\begin{frame} \frametitle{Exponential Functions vs. Power Functions} \begin{exampleblock}{} Which functions grows quicker when $x$ is large: \begin{talign} f(x) &= 10\cdot x^5 & g(x) = 1.1^x \end{talign} \end{exampleblock} \bigskip\pause \begin{center} \only<-3>{ \begin{minipage}{.59\textwidth} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=0cm,xscale=1,yscale=.1,nodes={scale=.9}] {\def\diaborderx{1cm} \def\diabordery{5cm} \diagram[5]{-0.5}{6}{-.5}{40}{1} } \diagramannotatez \diagramannotatex{5} \diagramannotatey{10,20,30} \begin{scope}[ultra thick] \draw[cblue] plot[smooth,domain=-0:5.31,samples=100] (\x,{1.1^\x}) node[above] {$1.1^x$}; \draw[cgreen] plot[smooth,domain=-0:1.32,samples=100] (\x,{10*\x^5}) node[above] {$10*x^5$}; \end{scope} \end{tikzpicture} } \end{minipage}\pause \begin{minipage}{.4\textwidth} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=0cm,xscale=.05,yscale=.01,nodes={scale=.9}] {\def\diaborderx{15cm} \def\diabordery{50cm} \diagram[50]{-0.5}{65}{-.5}{400}{1} } \diagramannotatez \diagramannotatex{50} \diagramannotatey{100,200,300} \begin{scope}[ultra thick] \draw[cblue] plot[id=onedotone,smooth,domain=-0:63,samples=100] function{1.1**x} node[above] {$1.1^x$}; \draw[cgreen] plot[id=xfive,smooth,domain=-0:2.1,samples=100] function{10*x**5} node[above] {$10*x^5$}; \end{scope} \end{tikzpicture} } \end{minipage} } \pause \begin{minipage}{.6\textwidth} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=0cm,xscale=.1,yscale=.01,nodes={scale=.9}] {\def\diaborderx{12cm} \def\diabordery{50cm} \diagram[50]{-0.5}{50}{-.5}{400}{1} } \diagramannotatez \diagramannotatexx{10/$100$,20/$200$,30/$300$,40/$400$} \diagramannotateyy{100/$1\cdot 10^{15}$,200/$2\cdot 10^{15}$,300/$3\cdot 10^{15}$} \begin{scope}[ultra thick] \draw[cblue] plot[id=onedotone,smooth,domain=-0:37.7,samples=100] function{1.1**(10*x)/10000000000000} node[above] {$1.1^x$}; \draw[cgreen] plot[id=xfive,smooth,domain=-0:50,samples=100] function{10*(10*x)**5/10000000000000} node[above,xshift=5mm] {$10*x^5$}; \end{scope} \end{tikzpicture} } \end{minipage} \end{center} \pause\pause\bigskip \begin{block}{} For any $1 < a$, the \emph{exponential function} $f(x) = a^x$ grows for large $x$ much \emph{faster than any polynomial}. \end{block} \vspace{10cm} \end{frame}