\begin{frame}
  \frametitle{Velocities}
  
  Let $f(t)$ be a \emph{position function} of an object:
  \begin{itemize}
  \pause
    \item $f(t)$ is the position (distance form the origin) after time $t$
  \end{itemize}
  \pause\medskip
  The average velocity in the time interval $(a,a+h)$ is 
  \begin{talign}
    \text{average velocity} 
    \;=\; \frac{\text{difference in position}}{\text{time difference}}
    \mpause[1]{
    \;=\; \frac{f(a+h) - f(a)}{h}
    }
  \end{talign}\vspace{-1ex}
  \pause\pause
%   which is the slope the line through $(a,f(a))$ and $(a+h,f(a+h))$.
%   \pause
  
  \begin{block}{}
    The (instantaneous) \emph{velocity} $v(a)$ at time $t=a$ is:
    \begin{talign}
      v(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h}
    \end{talign}
    \pause
    which is the slope of the tangent at point $(a,f(a))$.
  \end{block}
  \pause
  
  \begin{exampleblock}{}
    Let $f(t) = 2t^2$.
    What is the speed of the object after $n$ seconds?\pause
    \begin{talign}
      v(n) 
      &\mpause[1]{= \lim_{h\to 0} \frac{2\cdot (n+h)^2 - 2\cdot n^2}{h}}
      \mpause[2]{= \lim_{h\to 0} \frac{4nh+2\cdot h^2}{h}}\\
      &\mpause[3]{= \lim_{h\to 0} (4n+2\cdot h)}
      \mpause[4]{= 4n}
    \end{talign}
  \end{exampleblock}
  \vspace{15cm}
\end{frame}