Skip to main content Contents Prev Up Next \(\def\vs{{\it vs. }}
\def\cf{{\it cf. }}
\def\viz{{\it viz. }}
\def\ie{{\it i.e. }}
\def\etc{{\it etc. }}
\def\eg{{\it e.g. }}
\def\etal{{\it et al .}}
\def\via{{\it via }}
\def\adhoc{{\it ad hoc }}
\def\apriori{{\it apriori }}
\def\Afrak{\mathfrak{A}}
\def\Bfrak{\mathfrak{B}}
\def\Cfrak{\mathfrak{C}}
\def\Dfrak{\mathfrak{D}}
\def\Efrak{\mathfrak{E}}
\def\Ffrak{\mathfrak{F}}
\def\Gfrak{\mathfrak{G}}
\def\Hfrak{\mathfrak{H}}
\def\Ifrak{\mathfrak{I}}
\def\Jfrak{\mathfrak{J}}
\def\Kfrak{\mathfrak{K}}
\def\Lfrak{\mathfrak{L}}
\def\Mfrak{\mathfrak{M}}
\def\Nfrak{\mathfrak{N}}
\def\Ofrak{\mathfrak{O}}
\def\Pfrak{\mathfrak{P}}
\def\Qfrak{\mathfrak{Q}}
\def\Rfrak{\mathfrak{R}}
\def\Sfrak{\mathfrak{S}}
\def\Tfrak{\mathfrak{T}}
\def\Ufrak{\mathfrak{U}}
\def\Vfrak{\mathfrak{V}}
\def\Wfrak{\mathfrak{W}}
\def\Xfrak{\mathfrak{X}}
\def\Yfrak{\mathfrak{Y}}
\def\Zfrak{\mathfrak{Z}}
\def\afrak{\mathfrak{a}}
\def\bfrak{\mathfrak{b}}
\def\cfrak{\mathfrak{c}}
\def\dfrak{\mathfrak{d}}
\def\efrak{\mathfrak{e}}
\def\ffrak{\mathfrak{f}}
\def\gfrak{\mathfrak{g}}
\def\hfrak{\mathfrak{h}}
\def\ifrak{\mathfrak{i}}
\def\jfrak{\mathfrak{j}}
\def\kfrak{\mathfrak{k}}
\def\lfrak{\mathfrak{l}}
\def\mfrak{\mathfrak{m}}
\def\nfrak{\mathfrak{n}}
\def\ofrak{\mathfrak{o}}
\def\pfrak{\mathfrak{p}}
\def\qfrak{\mathfrak{q}}
\def\rfrak{\mathfrak{r}}
\def\sfrak{\mathfrak{s}}
\def\tfrak{\mathfrak{t}}
\def\ufrak{\mathfrak{u}}
\def\vfrak{\mathfrak{v}}
\def\wfrak{\mathfrak{w}}
\def\xfrak{\mathfrak{x}}
\def\yfrak{\mathfrak{y}}
\def\zfrak{\mathfrak{z}}
\def\AA{\mathbb{A}}
\def\BB{\mathbb{B}}
\def\CC{\mathbb{C}}
\def\DD{\mathbb{D}}
\def\EE{\mathbb{E}}
\def\FF{\mathbb{F}}
\def\GG{\mathbb{G}}
\def\HH{\mathbb{H}}
\def\II{\mathbb{I}}
\def\JJ{\mathbb{J}}
\def\KK{\mathbb{K}}
\def\LL{\mathbb{L}}
\def\MM{\mathbb{M}}
\def\NN{\mathbb{N}}
\def\OO{\mathbb{O}}
\def\PP{\mathbb{P}}
\def\QQ{\mathbb{Q}}
\def\RR{\mathbb{R}}
\def\SS{\mathbb{S}}
\def\TT{\mathbb{T}}
\def\UU{\mathbb{U}}
\def\VV{\mathbb{V}}
\def\WW{\mathbb{W}}
\def\XX{\mathbb{X}}
\def\YY{\mathbb{Y}}
\def\ZZ{\mathbb{Z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\Ap{A^\prime}
\def\Bp{B^\prime}
\def\Cp{C^\prime}
\def\Dp{D^\prime}
\def\Ep{E^\prime}
\def\Fp{F^\prime}
\def\Gp{G^\prime}
\def\Hp{H^\prime}
\def\Ip{I^\prime}
\def\Jp{J^\prime}
\def\Kp{K^\prime}
\def\Lp{L^\prime}
\def\Mp{M^\prime}
\def\Mp{N^\prime}
\def\Op{O^\prime}
\def\Pp{P^\prime}
\def\Qp{Q^\prime}
\def\Rp{R^\prime}
\def\Sp{S^\prime}
\def\Tp{T^\prime}
\def\Up{U^\prime}
\def\Vp{V^\prime}
\def\Wp{W^\prime}
\def\Xp{X^\prime}
\def\Yp{Y^\prime}
\def\Zp{Z^\prime}
\def\ap{a^\prime}
\def\bp{b^\prime}
\def\cp{c^\prime}
\def\dprime{d^\prime}
\def\ep{e^\prime}
\def\fp{f^\prime}
\def\gp{g^\prime}
\def\hp{h^\prime}
\def\ip{i^\prime}
\def\jp{j^\prime}
\def\kp{k^\prime}
\def\lp{l^\prime}
\def\mp{m^\prime}
\def\np{n^\prime}
