\begin{frame} \frametitle{Rates of Change} Suppose $y$ is a quantity that depends on $x$. That is $y = f(x)$. \pause\medskip If $x$ changes from $x_1$ to $x_2$, the change (increment) of $x$ is \begin{talign} \Delta x = x_2 - x_1 \end{talign} \pause and the corresponding change in $y$ is \begin{talign} \Delta y = f(x_2) - f(x_1) \end{talign} \pause The \emph{average rate of change over the interval $[x_1,x_2]$} is \begin{talign} \frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1) }{x_2 - x_1} \end{talign} \pause The \emph{instantaneous rate of change} by letting $\Delta x$ go to $0$: \begin{talign} \text{instantaneous rate of change} \;=\; \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \;=\; \lim_{x_2 \to x_1} \frac{f(x_2) - f(x_1) }{x_2 - x_1} \end{talign}\vspace{-2ex} \pause \begin{alertblock}{} This is the derivative $f'(x_1)$! \end{alertblock} \pause (Note that large derivative $f'(x_1)$ means rapid change.) \end{frame}