%fa07 problem set 1


\documentclass[11pt]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[nomarginpar]{geometry}
\usepackage{color}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{fancyhdr}
\usepackage{multicol}
\usepackage{hyperref}

\setcounter{MaxMatrixCols}{10}
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.00.0.2570}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{Created=Wednesday, July 12, 2006 00:27:03}
%TCIDATA{LastRevised=Wednesday, October 29, 2014 16:19:40}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="Scientific Notebook\Booklet #1 - with Instructions">}
%TCIDATA{CSTFile=40 LaTeX article.cst}
%TCIDATA{PageSetup=72,72,72,72,1}
%TCIDATA{ComputeGeneralSettings=0,15,15,0,0,0,0}
%TCIDATA{Counters=arabic,1}
%TCIDATA{<META NAME="PrintViewPercent" CONTENT="100">}
%TCIDATA{ComputeDefs=
%$f\left( x\right) =x^{3}-3x^{2}$
%}

%TCIDATA{AllPages=
%H=36
%F=36,\PARA{038<p type="texpara" tag="Body Text" >\hfill \hfill }
%}


\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}
\geometry{left=0.4in,right=0.5in,top=0.5in,bottom=0.4in}
\pagestyle{fancy}
\lhead{\color{blue} \Large Lecture Notes}
\chead{\color{black} \LARGE Complete Analysis of a Function - Part 2}
\rhead{\large page   \ \thepage}
\cfoot{}
\lfoot{\small   \copyright $\;$ copyright  Hidegkuti,  2014}
\rfoot{\small   Last revised:  October 2, 2014}
\textwidth 7.5in 
\textheight 9.6in 
\setlength{\headheight}{29pt}
\setlength{\parindent}{0in}

\begin{document}


Equation: $f\left( x\right) =\rule{90pt}{1pt}$\bigskip \bigskip \bigskip

\begin{tabular}[t]{llll}
1. & domain: $\rule{90pt}{1pt}$ & 5. & relative maximum(s): $\rule{90pt}{1pt}%
$ \\ 
&  &  &  \\ 
& range: $\rule{90pt}{1pt}$ &  & absolute maximum(s): $\rule{90pt}{1pt}$ \\ 
&  &  &  \\ 
2. & $y-$intercept: $\rule{90pt}{1pt}$ &  & relative minimum(s): $\rule%
{90pt}{1pt}$ \\ 
&  &  &  \\ 
& $x-$intercept(s): $\rule{90pt}{1pt}$ &  & absolute minimum(s): $\rule%
{90pt}{1pt}$ \\ 
&  &  &  \\ 
3. & horizontal asymptote(s): $\rule{90pt}{1pt}$ & 6. & continuous: $\rule%
{90pt}{1pt}$ \\ 
&  &  &  \\ 
& vertical asymptote(s): $\rule{90pt}{1pt}$ & 7. & one-to-one: $\rule%
{90pt}{1pt}$ \\ 
&  &  &  \\ 
4. & increasing: $\rule{90pt}{1pt}$ & 8. & end-behavior: \\ 
&  &  & $\lim\limits_{x\rightarrow -\infty }f\left( x\right) $\ \ \ \ and \
\ \ $\lim\limits_{x\rightarrow \infty }f\left( x\right) $ \\ 
& decreasing: $\rule{90pt}{1pt}$ &  &  \\ 
&  &  &  \\ 
&  & 9. & Graph%
\end{tabular}

\FRAME{dtbpF}{4.6942in}{4.7167in}{0pt}{}{}{10by10w.bmp}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 4.6942in;height 4.7167in;depth
0pt;original-width 12.7603in;original-height 12.8226in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename '10by10w.bmp';file-properties
"XNPEU";}}

\pagebreak

\begin{center}
{\LARGE Definitions}\bigskip
\end{center}

\begin{enumerate}
\item The \textbf{domain} of a function is a non-empty set of elements to
which we assign things.

The \textbf{range} of a function is the set of elements that we assign to
elements of the domain.\bigskip

\item The $y-$\textbf{intercept} of a function is the point where the graph
of the function intersects the $y-$axis. \ \newline
A function can have at most one $y-$intercept.

The $x-$\textbf{intercept} of a function is the point where the graph of the
function intersects the $x-$axis. \ \newline
A function can have several $x-$intercepts.\bigskip

\item Asymptotes. \ At this point we will not rigorously define asymptotes
yet, just provide with an intuitive idea. \ There are two types of
asymptotes.

A function can have at most two \textbf{horizontal asymptote}s. \ A
horizontal asymptote is the graphical representation of the property that as
the numbers in the domain get larger and larger (positive or negative),
their assigned values get closer and closer to a fixed number.

