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\begin{document}
%MODULE 1: Curve sketching


%USER DEFINED COMMANDS
\global\long\def\ds{{\displaystyle }}
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%INPUT THE INSTRUCTIONS PAGE 


%BUILD  TITLE PAGE 



\title{Curve Sketching}


\subtitle{We use the principles of Calculus to help sketch the graphs of functions.}

\makebeamertitle

\lyxframeend{}\lyxframe{{Introduction}}

The graph of a function can help us visualize how a function behaves.
Graphing calculators and computer algebra systems allow us to see
many aspects of a graph easily, but before using a graphing calculator
or a computer algebra system you need to know \emph{where} to look
for interesting features. For this we need a tool called \emph{\hilite
curve sketching}.


\lyxframeend{}\lyxframe{\noindent {Sample problems}}

\noindent \begin{question} 
\begin{itemize}
\item Make a sketch of the curve $\ec x^{3}+2x-1$. 
\item Sketch the curve of $\ec-10x^{3}+\frac{x}{10}+\frac{1}{10}$, labeling
every point where 

\begin{itemize}
\item the curve starts and stops increasing to decreasing; and 
\item the curve's slope starts and stops increasing and decreasing. 
\end{itemize}
\end{itemize}
\end{question}


\lyxframeend{}\lyxframe{{SAGE worksheets}}

You will not need a SAGE worksheet for this module.


\lyxframeend{}\lyxframe{{Four components of curves}}

\noindent We are interested in four general shapes that a curve can
take:\label{pg:four shapes} 
\begin{figure}[h!]
 

\centering{}%
\begin{tabular}{cc}
\includegraphics[scale=0.2]{figures/inc_concavedown}  & \includegraphics[scale=0.2]{figures/dec_concavedown}\tabularnewline
\includegraphics[scale=0.2]{figures/dec_concaveup}  & \includegraphics[scale=0.2]{figures/inc_concaveup} \tabularnewline
\end{tabular}
\end{figure}


\noindent We can assemble these four shapes, along with lines and
asymptotes, to provide a general idea of the shape of any curve.


\lyxframeend{}\lyxframe{{Four components of curves}}
\begin{overlayarea}
{\textwidth}{6in}

What is interesting about those four shapes? We need to discuss two
basic ideas.


\pause{}
\begin{definition}%{}
[(Increasing and decreasing)]
\begin{itemize}
\item We say that a function is \emph{\hilite increasing} on the interval
$\ec I=(a,b)$ when for any $\ec x_{1},x_{2}\in I$, if $\ec x_{2}>x_{1}$,
then $\ec f(x_{2})>f(x_{1})$. 
\item We say that a function is \emph{\hilite decreasing} on the interval
$\ec I=(a,b)$ when for any $\ec x_{1},x_{2}\in I$, if $\ec x_{2}>x_{1}$,
then $\ec f(x_{2})<f(x_{1})$. 
\end{itemize}
\end{definition}%{}

\pause{}
\begin{onlyenv}%{}
<3-4>

Which of the four basic shapes correspond to \emph{increasing} functions?

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<4>

\begin{figure}[h!]
 

\centering{}%
\begin{tabular}{cc}
\includegraphics[scale=0.3]{figures/inc_concavedown}  & \includegraphics[scale=0.3]{figures/inc_concaveup} \tabularnewline
\end{tabular}
\end{figure}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<5-6>

Which of the four basic shapes correspond to \emph{decreasing} functions?

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<6>

\begin{figure}[h!]
 

\centering{}%
\begin{tabular}{cc}
\includegraphics[scale=0.3]{figures/dec_concaveup}  & \includegraphics[scale=0.3]{figures/dec_concavedown} \tabularnewline
\end{tabular}
\end{figure}


\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{Derivatives and the direction of a function} }

Finding extrema allows us to determine when a function is increasing
or decreasing. To review, \begin{method} 
\begin{itemize}
\item Find all places where $\ec f'(x)=0$ or $\ec f'(x)$ does not exist.
These $x$ values are called {\hilite critical points}. 
\item Find the $\ec y$ values at each critical points \emph{and} at the
endpoints. 
\item The smallest $\ec y$ value found is the minimum, and the largest
$\ec y$ value found is the maximum. 
\end{itemize}
\noindent \end{method} Once we locate the extrema, continuity tells
us that we can connect them with one of the curves give on \RefPage{pg:four
shapes}. 
\begin{itemize}
\item An \emph{increasing} curve has a minimum on the left and a maximum
on the right; and 
\item a \emph{decreasing} curve has a maximum on the left and a minimum
on the right. 
\end{itemize}

\pause{}

\emph{Which} shape do we use to connect a minimum to a maximum, or
vice versa? To answer that, we need a new concept.


