\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}