\def\op{o^\prime}
\def\pp{p^\prime}
\def\qp{q^\prime}
\def\rp{r^\prime}
\def\sp{s^\prime}
\def\tp{t^\prime}
\def\up{u^\prime}
\def\vp{v^\prime}
\def\wp{w^\prime}
\def\xp{x^\prime}
\def\yp{y^\prime}
\def\zp{z^\prime}
\def\App{A^{\prime\prime}}
\def\Bpp{B^{\prime\prime}}
\def\Cpp{C^{\prime\prime}}
\def\Dpp{D^{\prime\prime}}
\def\Epp{E^{\prime\prime}}
\def\Fpp{F^{\prime\prime}}
\def\Gpp{G^{\prime\prime}}
\def\Hpp{H^{\prime\prime}}
\def\Ipp{I^{\prime\prime}}
\def\Jpp{J^{\prime\prime}}
\def\Kpp{K^{\prime\prime}}
\def\Lpp{L^{\prime\prime}}
\def\Mpp{M^{\prime\prime}}
\def\Mpp{N^{\prime\prime}}
\def\Opp{O^{\prime\prime}}
\def\Ppp{P^{\prime\prime}}
\def\Qpp{Q^{\prime\prime}}
\def\Rpp{R^{\prime\prime}}
\def\Spp{S^{\prime\prime}}
\def\Tpp{T^{\prime\prime}}
\def\Upp{U^{\prime\prime}}
\def\Vpp{V^{\prime\prime}}
\def\Wpp{W^{\prime\prime}}
\def\Xpp{X^{\prime\prime}}
\def\Ypp{Y^{\prime\prime}}
\def\Zpp{Z^{\prime\prime}}
\def\app{a^{\prime\prime}}
\def\bpp{b^{\prime\prime}}
\def\cpp{c^{\prime\prime}}
\def\dpp{d^{\prime\prime}}
\def\epp{e^{\prime\prime}}
\def\fpp{f^{\prime\prime}}
\def\gpp{g^{\prime\prime}}
\def\hpp{h^{\prime\prime}}
\def\ipp{i^{\prime\prime}}
\def\jpp{j^{\prime\prime}}
\def\kpp{k^{\prime\prime}}
\def\lpp{l^{\prime\prime}}
\def\mpp{m^{\prime\prime}}
\def\npp{n^{\prime\prime}}
\def\opp{o^{\prime\prime}}
\def\ppp{p^{\prime\prime}}
\def\qpp{q^{\prime\prime}}
\def\rpp{r^{\prime\prime}}
\def\spp{s^{\prime\prime}}
\def\tpp{t^{\prime\prime}}
\def\upp{u^{\prime\prime}}
\def\vpp{v^{\prime\prime}}
\def\wpp{w^{\prime\prime}}
\def\xpp{x^{\prime\prime}}
\def\ypp{y^{\prime\prime}}
\def\zpp{z^{\prime\prime}}
\def\abar{\overline{a}}
\def\bbar{\overline{b}}
\def\cbar{\overline{c}}
\def\dbar{\overline{d}}
\def\ebar{\overline{e}}
\def\fbar{\overline{f}}
\def\gbar{\overline{g}}
\def\ibar{\overline{i}}
\def\jbar{\overline{j}}
\def\kbar{\overline{k}}
\def\lbar{\overline{l}}
\def\mbar{\overline{m}}
\def\nbar{\overline{n}}
\def\obar{\overline{o}}
\def\pbar{\overline{p}}
\def\qbar{\overline{q}}
\def\rbar{\overline{r}}
\def\sbar{\overline{s}}
\def\tbar{\overline{t}}
\def\ubar{\overline{u}}
\def\vbar{\overline{v}}
\def\wbar{\overline{w}}
\def\xbar{\overline{x}}
\def\ybar{\overline{y}}
\def\zbar{\overline{z}}
\def\Abar{\overline{A}}
\def\Bbar{\overline{B}}
\def\Cbar{\overline{C}}
\def\Dbar{\overline{D}}
\def\Ebar{\overline{E}}
\def\Fbar{\overline{F}}
\def\Gbar{\overline{G}}
\def\Hbar{\overline{H}}
\def\Ibar{\overline{I}}
\def\Jbar{\overline{J}}
\def\Kbar{\overline{K}}
\def\Lbar{\overline{L}}
\def\Mbar{\overline{M}}
\def\Nbar{\overline{N}}
\def\Obar{\overline{O}}
\def\Pbar{\overline{P}}
\def\Qbar{\overline{Q}}
\def\Rbar{\overline{R}}
\def\Sbar{\overline{S}}
\def\Tbar{\overline{T}}
\def\Ubar{\overline{U}}
\def\Vbar{\overline{V}}
\def\Wbar{\overline{W}}
\def\Xbar{\overline{X}}
\def\Ybar{\overline{Y}}
\def\Zbar{\overline{Z}}
