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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}

The idea behind curve sketching is to translate the quantitative information
you can determine from the formula describing a function into qualitative information,
interpreting the quantitative information in terms of the dynamics of the function
(expressed as the graph, or picture, of the function). Conversely, given that
qualitative information, you can conclude a number of things about the function
that have quantatitive meaning. The two ways of understanding a function complement
each other.

Because we want to extract the qualitative understanding of the function from
the graph, we draw the graph by hand, not using a computer. Of course, Maple
can draw all these graphs, very well (although some skill is required to get
the domain and range appropriate to see all the features of the function). Certainly,
it's a good idea to use Maple to check your work.


\subsubsection{The Derivative and Graphing }

The primary use of the first derivative in graphing is that, if \( f'(x)>0 \),
then the function is increasing, and if \( f'(x)<0 \), the function is decreasing
(as you move from left to right). 

\vspace{0.3cm}
{\par\centering \includegraphics{pos-neg.ps} \par}
\vspace{0.3cm}

So, you can tell whether the curve is sloping up or down (as x increases) just
by figuring out whether \( f'(x) \) is positive or negative, which is usually
easier, since it's fairly easy to find where \( f'(x)=0 \), and between those
points, \( f' \) is either one (positive) or the other (negative). I like to
use a simple number-line, with the places where \( f'(x)=0 \) marked, indicating
the sign of the derivative in between:

\vspace{0.3cm}
{\par\centering \includegraphics{f-prime-number-line.ps} \par}
\vspace{0.3cm}


\subsubsection*{A few definitions }

We're going to deal with a function \( f \). A few points are of special interest,
called \textit{critical points}. As we mentioned earlier, a \textit{critical
point} of \( f \) is a value of \( x \) for which either:  

\begin{enumerate}
\item \( f'(x)=0 \), 
\item \( f'(x) \) doesn't exist, or 
\item \( x \) is an endpoint of the domain of the function. 
\end{enumerate}
From what we mentioned <a href="mvt.html">last time</a>, it follows that the
maximum or minimum value of \( f \) occurs at a critical point. 

We can, however, refine the idea of a critical point. If, for \( x<x_{0} \),
but \( x \) close to \( x_{0} \), \( f'(x)<0 \), and if for \( x>x_{0} \),
but \( x \) close to \( x_{0} \), \( f'(x)>0 \), and (of course) \( f'(x_{0})=0 \),
then the graph of \( f \) is dipping down to \( f(x_{0}) \) and rising away
from there, that is, \( f \) has a \textit{local minimum} or \textit{relative
minimum} at \( x_{0} \).  

On the other hand, if, for \( x<x_{0} \), but \( x \) close to \( x_{0} \),
\( f'(x)>0 \), and if for \( x>x_{0} \), but \( x \) close to \( x_{0} \),
\( f'(x)<0 \), and (of course) \( f'(x_{0})=0 \), then the graph of \( f \)
is rising up to \( f(x_{0}) \) and falling away from there, that is, \( f \)
has a \textit{local maximum} or \textit{relative maximum} at \( x_{0} \). 

\vspace{0.3cm}
{\par\centering \includegraphics{max-min.ps} \par}
\vspace{0.3cm}

A point \( x_{0} \) is a \textit{global} (or \emph{absolute}) maximum or minimum
if that is the point where \( f \) has its overall maximum (or minimum) on
its domain. 

You should try to keep straight that when we talk about a critical \textit{point},
or a relative minimum \textit{point}, we refer to the \( x \) (or whatever
the independent variable is called), not the value of the function. Those are
either called critical \textit{values} (or relative minimum \textit{values}),
or critical \textit{numbers} (\textit{et. cetera}). 

