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\(\newcommand{\markpoint}{node[point]{}} \renewcommand{\arraystretch}{1.5} \newcommand{\deriv}[2]{\frac{d#1}{d#2}} \renewcommand{\arraystretch}{10} \newcommand{\ddx}{\deriv {}x} \deflocal maximum{at $x=c$ if $f(c) \ge f(x)$ for $x$ near $c$.} \newcommand{\ddq}{\deriv {}q} \deflocal minimum{at $x=c$ if $f(c) \le f(x)$ for $x$ near $c$.} \newcommand{\ddt}{\deriv {}t} \defcritical point{of a function $f$ if either} \newcommand{\dx}{\,dx} \defcritical number{of $f$.\\} \newcommand{\du}{\,du} \defcritical value{of $f$. } \newcommand{\sfrac}[2]{\frac{\,#1\,}{#2}} \definflection point{for a function $f(x)$ is a point on the} \newcommand{\inv}{^{-1}} \defelasticity{$E$ defined as} \newcommand{\eval}{\Big|} \defelastic demand{\\} \newcommand{\abs}[1]{\left\lvert {#1} \right\rvert} \definelastic demand{\end{itemize}} \newcommand{\inside}[1]{{\color{insidecolor}#1}} \newcommand{\chain}[4][(0ex,0ex)]{ \begin{tikzpicture}[every node/.style={font=\normalsize}] \node[name = lhs, anchor = base west] at (0,0) {$#2=$}; \node[name = outside, anchor = base west, node distance = 0pt, inner sep = 0pt] at (lhs.base east) {$\color{outsidecolor}#3$}; \node[name = deriv inside, anchor = base west, node distance = 0pt, inner sep = 0pt] at (outside.base east) {${}\cdot {\color{derivcolor}#4}$}; \node[name = outside target] at (outside.south west){}; \node[shift={#1},name = inside target] at (outside.south){}; \node[name = deriv inside target] at (deriv inside.south){}; \node[name = text for outside, node distance = 2cm, below left of = outside target] {\begin{tabular}{c}deriv. of\\ outside\end{tabular}}; \node[name = text for inside, node distance = 2cm, below of = inside target] {\begin{tabular}{c}don't\\ change\\inside\end{tabular}}; \node[name = text for deriv inside, node distance = 2cm, below right of = deriv inside target] {\begin{tabular}{c}deriv. of\\ inside\end{tabular}}; \draw[->] (text for outside) -- (outside target.south west); \draw[->] (text for inside) -- (inside target); \draw[->] (text for deriv inside) -- (deriv inside target); \end{tikzpicture} } \newcommand{\lnchain}[4][(0ex,0ex)]{ \begin{tikzpicture}[every node/.style={font=\normalsize}] \node[name = lhs, anchor = base west] at (0,0) {$#2=$}; \node[name = outside, anchor = base west, right of = lhs, inner sep = 0pt] at (lhs.base east) {$\color{outsidecolor}#3$}; \node[name = deriv inside, anchor = base west, node distance = 0pt, inner sep = 0pt, ] at (outside.base east) {${}\cdot {\color{derivcolor}#4}$}; \node[name = outside target] at (outside.west){}; \node[shift={#1},name = inside target] at (outside.south){}; \node[name = deriv inside target] at (deriv inside.south){}; \node[name = text for outside, node distance = 2cm, below left of = outside target] {\begin{tabular}{c}deriv. of\\ outside\end{tabular}}; \node[name = text for inside, node distance = 2cm, below of = inside target] {\begin{tabular}{c}don't\\ change\\inside\end{tabular}}; \node[name = text for deriv inside, node distance = 2cm, below right of = deriv inside target] {\begin{tabular}{c}deriv. of\\ inside\end{tabular}}; \draw[->,bend right] (text for outside) -- (outside target.west); \draw[->] (text for inside) -- (inside target); \draw[->] (text for deriv inside) -- (deriv inside target); \end{tikzpicture} } \newcommand{\IncreaseStraight}{\begin{tikzpicture}[very thick,baseline=0.5cm] \draw[scale=0.9] (0,0) -- (1,1); \end{tikzpicture}} \newcommand{\IncreaseConcaveUp}{\begin{tikzpicture}[very thick,baseline=0.35cm] \draw (0,0) arc (280:350:1); \end{tikzpicture}} \newcommand{\IncreaseConcaveDown}{\begin{tikzpicture}[very thick,baseline=-0.5cm] \draw[scale=-1] (0,0) arc (280:350:1); \end{tikzpicture}} \newcommand{\DecreaseStraight}{\begin{tikzpicture}[very thick,baseline=-0.5cm] \draw[scale=0.9] (0,0) -- (1,-1); \end{tikzpicture}} \newcommand{\DecreaseConcaveUp}{\begin{tikzpicture}[very thick,baseline=-0.5cm] \draw (0,0) arc (190:260:1); \end{tikzpicture}} \newcommand{\DecreaseConcaveDown}{\begin{tikzpicture}[very thick,baseline=0.35cm] \draw[scale=-1] (0,0) arc (190:260:1); \end{tikzpicture}} \newcommand{\GenericConcaveDown}{\begin{tikzpicture}[very thick] \draw (0,0) parabola bend (0.5,0.5) (1,0); \end{tikzpicture}} \newcommand{\GenericConcaveUp}{\begin{tikzpicture}[very thick,yscale=-1] \draw (0,0) parabola bend (0.5,0.5) (1,0); \end{tikzpicture}} \newcommand{\blank}{\underline{\hspace{0.5in}}} \newcommand{\def}[1]{\textbf{#1}} \newcommand{\handoutpagebreak}{} \newcommand{\graphlabel}[2]{\makebox[0in][r]{#1}{#2}} \newcommand{\handoutfill}{} \newcommand{\handoutitemsep}{} \newcommand{\handoutpagebreak}{} \newcommand{\graphlabel}[2]{#2} \newcommand{\handoutfill}{\vfill} \newcommand{\handoutitemsep}{\itemsep=\fill} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)