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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}
\( f(x)=2x^{3}-3x^{2}-12x \). 
\end{example}

\subsubsection*{Solution}


\subparagraph*{Intercepts}

Here we get two for one. The \( y \)-intercept is at \( f(0)=0 \), so \( (0,0) \)
is both an \( x \)- and \( y \)-intercept. To find the other \( x \)-intercepts,
we solve \( 2x^{3}-3x^{2}-12x=0 \).\begin{eqnarray*}
0 & = & 2x^{3}-3x^{2}-12x\\
 & = & x(2x^{2}-3x-12)\\
 & = & x(2x-?)(x+??).
\end{eqnarray*}
Again I can't factor this. But, plugging in to the quadratic formula, \( 2x^{2}-3x-12=0 \)
when \begin{eqnarray*}
x & = & \frac{3\pm \sqrt{9-4\cdot 2\cdot (-12)}}{4}\\
 & = & \frac{3\pm \sqrt{105}}{4}\\
 & \cong  & 3.3\, \textrm{or}\, -1.8,
\end{eqnarray*}
where the last are just rough estimates, based on \( \sqrt{105}\cong 10.1 \).
At any rate, then \( (3.3,0) \) and \( (-1.8,0) \) are the approximate coordinates
of the other \( x \)-intercepts.


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

Computing the derivative, and factoring it, \begin{eqnarray*}
f'(x) & = & 6x^{2}-6x-12\\
 & = & 6(x^{2}-x-2)\\
 & = & 6(x-2)(x+1),
\end{eqnarray*}
so the derivative is 0 at \( x=2 \) and \( x=-1 \). Now, as before I'm going
to draw the number line, and see where \( f' \) is positive, and where it is
negative. 

\vspace{0.3cm}
{\par\centering \includegraphics{ex2-num-line.ps} \par}
\vspace{0.3cm}

Don't forget to find the critical values. \begin{eqnarray*}
f(-1) & = & 2(-1)^{3}-3(-1)^{2}-12(-1)\\
 & = & -2-3+12\\
 & = & 7,
\end{eqnarray*}
and \begin{eqnarray*}
f(2) & = & 2(2^{3})-3(2^{2})-12(2)\\
 & = & 16-12-24\\
 & = & -20,
\end{eqnarray*}
so the critical points and values are \( (-1,7) \) and \( (2,-20) \).


\subparagraph*{Draw the graph}

Then use this information to draw a fair representation of the graph. 

\begin{itemize}
\item The first information you plot are the intercepts; {[}Link to ex2-1.gif{]}
\item Then plot the critical points (and values). I always mark it with a horizontal
line to remind myself that it is a critical point. {[}Link to ex2-2.gif{]}
\item Then fill in the graph, connecting the dots and making sure that the graph you
draw is increasing/decreasing where the number line indicates it should be.
{[}Link to ex2-3.gif{]}
\end{itemize}
{[}Each of these items should trigger the appearance of a new drawing with that
information added.

\vspace{0.3cm}
{\par\centering \includegraphics{ex2-3.ps} \par}\vspace{0.3cm}

\end{document}