\lyxframeend{}\lyxframe{\noindent {The rate of change of the rate of change}}

\noindent Reexamine the four basic shapes that interest us. In each
case, the derivative--the slope of the tangent line--is either increasing
or decreasing. 
\begin{figure}[h!]
 

\centering{}%
\begin{tabular}{cc}
\includegraphics[scale=0.1]{figures/concdn_decrderiv2}  & \includegraphics[scale=0.1]{figures/concdn_decrderiv}\tabularnewline
\includegraphics[scale=0.1]{figures/concup_incrderiv2}  & \includegraphics[scale=0.1]{figures/concup_incrderiv} \tabularnewline
\end{tabular}
\end{figure}


\noindent What is the \emph{derivative} (\textbf{not} the \emph{function})
doing in each plot?


\pause{}

\begin{center}
\begin{tabular}{c|c}
decreasing  & decreasing \tabularnewline
\hline 
increasing  & increasing \tabularnewline
\end{tabular}
\par\end{center}


\pause{}

We can already narrow the shape of a curve to increasing or decreasing
curves. If we can decide whether the \emph{derivative} is increasing
or decreasing, then we can decide which of the two remaining shapes
to choose.


\lyxframeend{}\lyxframe{\noindent {Concavity}}

\noindent How do we decide whether the derivative is increasing or
decreasing? We use the derivative of the derivative, or the \emph{second}
derivative. If $\ec f'$ is continuous, then the maximum and minimum
values of $\ec f'$ tell you whether $\ec f'$ is increasing or decreasing.
How do you find the minimum and maximum values of $\ec f'$? The same
way that you find the minimum and maximum values of $\ec f$.
\begin{definition}%{}
[(Concavity)]
\begin{itemize}
\item If $\ec f'$ is increasing on the interval $\ec I$, then $\ec f$
is \emph{\hilite concave up} on $\ec I$. 
\item If $\ec f'$ is decreasing on the interval $\ec I$, then $\ec f$
is \emph{\hilite concave up} on $\ec I$. 
\item if $\ec f$ changes concavity at $\ec x=a$, then the point $\ec(a,f(a))$
is called an \emph{\hilite inflection point}. 
\end{itemize}
\end{definition}%{}

\lyxframeend{}\lyxframe{{Concavity and curve sketching}}

We can classify each of the four basic shapes according to the curve's
(a) rate of change and (b) concavity. 
\begin{figure}[h!]
 

\centering{}%
\begin{tabular}{cc}
\includegraphics[scale=0.1]{figures/concdn_decrderiv2}  & \includegraphics[scale=0.1]{figures/concdn_decrderiv}\tabularnewline
increasing, concave down  & decreasing, concave down \tabularnewline
\includegraphics[scale=0.1]{figures/concup_incrderiv2}  & \includegraphics[scale=0.1]{figures/concup_incrderiv}\tabularnewline
decreasing, concave up  & increasing, concave up \tabularnewline
\end{tabular}
\end{figure}



\lyxframeend{}\lyxframe{\noindent {Curve Sketching}}

\noindent We can now summarize a method for curve sketching by extrema. 

\begin{method}
\begin{itemize}
\item Determine asymptotes. 
\item Determine intervals where the curve is \emph{increasing} or \emph{decreasing}: 

\begin{itemize}
\item Find all critical points: that is, places where $\ec f'(x)=0$ or
$\ec f'(x)$ does not exist. These $x$ values are called {\hilite
critical points}. 
\item Classify the critical points as local minima or maxima. This indicates
intervals where the curve is increasing or decreasing. 
\end{itemize}
\item Determine intervals where the curve is concave \emph{up} or \emph{down}: 

\begin{itemize}
\item Find all places where $\ec f''(x)=0$ or $\ec f''(x)$ does not exist.
These $x$ values are called {\hilite possible inflection points}. 
\item Find the values of $\ec f'$ at each possible inflection point \emph{and}
at the endpoints. 
\item Classify the possible inflection points as minima or maxima of $\ec f'$.
This indicates intervals the curve is concave up or down. 
\end{itemize}
\item Combine the information from the first and second derivatives to determine
the shape of the curve on each interval. 
\end{itemize}
\noindent \end{method}