We can emphasize the idea of what a local maximum or minimum looks like in the
following ``test''.

\begin{thm}
\textbf{\textup{{[}First Derivative Test for Extrema{]}.}} If, at a critical
point \( x_{0} \), then:  
\begin{enumerate}
\item If, for \( x<x_{0} \), but \( x \) close to \( x_{0} \), \( f'(x)<0 \),
and if for \( x>x_{0} \), but \( x \) close to \( x_{0} \), \( f'(x)>0 \),
\( f \) has a local minimum or at \( x_{0} \). 
\item If, for \( x<x_{0} \), but \( x \) close to \( x_{0} \), \( f'(x)>0 \),
and if for \( x>x_{0} \), but \( x \) close to \( x_{0} \), \( f'(x)<0 \),
\( f \) has a local maximum at \( x_{0} \). 
\end{enumerate}
\end{thm}

\subsection*{ Graphing Examples }

 We're going to use the information we just talked about theoretically regarding
the first derivative to draw reasonable graphs of the functions. Here are the
examples: 

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

\subsubsection*{Solution}

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

\subsubsection*{Solution}
\vspace{0.3cm}

\begin{xca}
\( f(x)=3x^{4}-4x^{3} \). 
\end{xca}

\subsubsection*{Answers}


\subparagraph*{Intercepts}


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

Here you should describe the range where \( f' \) is positive, and where it
is negative.


\subparagraph*{Draw the graph}

Well, we can't do that over the net so easily. So, instead, pick which of the
following graphs looks most like the graph of that function. Choose the closest:

\vspace{0.3cm}
{\par\centering \includegraphics{exercise-1-graphs.ps} \par}
\vspace{0.3cm}

\begin{example}
\( f(x)=\sin (x)+\cos (x) \), \( x\in [0,2\pi ] \). 
\end{example}

\subsubsection*{Solution}
\vspace{0.3cm}


\subsubsection{The Second Derivative, Concavity, and Inflection Points}

The second derivative, which measures the rate of change of the slope, tells
about the ``curvature'' of the curve. It curves upward (\textit{concave up})
where \( f''(x)>0 \), and curves downward (\textit{concave down}) where \( f''(x)<0 \). 

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

A place where the concavity changes from concave up to concave down is called
an \textit{inflection point}. It usually will be found where \( f''(x)=0 \),
but can also be so called at a place where the derivative doesn't exist. We
do want to restrict to continuous functions, though. The \( x \) value, as
before, is the \textit{inflection point}. 

\begin{thm}
\textbf{\emph{{[}The Second Derivative Test{]}}}. If \( f(x) \) has a critical
point at \( c \), then:
\begin{enumerate}
\item If \( f''(c)>0 \), then \( f(x) \) has a relative minimum at \( c \),
\item If \( f''(c)<0 \), then \( f(x) \) has a relative maximum at \( c \), and 
\item If \( f''(c)=0 \), then you can't tell whether \( f(x) \) has a relative maximum,
or minimum, or neither, at \( c \).
\end{enumerate}
\end{thm}
\textbf{\emph{Proof.}}


\subsubsection*{Examples }

Find the inflection points and concavity; then sketch the curve, showing these
points as well as slope and critical points. 

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

\subsubsection*{Solution}
\vspace{0.3cm}

\begin{example}
\( f(x)=x^{4}+4x^{3}+4x^{2}-2 \). 
\end{example}

\subsubsection*{Solution}
\vspace{0.3cm}

\begin{xca}
\( f(x)=x^{4}-2x^{3} \). 
\end{xca}

\subparagraph*{Intercepts}

Find all intecepts


\subparagraph*{Increasing/decreasing, critical points}

Find where the curve is increasing/decreaing, and find all critical points and
values.


\subparagraph*{Concavity and points of inflection}

Find where the curve is concave up or concave down, and find all inflection
points (sometimes, finding the values at the critical points is messy, but do
what you can). 


\subsubsection{Complete Curve Sketching}

There are a number of techniques to apply to produce a reasonable qualitative
picture of the graph of a function. The idea is to be able to translate analytic
properties of the function to the qualitative behavior illustrated in the graph.
Here are the points (pun unintended) you should make sure to include in any
sketch. 

