\begin{frame}
  \frametitle{Exponential Growth and Decay}
  
  Often quantities grow or decay proportional to their size:
  \begin{itemize}
  \pause
    \item growth of a population (animals, bacteria,\ldots)
  \pause
    \item decay of radioactive material
  \pause
    \item growth of savings on your bank account (interest rates)
  \end{itemize} 
  \pause\bigskip
  Assume that
  \begin{itemize}
  \pause
    \item $y(t)$ be a quantity depending on time $t$
  \pause
    \item rate of change of $y(t)$ is proportional to $y(t)$
  \end{itemize}
  \pause
  Then
  \begin{talign}
    y' = ky &&\mpause[1]{\text{or equivalently}} && \mpause[1]{\frac{d}{dt}y = ky}
  \end{talign}
  where $k$ is a constant. \pause\pause This equation is called:
  \begin{itemize}
  \pause
    \item \emph{law of natural growth} if $k > 0$
  \pause
    \item \emph{law of natural decay} if $k < 0$
  \end{itemize}
\end{frame}