\begin{frame} \frametitle{Newtons Law of Cooling/Warming} \begin{block}{Newtons Law of Cooling} The rate of cooling of an object is proportional to the temperature difference of the object and surrounding temperature. \end{block} \pause\medskip Let \begin{itemize} \pause \item $T(t)$ be the temperature after time $t$, and \pause \item $T_s$ the temperature of the surroundings. \end{itemize} \pause\medskip Then the law can be written as differential equation: \begin{talign} T'(t) = k(T(t) - T_s) \end{talign} where $k$ is constant. \pause\medskip This is not yet the form that we need. Let \begin{talign} y(t) = T(t) - T_s &&\mpause[1]{then} &&\mpause[1]{y'(t) = T'(t)} &&\mpause[2]{thus} &&\mpause[2]{y'(t) = ky(t)} \end{talign} \pause\pause\pause Thus the solution for $y$ is an exponential function $Ce^{kt}$. \end{frame}