\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}