\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}