\FRAME{itbpFUX}{3.0113in}{1.4053in}{0in}{\Qcb{As $x$ gets larger and larger
in the negative, $f\left( x\right) $ gets closer and closer to $-3$.}}{}{Plot%
}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.0113in;height
1.4053in;depth 0in;display "USEDEF";plot_snapshots TRUE;mustRecompute
FALSE;lastEngine "MuPAD";xmin "-15.01";xmax "10.01";xviewmin
"-15.01";xviewmax "10.01";yviewmin "-7";yviewmax
"10";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-y-ticks 8;numpoints
100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function
\TEXUX{$-2.8+\dfrac{8}{x^{2}+1}\left( 2^{x}\right) $};linecolor
"black";linestyle 2;pointstyle "point";linethickness 3;lineAttributes
"Dash";var1range "-15.01,10.01";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";function \TEXUX{$-3$};linecolor
"gray";linestyle 2;pointstyle "point";linethickness 1;lineAttributes
"Dash";var1range "-15.01,10.01";num-x-gridlines 100;curveColor
"[flat::RGB:0x00c0c0c0]";curveStyle "Line";valid_file "T";tempfilename
'NCTLEX00.wmf';tempfile-properties "XPR";}} \ \ \ \ \ \ \FRAME{itbpFUX}{%
3.0113in}{1.4053in}{0.0156in}{\Qcb{As $x$ gets larger and larger in the
positive, $f\left( x\right) $ gets closer and closer to $2$.}}{}{Plot}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 3.0113in;height
1.4053in;depth 0.0156in;display "USEDEF";plot_snapshots TRUE;mustRecompute
FALSE;lastEngine "MuPAD";xmin "-15.01";xmax "10.01";xviewmin
"-15.01";xviewmax "10.01";yviewmin "-6";yviewmax
"10";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-y-ticks 8;numpoints
100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function
\TEXUX{$1.8-\dfrac{1}{4x^{2}+1}\left( 2^{-x}\right) \left( \sin x\right)
$};linecolor "black";linestyle 2;pointstyle "point";linethickness
3;lineAttributes "Dash";var1range "-15.01,10.01";num-x-gridlines
200;curveColor "[flat::RGB:0000000000]";curveStyle "Line";function
\TEXUX{$2$};linecolor "gray";linestyle 2;pointstyle "point";linethickness
1;lineAttributes "Dash";var1range "-15.01,10.01";num-x-gridlines
100;curveColor "[flat::RGB:0x00c0c0c0]";curveStyle "Line";valid_file
"T";tempfilename 'NCTLEX01.wmf';tempfile-properties "XPR";}}\FRAME{dtbpFUX}{%
3.0113in}{1.4053in}{0pt}{\Qcb{As $x$ gets larger and larger, $f\left(
x\right) $ gets closer and closer to $1$.}}{}{Plot}{\special{language
"Scientific Word";type "MAPLEPLOT";width 3.0113in;height 1.4053in;depth
0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine
"MuPAD";xmin "-10.01";xmax "10.01";xviewmin "-10.01";xviewmax
"10.01";yviewmin "-7";yviewmax "9";viewset"XY";rangeset"X";plottype
4;plotticks 1;num-y-ticks 8;numpoints 100;plotstyle "patch";axesstyle
"normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name
\TEXUX{$y$};function \TEXUX{$1+\dfrac{10\sin \left( 2x\right)
}{x^{2}+1}$};linecolor "black";linestyle 2;pointstyle "point";linethickness
3;lineAttributes "Dash";var1range "-10.01,10.01";num-x-gridlines
300;curveColor "[flat::RGB:0000000000]";curveStyle "Line";function
\TEXUX{$1$};linecolor "gray";linestyle 2;pointstyle "point";linethickness
1;lineAttributes "Dash";var1range "-10.01,10.01";num-x-gridlines
100;curveColor "[flat::RGB:0x00c0c0c0]";curveStyle "Line";valid_file
"T";tempfilename 'NCTLJ102.wmf';tempfile-properties "XPR";}}

The three examples above show different ways a function can approach a
horizontal asymptote. \ The first picture shows a graph approaching a
horizontal asympotote from above. \ The second graph approaches a horizontal
asympotote from below. \ \ The third graph crosses it many times as it
oscillates around it.\bigskip

A \textbf{vertical asymptote} occurs when numbers in the domain \ close to a
fixed number take extremely large values. \ A function can have many
different vertical asymptotes. \ The function may behave in different ways
at the left-hand side and at the right-hand side of the asymptote. \
Sometimes the function is not even defined on both sides of a vertical
asymptote.

