Skip to main content\(\DeclareMathOperator{\erf}{erf}
\DeclareMathOperator{\arccot}{arccot}
\DeclareMathOperator{\Arctan}{Arctan}
\DeclareMathOperator{\Arcsin}{Arcsin}
\DeclareMathOperator{\arcsec}{arcsec}
\DeclareMathOperator{\arccsc}{arccsc}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\csch}{csch}
\def\abs#1{\left|#1\right|}
\def\tallstrut{\rule{0mm}{1.1\baselineskip}}
\def\approach#1{{\left(\rightarrow{}#1\right)}}
\def\CancelToRed#1#2{\textcolor{red}{{\cancelto{#1}{#2}}}}
\newcommand{\eps}{\varepsilon}
\newcommand{\RR}{\mathbb {R}}
\newcommand{\QQ}{\mathbb {Q}}
\newcommand{\NN}{\mathbb {N}}
\newcommand{\ZZ}{\mathbb {Z}}
\def\DD{{\bf{}D}}
\def\arccot{{\rm{}arccot}{}}
\def\Arctan{{\rm{}Arctan}{}}
\def\Arcsin{{\rm{}Arcsin}{}}
\def\arcsec{{\rm{}arcsec}{}}
\def\arccsc{{\rm{}arccsc}{}}
\def\halmos{\mbox{\raggedright\rule{0.1in}{0.1in}}}
\def\eval#1#2#3{\left.\strut{}#1\right|_{#2=#3}}
\def\geneval#1#2#3{\left.\strut{}#1\right|_{#2#3}}
\def\mineval#1#2{\left.\strut{}#1\right|_{#2}}
\def\bigeval#1#2#3{\left.\strut{}#1\right|_{#2=#3}}
\def\ftceval#1#2#3#4{\eval{#1}{#2}{#3}^{#4}}
\def\inverse#1{#1^{-1}}
\def\dx#1{\thinspace{\mathrm d}#1}
\def\dfdx#1#2{\frac{\dx{#1}}{\dx{#2}}}
\def\dfdxat#1#2#3{\eval{\dfdx{#1}{#2}}{#2}{#3}}
\def\partialfx#1#2{\frac{\partial{#1}}{\partial{#2}}}
\def\partialfxn#1#2#3{\frac{\partial^{#3}{#1}}{\partial{#2}^{#3}}}
\def\dfdxn#1#2#3{\frac{\text{d}^{#3}{#1}}{\text{d}{#2}^{#3}}}
\def\limit#1#2#3{{\displaystyle\lim_{#1\rightarrow #2}#3}}
\def\tlimit#1#2#3{{\displaystyle\lim_{#1\rightarrow #2}}#3}
\def\tlimitX#1#2#3#4{{\displaystyle\lim_{\underset{#4}{#1\rightarrow #2}}}#3}
\def\rtlimit#1#2#3{{\tlimitX{#1}{#2}{#3}{#1\gt{}#2}{}}}
\def\ltlimit#1#2#3{{\tlimitX{#1}{#2}{#3}{#1\lt{}#2}{}}}
\def\limitX#1#2#3#4{{\displaystyle\lim_{\underset{#4}{#1\rightarrow #2}}#3}}
\def\rlimit#1#2#3{{\limitX{#1}{#2}{#3}{#1\gt{}#2}{}}}
\def\llimit#1#2#3{{\limitX{#1}{#2}{#3}{#1\lt{}#2}{}}}
\def\limitatinf#1#2{\limit{#1}{\infty}{#2}}
\def\limitatninf#1#2{\limit{#1}{-\infty}{#2}}
\def\lprime#1{{#1^{\prime}_{\scriptscriptstyle L}}}
\def\rprime#1{{#1^{\prime}_{\scriptscriptstyle R}}}
\def\lhopeq{\stackrel{L'H}{=}}
\newcommand{\ParamEqTwo}[2]
{
\left\{
\begin{array}{c}
{#1}\\
{#2}
\end{array}
\right\}
}
\newcommand{\ParamEqThree}[3]
{
\left\{
\begin{array}{c}
{#1}\\
{#2}\\
{#3}
\end{array}
\right\}
}
\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\,}$}}}
\)
Chapter 11 Modeling with Calculus
