\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} 
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  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}