\textbf{Domain:} Your graph should give an idea of the domain of the function.
This does not mean that you have to include \( x \)'s out to \( +\infty  \),
but certainly if the domain is restricted, the picture needs to illustrate that. 

\textbf{Intercepts:} Finding the \( y \)-intercept of a graph, where the function
crosses the \( y \)-axis, gives you a good place to start drawing the graph.
Finding the \( x \)-intercepts, where the function has value \( 0 \), is often
harder, but it is a good thing to include if you can find them.

\textbf{Symmetry/Periodicity:} Look for whether the function is \textit{even}
\( f(-x)=f(x) \) or \textit{odd} \( f(-x)=-f(x) \). Even functions are symmetric
about the \( y \)-axis, so whatever the graph is for \( x>0 \) is mirrored
for \( x<0 \). Odd functions are upside-down symmetric about the \( y \)-axis,
or are {}``symmetric about the origin{}''. \\
Also look for \textit{periodicity}, such as with trigonometric functions, that
is, for some number \( a \), called the period of the function, for which \( f(x+a)=f(x) \).
Then, the graph is simply repeated period after period.

\textbf{Asymptotes:} An \textit{asymptote} is a line, vertical, horizontal,
or slanted, which the graph of the function approximates {}``near infinity{}''.
This was already discussed somewhat back in <a href="../Unit1/limits-infinite.html">Chapter 1</a>,
when we were dealing with infinite limits. 

\emph{<a href="curve-sketching-asymptotes.html">}\textbf{\emph{More}}\emph{ }\textbf{\emph{about Asymptotes</a>}}

\begin{description}
\item [Increasing/Decreasing]Find the intervals where \( f'(x) \) is positive, and
where it is negative, and the critical numbers. This would also imply checking
which are local extrema, using the first derivative test. Draw a number line
with the sign of \( f'(x) \) indicated.
\item [Concavity]Find the intervals where \( f''(x) \) is positive, and where it
is negative, and the inflection points. Draw a number line with the sign of
\( f''(x) \) indicated.
\end{description}

\subsection*{Then sketch the curve}

\begin{enumerate}
\item The first thing I use is the asymptotes, if any. They give the \emph{tails}
of the graph. With a little care, checking on which side of the lines the graph
lies on, you can learn a great deal of information from the asymptotes alone.
Remember that, while a graph can never cross a vertical asymptote, it \textit{can}
cross a horizontal one. 
\item Find the intercepts that you can, which gives a start. Find endpoints, and the
value of the function there. Check for symmetry.
\item Compute the derivative, and find where it is positive/negative. Find all critical
points, and the values at those points. Draw each of them, with a horizontal
line to indicate that it has a horizontal tangent.
\item Find the second derivative, and again find its sign, and find inflection points.
\end{enumerate}
Then draw the curve, showing these points.


\subsection*{Examples}

\begin{example}
\[
f(x)=\frac{2x+3}{x+1}\]

\end{example}

\subsubsection*{Solution}
\vspace{0.3cm}

\begin{xca}
\[
f(x)=\frac{e^{x}}{x}.\]

\end{xca}

\subparagraph*{Asymptotes}


\subparagraph*{Intercepts}


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


\subparagraph*{Concavity, and inflection points}

\begin{xca}
\[
\displaystyle y=\frac{x}{x+1}.\]

\end{xca}

\subparagraph*{Asymptotes}

Look for a slant asymptote for this one.


\subparagraph*{Intercepts}


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


\subparagraph*{Concavity, and inflection points}

\begin{example}
\[
\displaystyle y=\sqrt{x}-\sqrt{x-1}.\]

\end{example}

\subsubsection*{Solution}
\vspace{0.3cm}

\begin{xca}
\[
\displaystyle y=2\cos x+\sin (2x).\]

\end{xca}

\subparagraph*{Asymptotes}


\subparagraph*{Intercepts}


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


\subparagraph*{Concavity, and inflection points}

\begin{example}
\[
\displaystyle y=xe^{1/x}.\]

\end{example}

\subsubsection*{Solution}\vspace{0.3cm}

\end{document}