\def\aunder{\underline{a}}
\def\bunder{\underline{b}}
\def\cunder{\underline{c}}
\def\dunder{\underline{d}}
\def\eunder{\underline{e}}
\def\funder{\underline{f}}
\def\gunder{\underline{g}}
\def\hunder{\underline{h}}
\def\iunder{\underline{i}}
\def\junder{\underline{j}}
\def\kunder{\underline{k}}
\def\lunder{\underline{l}}
\def\munder{\underline{m}}
\def\nunder{\underline{n}}
\def\ounder{\underline{o}}
\def\punder{\underline{p}}
\def\qunder{\underline{q}}
\def\runder{\underline{r}}
\def\sunder{\underline{s}}
\def\tunder{\underline{t}}
\def\uunder{\underline{u}}
\def\vunder{\underline{v}}
\def\wunder{\underline{w}}
\def\xunder{\underline{x}}
\def\yunder{\underline{y}}
\def\zunder{\underline{z}}
\def\Aunder{\underline{A}}
\def\atilde{\widetilde{a}}
\def\btilde{\widetilde{b}}
\def\ctilde{\widetilde{c}}
\def\dtilde{\widetilde{d}}
\def\etilde{\widetilde{e}}
\def\ftilde{\widetilde{f}}
\def\gtilde{\widetilde{g}}
\def\htilde{\widetilde{h}}
\def\itilde{\widetilde{i}}
\def\jtilde{\widetilde{j}}
\def\ktilde{\widetilde{k}}
\def\ltilde{\widetilde{l}}
\def\mtilde{\widetilde{m}}
\def\ntilde{\widetilde{n}}
\def\otilde{\widetilde{o}}
\def\ptilde{\widetilde{p}}
\def\qtilde{\widetilde{q}}
\def\rtilde{\widetilde{r}}
\def\stilde{\widetilde{s}}
\def\ttilde{\widetilde{t}}
\def\utilde{\widetilde{u}}
\def\vtilde{\widetilde{v}}
\def\wtilde{\widetilde{w}}
\def\xtilde{\widetilde{x}}
\def\ytilde{\widetilde{y}}
\def\ztilde{\widetilde{z}}
\def\Atilde{\widetilde{A}}
\def\Btilde{\widetilde{B}}
\def\Ctilde{\widetilde{C}}
\def\Dtilde{\widetilde{D}}
\def\Etilde{\widetilde{E}}
\def\Ftilde{\widetilde{F}}
\def\Gtilde{\widetilde{G}}
\def\Htilde{\widetilde{H}}
\def\Itilde{\widetilde{I}}
\def\Jtilde{\widetilde{J}}
\def\Ktilde{\widetilde{K}}
\def\Ltilde{\widetilde{L}}
\def\Mtilde{\widetilde{M}}
\def\Ntilde{\widetilde{N}}
\def\Otilde{\widetilde{O}}
\def\Ptilde{\widetilde{P}}
\def\Qtilde{\widetilde{Q}}
\def\Rtilde{\widetilde{R}}
\def\Stilde{\widetilde{S}}
\def\Ttilde{\widetilde{T}}
\def\Utilde{\widetilde{U}}
\def\Vtilde{\widetilde{V}}
\def\Wtilde{\widetilde{W}}
\def\Xtilde{\widetilde{X}}
\def\Ytilde{\widetilde{Y}}
\def\Ztilde{\widetilde{Z}}
\def\Alphatilde{\widetilde{\Alpha}}
\def\Betatilde{\widetilde{\Beta}}
\def\Gammatilde{\widetilde{\Gamma}}
\def\Deltatilde{\widetilde{\Delta}}
\def\Epsilontilde{\widetilde{\Epsilon}}
\def\Zetatilde{\widetilde{\Zeta}}
\def\Etatilde{\widetilde{\Eta}}
\def\Thetatilde{\widetilde{\Theta}}
\def\Iotatilde{\widetilde{\Iota}}
\def\Kappatilde{\widetilde{\Kappa}}
\def\Lambdatilde{\widetilde{\Lamdba}}
\def\Mutilde{\widetilde{\Mu}}
\def\Nutilde{\widetilde{\Nu}}
\def\Xitilde{\widetilde{\Xi}}
\def\Omicrontilde{\widetilde{\Omicron}}
\def\Pitilde{\widetilde{\Pi}}
\def\Rhotilde{\widetilde{\Rho}}
\def\Sigmatilde{\widetilde{\Sigma}}
\def\Tautilde{\widetilde{\Tau}}
\def\Upsilontilde{\widetilde{\Upsilon}}
\def\Phitilde{\widetilde{\Phi}}