\lyxframeend{}\lyxframe{{First example}}

We apply this method to the first question posed at the beginning
of this module: \begin{question}Make a sketch of the curve $\ec x^{3}+2x-1$.\end{question}


\pause{}

\noindent There are no asymptotes (\emph{why not?}), so first we find
the intervals where the curve is increasing or decreasing.


\pause{}

\noindent Setting the derivative to zero, we have
\[
{\ec3x^{2}+2=0}.
\]
 The derivative has \emph{no real roots}, so it is either \emph{always
increasing} or \emph{always decreasing}.


\pause{}

Which is it? There are several ways to decide, but one easy way is
to check that 
\begin{align*}
\mathrm{at\ }{\ec x=0},\  & {\ec x^{3}+2x-1=-1}\mathrm{\ and}\\
\mathrm{at\ }{\ec x=1},\  & {\ec x^{3}+2x-1=2}.
\end{align*}
 Since the curve must increase to rise from 0 to 1, the curve is \emph{always
increasing}.


\lyxframeend{}\lyxframe{First example}
\begin{overlayarea}
{\textwidth}{6in}

\noindent We apply this method to the first question posed at the
beginning of this module: \begin{question}Make a sketch of the curve
$x^{3}+2x-1$.\end{question} There are no asymptotes (\emph{why not?}).
The curve \emph{always decreases}.


\noindent Next we find the intervals where the curve is concave up
or down.


\pause{}
\begin{onlyenv}%{}
<2-3>

\noindent Setting the second derivative to zero, we have ${\ec6x=0}$.
There is a \emph{possible} inflection point at $\ec x=0$.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>

Since no endpoints have been specified, we check the first derivative
around $\ec x=0$ to see whether it's a maximum or a minimum of the
first derivative. 
\begin{align*}
\mathrm{at\ } & {\ec x=-1},\ {\ec3x^{2}+2=5},\\
\mathrm{at\ } & {\ec x=0},\ {\ec3x^{2}+2=2},\mathrm{\ and}\\
\mathrm{at\ } & {\ec x=1},\ {\ec3x^{2}+2=5}.
\end{align*}


\end{onlyenv}%{}


\begin{onlyenv}%{}
<3->

Thus $\ec x=0$ is a \emph{minimum} of the first derivative.

\end{onlyenv}%{}

\pause{}


\noindent This tells us that
\begin{itemize}
\item $\ec f'(x)$ is \emph{decreasing} on $\ec(-\infty,0)$ $\Longrightarrow$
$\ec f(x)$ is concave \emph{down} on $\ec(-\infty,0)$; and 
\item $\ec f'(x)$ is \emph{increasing} on $\ec(0,\infty)$ $\Longrightarrow$
$\ec f(x)$ is concave \emph{up} on $\ec(0,\infty)$. 
\end{itemize}
\end{overlayarea}

\lyxframeend{}\lyxframe{First example}

\noindent We apply this method to the first question posed at the
beginning of this module: \begin{question}Make a sketch of the curve
$x^{3}+2x-1$.\end{question} There are no asymptotes (\emph{why not?}).
The curve is \emph{always decreasing}. It is 
\begin{itemize}
\item concave \emph{down} on $(-\infty,0)$; and 
\item concave \emph{up} on $(0,\infty)$. 
\end{itemize}
 When $x=0$, $y=-1$. That suggests the following shape of the curve:
\begin{figure}[h!]
 

