\begin{frame} \frametitle{Exponential Functions: Applications} \begin{exampleblock}{} We consider a population of bacteria: \begin{itemize} \item suppose the population doubles every hour \item we write $p(t)$ for the population after $t$ hours \item initial population is $p(0) = 1000$ \end{itemize} \end{exampleblock} \pause We have: \begin{talign} &\mpause[1]{p(1) = 2\cdot p(0) = 2\cdot 1000}\\ &\mpause[2]{p(2) = 2\cdot p(1) = 2^2\cdot 1000}\\ &\mpause[3]{p(3) = 2\cdot p(2) = 2^3\cdot 1000}\\[-1ex] &\mpause[4]{\hspace{2cm}\vdots} \end{talign} \pause[6] Thus in general \begin{talign} p(t) = 1000\cdot 2^t \end{talign} \vspace{-2ex}\pause \begin{exampleblock}{} Under ideal conditions such rapid growth occurs in nature. \end{exampleblock} \end{frame}