\def\Chitilde{\widetilde{\Chi}}
\def\Psitilde{\widetilde{\Psi}}
\def\Omegatilde{\widetilde{\Omega}}
\def\alphatilde{\widetilde{\alpha}}
\def\betatilde{\widetilde{\beta}}
\def\gammatilde{\widetilde{\gamma}}
\def\deltatilde{\widetilde{\delta}}
\def\epsilontilde{\widetilde{\epsilon}}
\def\zetatilde{\widetilde{\zeta}}
\def\etatilde{\widetilde{\eta}}
\def\thetatilde{\widetilde{\theta}}
\def\iotatilde{\widetilde{\iota}}
\def\kappatilde{\widetilde{\kappa}}
\def\lambdatilde{\widetilde{\lamdba}}
\def\mutilde{\widetilde{\mu}}
\def\nutilde{\widetilde{\nu}}
\def\xitilde{\widetilde{\xi}}
\def\omicrontilde{\widetilde{\omicron}}
\def\pitilde{\widetilde{\pi}}
\def\rhotilde{\widetilde{\rho}}
\def\sigmatilde{\widetilde{\sigma}}
\def\tautilde{\widetilde{\tau}}
\def\upsilontilde{\widetilde{\upsilon}}
\def\phitilde{\widetilde{\phi}}
\def\chitilde{\widetilde{\chi}}
\def\psitilde{\widetilde{\psi}}
\def\omegatilde{\widetilde{\omega}}
\def\Alphabar{\bar{\Alpha}}
\def\Betabar{\bar{\Beta}}
\def\Gammabar{\bar{\Gamma}}
\def\Deltabar{\bar{\Delta}}
\def\Epsilonbar{\bar{\Epsilon}}
\def\Zetabar{\bar{\Zeta}}
\def\Etabar{\bar{\Eta}}
\def\Thetabar{\bar{\Theta}}
\def\Iotabar{\bar{\Iota}}
\def\Kappabar{\bar{\Kappa}}
\def\Lambdabar{\bar{\Lamdba}}
\def\Mubar{\bar{\Mu}}
\def\Nubar{\bar{\Nu}}
\def\Xibar{\bar{\Xi}}
\def\Omicronbar{\bar{\Omicron}}
\def\Pibar{\bar{\Pi}}
\def\Rhobar{\bar{\Rho}}
\def\Sigmabar{\bar{\Sigma}}
\def\Taubar{\bar{\Tau}}
\def\Upsilonbar{\bar{\Upsilon}}
\def\Phibar{\bar{\Phi}}
\def\Chibar{\bar{\Chi}}
\def\Psibar{\bar{\Psi}}
\def\Omegabar{\bar{\Omega}}
\def\alphabar{\bar{\alpha}}
\def\betabar{\bar{\beta}}
\def\gammabar{\bar{\gamma}}
\def\deltabar{\bar{\delta}}
\def\epsilonbar{\bar{\epsilon}}
\def\zetabar{\bar{\zeta}}
\def\etabar{\bar{\eta}}
\def\thetabar{\bar{\theta}}
\def\iotabar{\bar{\iota}}
\def\kappabar{\bar{\kappa}}
\def\lambdabar{\bar{\lamdba}}
\def\mubar{\bar{\mu}}
\def\nubar{\bar{\nu}}
\def\xibar{\bar{\xi}}
\def\omicronbar{\bar{\omicron}}
\def\pibar{\bar{\pi}}
\def\rhobar{\bar{\rho}}
\def\sigmabar{\bar{\sigma}}
\def\taubar{\bar{\tau}}
\def\upsilonbar{\bar{\upsilon}}
\def\phibar{\bar{\phi}}
\def\chibar{\bar{\chi}}
\def\psibar{\bar{\psi}}
\def\omegabar{\bar{\omega}}
\def\del{\partial}
\def\delbar{\overline{\partial}}
\def\Cech{\check{C}}
\def\half{\frac{1}{2}}
\def\defeq{\mathrel{\mathop:}=}
\def\alg{\mathrm{alg}}
\def\Alt{\mathrm{Alt}}
\def\Amp{\mathrm{Amp}}
\def\Arg{\mathrm{Arg}}
\def\an{\mathrm{an}}
\def\anti{\mathrm{anti}}
\def\Ap{\mathrm{Ap}}
\def\arcsinh{\mathrm{arcsinh\hspace{0.07cm}}}
\def\arccosh{\mathrm{arccosh\hspace{0.07cm}}}
\def\arctanh{\mathrm{arctanh\hspace{0.07cm}}}
\def\arccsch{\mathrm{arccsch\hspace{0.07cm}}}
\def\arcsech{\mathrm{arcsech\hspace{0.07cm}}}
\def\arccoth{\mathrm{arccoth\hspace{0.07cm}}}
\def\arccsc{\mathrm{arccsc\hspace{0.07cm}}}
\def\arcsec{\mathrm{arcsec\hspace{0.07cm}}}
\def\arccot{\mathrm{arccot\hspace{0.07cm}}}
