\begin{frame} \frametitle{Curve Sketching} \begin{block}{} For sketching a curve of $f(x)$: \begin{itemize} \pause \item determine the \emph{domain} \pause \item find the \emph{$y$-intercept} $f(0)$ and the \emph{$x$-intercepts} $f(x) = 0$ \pause \item find \emph{vertical asymptotes $x = a$}, that is:\vspace{-1ex} \begin{talign} \lim_{x\to a^-} = \pm\infty &&\text{ or }&& \lim_{x\to a^+} = \pm\infty \end{talign} \vspace{-2ex} \pause \item find \emph{horizontal asymptotes $y = L$}, that is:\vspace{-1ex} \begin{talign} \lim_{x\to \infty} = L &&\text{ or }&& \lim_{x\to -\infty} = L \end{talign} \vspace{-2ex} \pause \item find intervals of \emph{increase} $f'(x) > 0$ and \emph{decrease} $f'(x) < 0$ \pause \item find \emph{local maxima and minima} \pause \item determine \emph{concavity} on intervals and \emph{points of inflection} \begin{itemize} \item $f''(x) > 0$ concave upward \item $f''(x) < 0$ concave downward \item inflections points where $f''(x)$ changes the sign \end{itemize} \end{itemize} \end{block} \end{frame}