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\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}