\begin{frame}
  \frametitle{Average Value of a Function}

  \begin{exampleblock}{}
    We consider an object moving in a straight line:
    \begin{itemize}
      \item $s(t)$ the position at time $t$ 
      \item $v(t)$ the speed at time $t$
    \end{itemize}
    \bigskip
    \pause
    
    What is the average velocity over time interval $[a,b]$?\vspace{-.75ex}
    \pause
    \begin{talign}
      v_{\text{avg}} = \frac{s(b) - s(a)}{b-a}
    \end{talign}
    \pause
    By the Fundamental Theorem we have \pause\vspace{-.75ex}
    \begin{talign}
      s(b) - s(a) = \mpause[1]{ \int_a^b v(t) dt }
    \end{talign}
    \pause\pause
    Hence\vspace{-2ex}
    \begin{talign}
      v_{\text{avg}} = \frac{1}{b-a}\int_a^b v(t) dt
    \end{talign}
    \pause
    
    The Mean Value Theorem tell us that:\pause\vspace{-.75ex}
    \begin{center}
      There exists $c$ in $(a,b)$ such that $v(c) = v_{\text{avg}} = \frac{1}{b-a}\int_a^b v(t) dt$.
    \end{center}
  \end{exampleblock}

\end{frame}