\FRAME{itbpFUX}{1.8066in}{1.8066in}{0in}{\Qcb{Vertical asymptote at $x=2$}}{%
}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width
1.8066in;height 1.8066in;depth 0in;display "USEDEF";plot_snapshots
TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-3.01";xmax
"7.01";xviewmin "-3.01";xviewmax "7.01";yviewmin "-10";yviewmax
"20";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 11;numpoints
100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function
\TEXUX{$\dfrac{1}{x-2}+3$};linecolor "black";linestyle 1;pointstyle
"point";linethickness 3;lineAttributes "Solid";var1range
"-3.01,7.01";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";function \TEXUX{$\left( \left(
2,-100\right) ,\left( 2,100\right) \right) $};linecolor "gray";linestyle
5;pointstyle "point";linethickness 1;lineAttributes "DotDotDash";curveColor
"[flat::RGB:0x00808080]";curveStyle "Line";valid_file "T";tempfilename
'NCTLJ403.wmf';tempfile-properties "XPR";}} \ \ \ \ \ \ \ \ \ \ \ \FRAME{%
itbpFUX}{1.8066in}{1.8066in}{0in}{\Qcb{Vertical asymptote at $x=2$}}{}{Plot}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 1.8066in;height
1.8066in;depth 0in;display "USEDEF";plot_snapshots TRUE;mustRecompute
FALSE;lastEngine "MuPAD";xmin "-3.01";xmax "7.01";xviewmin "-3.01";xviewmax
"7.01";yviewmin "-10";yviewmax "20";viewset"XY";rangeset"X";plottype
4;plotticks 1;num-x-ticks 11;numpoints 100;plotstyle "patch";axesstyle
"normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name
\TEXUX{$y$};function \TEXUX{$-\left\vert \dfrac{1}{x-2}\right\vert
+7$};linecolor "black";linestyle 1;pointstyle "point";linethickness
3;lineAttributes "Solid";var1range "-3.01,7.01";num-x-gridlines
100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";function
\TEXUX{$\left( \left( 2,-100\right) ,\left( 2,100\right) \right)
$};linecolor "gray";linestyle 5;pointstyle "point";linethickness
1;lineAttributes "DotDotDash";curveColor "[flat::RGB:0x00808080]";curveStyle
"Line";valid_file "T";tempfilename 'NCTLJ404.wmf';tempfile-properties "XPR";}%
} \ \ \ \ \ \ \ \ \ \ \ \FRAME{itbpFUX}{1.8066in}{1.8066in}{0in}{\Qcb{%
Vertical asymptote at $x=2$}}{}{Plot}{\special{language "Scientific
Word";type "MAPLEPLOT";width 1.8066in;height 1.8066in;depth 0in;display
"USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin
"-3.01";xmax "1.99";xviewmin "-3.01";xviewmax "7.01";yviewmin "-10";yviewmax
"20";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 11;numpoints
100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function
\TEXUX{$-\dfrac{1}{x-2}+7$};linecolor "black";linestyle 1;pointstyle
"point";linethickness 3;lineAttributes "Solid";var1range
"-3.01,1.99";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$\left( \left( 2,-100\right) ,\left( 2,100\right) \right)
$};linecolor "gray";linestyle 5;pointstyle "point";linethickness
1;lineAttributes "DotDotDash";curveColor "[flat::RGB:0x00808080]";curveStyle
"Line";valid_file "T";tempfilename 'NCTLJ405.wmf';tempfile-properties "XPR";}%
}

\item Definition: \ A function $f$ is \textbf{increasing} on an interval $I$
if for all $a$ and $b$ in $I$, \ if $a<b$, then \ $f\left( a\right) \leq
f\left( b\right) $.

Definition: \ A function $f$ is \textbf{strictly increasing} on an interval $%
I$ if for all $a$ and $b$ in $I$,\ if $a<b$, then \ $f\left( a\right)
<f\left( b\right) $.

Definition: \ A function $f$ is \textbf{decreasing} on an interval $I$ if
for all $a$ and $b$ in $I$, if $a<b$, then \ $f\left( a\right) \geq f\left(
b\right) $.

Definition: \ A function $f$ is \textbf{strictly decreasing} on an interval $%
I$ if for all $a$ and $b$ in $I$,\ if $a<b$, then \ $f\left( a\right)
>f\left( b\right) $.\bigskip

\item Extremum is a common name for a maximum or a minimum.

A function $f$ has an \textbf{absolute maximum} at $x_{M}$ if for all $x$ in
the domain, $f\left( x_{M}\right) \geq f\left( x\right) $.

