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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Curve Sketching}

\begin{example}
\( f(x)=x^{1/3} \). 
\end{example}

\subsubsection*{Solution}


\subparagraph*{Intercepts}

Here the only intercept is at \( (0,0) \). 


\subparagraph*{Increasing/decreasing, and critical points}

Since \[
f'(x)=\frac{1}{3}x^{-2/3},\]
 \( f'(x) \) is always positive.


\subparagraph*{Concavity, and inflection points}

Again, since \[
f''(x)=-\frac{2}{9}x^{-5/3},\]
\( f''(x) \) is always negative. Note that \( f'(0) \) is undefined, and more
specifically \[
\lim _{x\rightarrow 0}f'(x)=\infty .\]


This information we just found is not quite as empty as it may seem. By saying
there are no critical points definitely says that the graph does not level off,
and since \( f(x) \) is never \( 0 \) except at the origin, you know it never
crosses the horizontal axis again.


\subparagraph*{Draw the graph}

Here the drawing is a bit simpler, you don't need the several stages of plotting
various information. Instead, start at 0 and maintain the proper slope and concavity. 

\vspace{0.3cm}
{\par\centering \includegraphics{ex4.ps} \par}\vspace{0.3cm}

\end{document}
