plot (\x,{pow(pow(4,.25),\x)});
\draw[<->] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}

\formatmm{Harmonic model}
\par \normalsize $\yy=A\cos\left(k\xx+\phi\right)$ or $\yy=A\sin\left(k\xx+\phi'\right)$ or $\yy=\mathfrak{Re}\left\{Ae^{i(k\xx+\phi)}\right\}$
\par $\dfrac{d^{2}\yy}{d\xx^{2}}=-k^{2}\yy$ &
Rate of change of rate of change is proportional to amount. &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
\draw[color=plot] plot (\x,{2*cos((\x*6.28-1)r)});
\draw[->] (0,-2) -- (0,2);
\draw[->] (0,0) -- (4,0);
\end{tikzpicture}}

\formatmm{Sudden change model}
\par \normalsize $\yy=A\ \theta(\xx-x_{0})+B$
\par $\dfrac{d\yy}{d\xx}=A\ \delta(\xx-x_{0})$ &
Change is finite and instantaneous. &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
\draw[color=plot, domain=0:2] plot (\x,1);
\draw[color=plot, domain=2:4] plot (\x,3);
\draw[<->] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}

\bottomrule[.1em]
\end{tabular}
\end{center}
\end{frame}
\end{document}


# Graphical Methods Summary

Linearizing and analyzing graphs are tricky skills for students to pick up when doing physics.  I enjoyed the discussion of a poster created by Paul Jebb, on the Modeling Physics listserve, but I wanted to create my own.  The result is here:

It would be nice to add the Hestenes’ descriptions of the change in the graph, but perhaps that deserves another poster.

Incidentally, I found the LaTeX beamerposter package quite easy to use.  As usual with LaTeX, the sizes of margins and things required tweaking, but it should work to print out as an 18″ by 24″ poster.

The source code is clearly cribbed from the beamerposter template.  PGF/TikZ is used for the graphs.

Update: You can now find source code for this and other posters in my GitHub repository.

  \documentclass[final]{beamer} % beamer 3.10: do NOT use option hyperref={pdfpagelabels=false} !
%\documentclass[final,hyperref={pdfpagelabels=false}]{beamer} % beamer 3.07: get rid of beamer warnings
\mode {  %% check http://www-i6.informatik.rwth-aachen.de/~dreuw/latexbeamerposter.php for examples
\definecolor{royalblue}{rgb}{0,0.13725490196078433,0.4}
\definecolor{royalblueweb}{rgb}{0.25490196078431371,0.41176470588235292,0.88235294117647056}
\definecolor{burntorange}{rgb}{0.8,0.3333333333333333,0}
\setbeamercolor{frametitle}{fg=blue!80!black}
\setbeamertemplate{frametitle} {
\begin{centering}
\vspace{-1.5cm}\textbf{\insertframetitle} \par
\end{centering}
}
}
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{amsmath,amsthm, amssymb, latexsym}
%\usepackage{times}\usefonttheme{professionalfonts}  % times is obsolete
\usefonttheme[onlymath]{serif}
\boldmath
%\usepackage[orientation=portrait,size=a0,scale=1.4,debug]{beamerposter}                       % e.g. for DIN-A0 poster
%\usepackage[orientation=portrait,size=a1,scale=1.4,grid,debug]{beamerposter}                  % e.g. for DIN-A1 poster, with optional grid and debug output
\usepackage[size=custom,width=45.72,height=60.96,scale=1.8,debug]{beamerposter}                     % e.g. for custom size poster (18in x 24in w/ printable 17in x 23in)
%\usepackage[orientation=portrait,size=a0,scale=1.0,printer=rwth-glossy-uv.df]{beamerposter}   % e.g. for DIN-A0 poster with rwth-glossy-uv printer check
% ...
%
\geometry{margin=.5in}
\usepackage{array}
\usepackage{booktabs}
\newcolumntype{P}[1]{>{\raggedright\large}p{#1}}
\def\imagetop#1{\vtop{\vspace{-1.5cm}\null\hbox{#1}\vspace{-1.5cm}}}
\usepackage{tikz}

\newcommand{\xx}{\textcolor{variable}{x}}
\newcommand{\yy}{\textcolor{variable}{y}}
\newcommand{\versus}{vs\ }
\newcommand{\plotscale}{1.5}
\newcommand{\plotline}{6pt}
\newcommand{\formatmm}[1]{\textcolor{royalblue}{\textbf{#1}}}
\colorlet{plot}{burntorange}
\colorlet{variable}{blue!80!black}

\title[Graph Methods]{Graphical Methods Summary}
\author[Vancil]{Brian Vancil}
\institute[Sumner]{Sumner Academy of Arts & Sciences}
\date{2012-04-07}

\begin{document}
\begin{frame}{Graphical Methods Summary}
\vspace{-2cm}
\normalsize Mathematical model & \normalsize Graph shape & \normalsize Written relationship & \normalsize To linearize, graph\ldots \\ \midrule[.1em] \addlinespace

\formatmm{constant}\ \ \  $\yy=b$ &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth]
\draw[color=plot] plot (\x,3);
\draw[] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}
& $\yy$ is constant. & \\ \addlinespace \midrule \addlinespace

\formatmm{proportional} $\yy=m\xx$ &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth]
\draw[color=plot] plot (\x,.8*\x);
\draw[] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}
& $\yy$ is directly proportional to $\xx$.  & \\ \addlinespace  \midrule \addlinespace

\formatmm{linear} $\yy=m\xx+b$ &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth]
\draw[color=plot] plot (\x,.6*\x+1);
\draw[] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}
& $\yy$ is linear in $\xx$.  & \\ \addlinespace \midrule \addlinespace

\formatmm{inversely proportional} $\yy=\frac{a}{\xx}$ &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=.1:4,line width=\plotline,smooth,samples=40]
\draw[color=plot] plot (\x,.4/\x);
\draw[] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}
& $\yy$ is inversely proportional to $\xx$.  & $\yy$ \versus $\frac{1}{\xx}$ \\ \addlinespace \midrule \addlinespace

\formatmm{power law} $\yy=a\xx^{n}$ &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
\draw[color=plot] plot (\x,.25*\x*\x);
\draw[] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}
& $\yy$ is proportional to $\xx^{n}$.  & $\yy$ \versus $\xx^{n}$ \\ \addlinespace \midrule \addlinespace

\formatmm{square root} $\yy=a\sqrt{\xx}$ &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
\draw[color=plot] plot (.25*\x*\x,\x);
\draw[] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}
%& $\yy^{2}$ is proportional to $\xx$. & Graph $\yy^{2}$ \versus $\xx$ \\ \addlinespace \midrule \addlinespace
& $\yy$ is proportional to the square root of $\xx$. & $\yy^{2}$ \versus $\xx$ \\ \addlinespace \midrule \addlinespace

\formatmm{exponential} $\yy=ab^{\xx}$ &
\imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
\draw[color=plot] plot (\x,{pow(pow(4,.25),\x)});
\draw[] (0,4) -- (0,0) -- (4,0);
\end{tikzpicture}}
& $\yy$ is exponential in $\xx$. & $\log \yy$ \versus $\xx$ \\  \addlinespace

\bottomrule[.1em]
\end{tabular}
\end{frame}
\end{document}