\def\arg{\mathrm{arg}}
\def\BC{\mathrm{BC}}
\def\Bel{\mathrm{Bel}}
\def\calCH{\mathcal{CH}}
\def\csch{\mathrm{csch}\hspace{0.07cm}}
\def\CH{\mathrm{CH}}
\def\ch{\mathrm{ch}}
\def\closed{\mathrm{closed}}
\def\codim{\mathrm{codim}}
\def\coth{\mathrm{coth}\hspace{0.07cm}}
\def\Coh{\mathfrak{Coh}}
\def\Coker{\mathrm{Coker}}
\def\Cone{\mathrm{Cone}}
\def\darg{d\mathrm{arg}}
\def\Db{\mathrm{Db}}
\def\dclosed{\mathrm{d-closed}}
\def\deg{\mathrm{deg}}
\def\dim{\mathrm{dim}}
\def\divisor{\mathrm{div}}
\def\dlog{d\mathrm{log}}
\def\DNE{\mathrm{DNE}}
\def\DR{\mathrm{DR}}
\def\DST{\mathrm{DST}}
\def\exp{\mathrm{exp}}
\def\FLB{\mathrm{FLB}}
\def\FLS{\mathrm{FLS}}
\def\Gr{\mathrm{Gr}}
\def\Hzar{H_{\mathrm{Zar}}}
\def\Hol{\mathrm{Hol}}
\def\Id{\mathrm{Id}}
\def\Image{\mathrm{Im}}
\def\Ka{\mathcal{K}_A}
\def\Ker{\mathrm{Ker}}
\def\kod{\mathrm{kod}}
\def\Kx{\mathcal{K}_X}
\def\Kz{\mathcal{K}_Z}
\def\log{\mathrm{log}}
\def\Log{\mathrm{Log}}
\def\Li{\mathrm{Li}}
\def\min{\mathrm{min}}
\def\Mon{\mathrm{Mon}}
\def\Nef{\mathrm{Nef}}
\def\NS{\mathrm{NS}}
\def\Oa{\mathcal{O}_A}
\def\Ox{\mathcal{O}_X}
\def\Oz{\mathcal{O}_Z}
\def\Perp{\mathrm{Perp}}
\def\Pic{\mathrm{Pic}}
\def\Proj{\mathrm{Proj}}
\def\rank{\mathrm{rank}}
\def\Rat{\mathrm{Rat}}
\def\Real{\mathrm{Re}}
\def\reg{\mathrm{reg}}
\def\Res{\mathrm{Res}}
\def\res{\mathrm{res}}
\def\Ric{\mathrm{Ric}}
\def\sech{\mathrm{sech}\hspace{0.07cm}}
\def\Span{\mathrm{Span}}
\def\Spec{\mathrm{Spec}}
\def\sing{\mathrm{sing}}
\def\Singx{\mathrm{Sing}(X)}
\def\sheafKer{\mathcal{\Ker}}
\def\sheafIm{\mathcal{\Im}}
\def\Span{\mathrm{Span}}
\def\Spin{\mathrm{Spin}}
\def\Str{\mathrm{Str}}
\def\td{\mathrm{td}}
\def\tr{\mathrm{tr}}
\def\Todd{\mathrm{Todd}}
\def\tor{\mathrm{tor}}
\def\trdeg{\mathrm{trdeg}}
\def\Zar{\mathrm{Zar}}
\def\ZFLS{\mathrm{ZFLS}}
\usepackage{tikz}
\usepackage{tkz-graph}
\usepackage{tkz-euclide}
\usetikzlibrary{patterns}
\usetikzlibrary{positioning}
\usetikzlibrary{matrix,arrows}
\usetikzlibrary{calc}
\usetikzlibrary{shapes}
\usetikzlibrary{through,intersections,decorations,shadows,fadings}
\usepackage{pgfplots}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 2.5 Models
Subsection 2.5.1 The Concept of a Model
In this course, I use the term ‘model’ in the broadest possible mathematical sense: a model is any application of a mathematical concept to an external situation. Since calculus is the study of functions, models will be functions chosen to describe an observed connection between two quantities. The branch of mathematics that deals with models is called applied mathematics.
Models are the mathematical versions of a scientific hypothesis. If I observe a relationship between two quantities, choosing a function to describe that relationship is making a testable claim. I can work with the mathematics to understand the function. The behaviour of the function can be then tested against data, either the original data or new data. Finally, I can re-evaluate to see if the model fits the data to an acceptable precision.