\centering{}%
\begin{tabular}{ccccc}
left: inc, conc dn  & +  & right: incr, conc up  & =  & together \tabularnewline
\includegraphics[scale=0.2]{figures/ex1left}  &  & \includegraphics[scale=0.2]{figures/ex1right}  &  & \includegraphics[scale=0.2]{figures/ex1together}\tabularnewline
\end{tabular}
\end{figure}



\lyxframeend{}\lyxframe{{Second example}}
\begin{overlayarea}
{\textwidth}{6in}

Now turn to the second question: \begin{question} Sketch the curve
of $\ec-10x^{3}+\frac{x}{10}+\frac{1}{10}$, labeling every point
where 
\begin{itemize}
\item the curve starts and stops increasing to decreasing; and 
\item the curve's slope starts and stops increasing and decreasing. 
\end{itemize}

\noindent \end{question}


\pause{}


Again there are no asymptotes \emph{(why not?)} so we look for critical
points. Take a moment to find the extrema of the function.


\pause{}
\begin{onlyenv}%{}
<3>

Checking $y$ values, we find these extrema: 
\begin{align*}
\mathrm{local\ max\ at\ } & {\ec x=\frac{1}{10\sqrt{3}}\approx0.058},\ {\ec y\approx0.104}\mathrm{\ and}\\
\mathrm{local\ min\ at\ } & {\ec x=-\frac{1}{10\sqrt{3}}\approx-0.058},\ {\ec y=\approx0.096}.
\end{align*}
 These are very close to each other---you probably would not notice
them if you merely glanced at a graph.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<4>

\noindent That implies that the function
\begin{itemize}
\item \emph{increases} on $\ec(-0.058,0.058)$, while it 
\item \emph{decreases} on $\ec(-\infty,-0.058)$ and $\ec(0.058,\infty)$. 
\end{itemize}
\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{Second example}

\noindent Now turn to the second question: \begin{question} Sketch
the curve of $-10x^{3}+\frac{x}{10}+\frac{1}{10}$, labeling every
point where 
\begin{itemize}
\item the curve starts and stops increasing to decreasing; and 
\item the curve's slope starts and stops increasing and decreasing. 
\end{itemize}
\end{question} Again there are no asymptotes. The curve increases
on $(-0.058,0.058)$, while it decreases on $(-\infty,-0.058)$ and
$(0.058,\infty)$. Now find the extrema of $\ec f'$.


\pause{}

\bigskip{}
There is only one extremum of $\ec f'$: at $\ec x=0$, there is a
local maximum of $\ec f'$. So $\ec f'$ is increasing on $\ec(-\infty,0)$
and decreasing on $\ec(0,\infty)$, which means that the $\ec f$
is concave up on $\ec(-\infty,0)$ and concave down on $\ec(0,\infty)$.


\lyxframeend{}\lyxframe{Second example}

\noindent Now turn to the second question: \begin{question} Sketch
the curve of $-10x^{3}+\frac{x}{10}+\frac{1}{10}$\ldots{}\end{question}
Again there are no asymptotes. The curve increases on $(-0.058,0.058)$,
and decreases on $(-\infty,-0.058)$ and $(0.058,\infty)$. It is
concave up on $\left(-\infty,0\right)$, concave down on $\left(0,\infty\right)$.

\bigskip{}
We divide the intervals as follows: 

\noindent \begin{center}
\begin{tabular}{c|c|c|c|c}
$x$  & $(-\infty,-0.058)$  & $(-0.058,0)$  & $(0,0.058)$  & $(0.058,\infty)$ \tabularnewline
\hline 
incr/decr  & decr  & incr  & incr  & decr \tabularnewline
\hline 
concavity  & up  & up  & down  & down \tabularnewline
\hline 
 &  &  &  & \tabularnewline
shape  & \includegraphics[scale=0.1]{figures/dec_concaveup}  & \includegraphics[scale=0.1]{figures/inc_concaveup}  & \includegraphics[scale=0.1]{figures/inc_concavedown}  & \includegraphics[scale=0.1]{figures/dec_concavedown} \tabularnewline
 &  &  &  & \tabularnewline
\end{tabular}
\par\end{center}


\lyxframeend{}\lyxframe{Second example}

\noindent Now turn to the second question: \begin{question} Sketch
the curve of $-10x^{3}+\frac{x}{10}+\frac{1}{10}$\ldots{}\end{question}
Again there are no asymptotes. The curve increases on $(-0.058,0.058)$,
and decreases on $(-\infty,-0.058)$ and $(0.058,\infty)$. It is
concave up on $\left(-\infty,0\right)$, concave down on $\left(0,\infty\right)$.

\bigskip{}
Evaluating $y$ values and assembling the pieces, we obtain: 
\begin{figure}[h!]
 

