\begin{frame} \frametitle{Average Value of a Function} \begin{block}{} The average value $f_{\text{avg}}$of a function $f$ on an interval $[a,b]$ is: \begin{talign} f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x) \, dx \end{talign} \end{block} \pause\medskip \begin{exampleblock}{} Find the average value of the function $f(x) = 1+ x^2$ on the interval $[-1,2]$. \pause \begin{talign} f_{\text{avg}} &= \mpause[1]{ \frac{1}{2-(-1)} \int_{-1}^2 f(x) dx}\\ \mpause{ &= \frac{1}{3} \left(x + \frac{1}{3}x^3\right)\Big]_{-1}^2} \\ \mpause{ &= \frac{1}{3} \left(2 + \frac{1}{3}2^3 - \left((-1) + \frac{1}{3}(-1)^3\right)\right) } \\ \mpause{ &= \frac{1}{3} \left(2 + \frac{8}{3} + 1 + \frac{1}{3}\right) } \mpause{ = 2 } \end{talign} \end{exampleblock} \vspace{10cm} \end{frame}