The analysis of functions helps me to understand models. I can ask questions about functions inspired by their applied context. I’ll start with three important questions.
Do I need to restrict the domain to make the model reasonable? A mathematical model where the independent variable is human population would reasonably have a restriction that the population must be positive and bounded above by some maximum population. These restrictions of the model exist in addition to whatever natural mathematical restrictions we already have on domain.
I can ask about starting values. If a model depends on time, I can set a starting time (often but not necessarily \(t=0\) ) and specify a value for the function at the starting time. Even though there is a mathematical description of what happens before the starting time, I can ignore it since it doesn’t apply to the situation I am modelling.
I can ask about the constants in the model. Functions often involve constants apart from the variables, such at the coefficients of a polynomial. In a model, the constants should have reasonable meaning and interpretation.
The analysis of models thus consists of everything I used to analyze functions in
Section 2.1 and
Section 2.4 as well as these new questions. Often, I will also want a qualitative analysis of the model. Understanding the general shape and behaviour of the functions involved is invaluable to this qualitative analysis. A major goal for this course is to develop tools and competencies to allow students to analyze functions and models both quantitatively and qualitatively. The qualitative analysis can often be expressed in a narrative: what story is the function telling about the relationship it represents? If the independent variable is time, as it often is, what is the story of the dependant variable over a time span? What happens to it?
Subsection 2.5.2 Regression
One of the greatest challenges of applied mathematics is choosing appropriate models. If the real world provides me with a set of data points, how do I choose a function? Often I can intuit what kind of function (linear, polynomial, exponential, etc) I think might fit the data. However, it is very difficult to be more specific just through intuition. Once I’ve chosen a type of function for the model, regression is a set of techniques that help to find the best specific function for the data. In this course, I will only be looking at regressions graphically. The hard work of calculating functions in a regression is left for future courses; regression is itself a major area of study in statistics.
Figure 2.5.1. A Linear Regression
Figure 2.5.2. A Sinusoidal Regression
Figure 2.5.3. A Square-Root Regression