\centering{}%
\begin{tabular}{c}
dec, conc up + inc, conc up + incr, conc dn + dec, conc dn = sketch \tabularnewline
\includegraphics[scale=0.3]{figures/ex2together} \tabularnewline
\end{tabular}
\end{figure}



\lyxframeend{}\lyxframe{\noindent {Alternate Method}}

\noindent An alternate method to determining the intervals where $\ec f$
is increasing, decreasing, concave up, and concave down is the following.
\begin{method} 
\begin{itemize}
\item Determine the critical points. 
\item Determine the possible points of inflection. 
\item Divide the real line into intervals at the critical points and possible
points of inflection. 
\item From each interval choose a sample point. Evaluate $\ec f'$ and $\ec f''$
at that sample point. 
\item If $\ec f'>0$ at the sample point, then $\ec f$ is increasing on
the entire interval; otherwise $\ec f$ is decreasing on the entire
interval. 
\item If $\ec f''>0$ at the sample point, then $\ec f$ is concave up on
the entire interval; otherwise $\ec f$ is concave down on the entire
interval. 
\end{itemize}
\noindent \end{method}


\lyxframeend{}\lyxframe{{Second example: alternate method}}

We illustrate the alternate method using the second example. We already
found the critical points and possible points of inflection, so we
can divide the real line into intervals: 

\noindent \begin{center}
\begin{tabular}{c|c|c|c|c|c}
\quad{} & $\ec-\infty$\quad{}$\ec-0.058$ & $\ec-0.058$\quad{}$\ec0$ & $\ec0$\quad{}$\ec0.058$ & $\ec0.058$\quad{}$\ec\infty$ & \quad{}\tabularnewline
\hline 
$\ec f'$  & neg  & pos  & pos  & neg  & \tabularnewline
\hline 
$\ec f''$  & pos  & pos  & neg  & neg  & \tabularnewline
\end{tabular}
\par\end{center}

\begin{explanation} We evaluate $f'$ and $f''$ at sample points
$-1$, $-0.01$, $0.01$, and $1$. \end{explanation}


\pause{}

\bigskip{}
We now assemble the pieces of the graph the same way as before: 

\begin{center}
dec, con up + inc, con up + incr, con dn + dec, con dn = sketch 
\par\end{center}

\begin{center}
\includegraphics[scale=0.2]{figures/ex2together}
\par\end{center}


\lyxframeend{}\lyxframe{\noindent {Proof of the alternate method}}

\noindent This alternate method is based on the \emph{\hilite Mean
Value Theorem}. Recall that it states
\begin{theorem}%{}
[(Mean Value Theorem)]Suppose that a function $\ec f$ is continuous
on $\ec[a,b]$ and differentiable on $\ec(a,b)$. Then there exists
$\ec c\in(a,b)$ such that
\[
{\ec f'(c)=\frac{f(b)-f(a)}{b-a}}.
\]
 
\end{theorem}%{}
\noindent Essentially, $f$ is increasing if, and only if, $f(b)-f(a)$
is positive. If $\ec f'(c)$ is \emph{negative} for all $\ec c\in(a,b)$,
then the fact that $\ec b-a$ is positive means that, by the Mean
Value Theorem, $\ec f(b)-f(a)$ must is negative.


\lyxframeend{}\lyxframe{\noindent {Conclusion} }
\begin{itemize}
\item We showed how to sketch the graphs of curves of a function. 
\item There are two different methods to use, one based on analysis of extrema
and the other based on the Mean Value Theorem. Both methods give identical
results. 
\item Curve sketching requires an identification of \emph{concavity}:\setlength{\shadeboxwidth}{4in}

\begin{definition}%{}
[(Concavity)]
\begin{itemize}
\item If $\ec f'$ is increasing on the interval $\ec I$, then $\ec f$
is \emph{\hilite concave up} on $\ec I$. 
\item If $\ec f'$ is decreasing on the interval $\ec I$, then $\ec f$
is \emph{\hilite concave up} on $\ec I$. 
\item if $\ec f$ changes concavity at $\ec x=a$, then the point $\ec(a,f(a))$
is called an \emph{\hilite inflection point}. 
\end{itemize}
\end{definition}%{}
\end{itemize}

\lyxframeend{}
\end{document}