A function $f$ has an \textbf{absolute minimum} at $x_{m}$ if for all $x$ in
the domain, $f\left( x_{m}\right) \leq f\left( x\right) $.

A function $f$ has a \textbf{relative (or local) maximum} at $x_{M}$ if\
there exists an open interval $I=\left( a,b\right) $ that contains $x_{M}$
such that

\qquad i) \ the function is defined on $I$ and

\qquad ii) \ if we restrict the function to $I$ as its domain, then $\left(
x_{M},f\left( x_{M}\right) \right) $ is an absolute maximum.\medskip

A function $f$ has a \textbf{relative (or local) minimum} at $x_{m}$ if\
there exists an open interval $I=\left( a,b\right) $ that contains $x_{m}$
such that

\qquad i) \ the function is defined on $I$ and

\qquad ii) \ if we restrict the function to $I$ as its domain, then $\left(
x_{m},f\left( x_{m}\right) \right) $ is an absolute minimum.\bigskip

\item Definition: A function $f$ is \textbf{continuous} at a point $c$ if $%
\lim\limits_{x\rightarrow c}f\left( x\right) =f\left( c\right) $.

Definition: \ A function $f$ is \textbf{continuous} on an open interval $%
\left( a,b\right) $ if for all $c$ with $a<c<b$, $f$ is continuous at $c$.

Definition: \ A function $f$ is \textbf{continuous} on a closed interval $%
\left[ a,b\right] $ if for all $c$ with $a<c<b$, $f$ is continuous at $c$,
and $\lim\limits_{x\rightarrow a^{+}}f\left( x\right) =f\left( a\right) $
and $\lim\limits_{x\rightarrow b^{-}}f\left( x\right) =f\left( b\right) $.
\bigskip 

\item Definition: \ A function $f$ is \textbf{one-to-one (or injective)} if
for all $a$ and $b$ in its domain,\ if $a\not=b$, then \ $f\left( a\right)
\not=f\left( b\right) $.\medskip

Alternative definition: \ A function $f$ is \textbf{one-to-one (or injective)%
} if for all $a$ and $b$ in its domain,\ if $f\left( a\right) =f\left(
b\right) $, then \ $a=b$.\bigskip

\item The end-behavior of a function $f$\ describes the behavior of $f$ for
very large negative and very large positive numbers. \ In short, 
\begin{equation*}
\lim\limits_{x\rightarrow -\infty }f\left( x\right) \text{ \ and }%
\lim\limits_{x\rightarrow \infty }f\left( x\right)
\end{equation*}%
\bigskip \bigskip
\end{enumerate}

\begin{center}
{\LARGE Sample Problems}\bigskip
\end{center}

Sketch the graph and give a complete analysis for each of the following
functions.

\begin{enumerate}
\item $f\left( x\right) =\sqrt{x+1}-2$

\item $\ f\left( x\right) =10x-x^{2}+11$ \ where the domain is the closed
interval $\left[ 3,8\right] $

\item $f\left( x\right) =x^{3}-3x^{2}$ on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$\bigskip \vspace{1in}\bigskip
\end{enumerate}

\begin{center}
{\LARGE Sample Problems - Answers}\bigskip
\end{center}

\begin{enumerate}
\item $f\left( x\right) =\sqrt{x+1}-2$

\begin{tabular}[t]{lll}
domain: \ $\left[ -1,\infty \right) $ & increasing on $\left( -1,\infty
\right) $ & continuous on $\left[ -1,\infty \right) $ \\ 
range: $\left[ -2,\infty \right) $ & decreasing nowhere & one-to-one \\ 
$y-$intercept: $\left( 0,-1\right) $ & no relative maximum & end-behavior:
\\ 
$x-$intercept: $\left( 3,0\right) $ & no absolute maximum & $%
\lim\limits_{x\rightarrow -\infty }f\left( x\right) =$ undefined\ \ \ \ \ \ 
\\ 
no asymptotes & no relative minimum & $\lim\limits_{x\rightarrow \infty
}f\left( x\right) =\infty $ \\ 
& absolute minimum: $\left( -1,-2\right) $ \ \ \ \ \ \ \  & 
\end{tabular}%
\FRAME{dtbpF}{4.8499in}{2.156in}{0pt}{}{}{Plot}{\special{language
"Scientific Word";type "MAPLEPLOT";width 4.8499in;height 2.156in;depth
0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine
"MuPAD";xmin "-2.006";xmax "20.006";xviewmin "-2.006";xviewmax
"20.006";yviewmin "-3.002449";yviewmax "5";viewset"XY";rangeset"X";plottype
4;plotticks 1;num-x-ticks 21;num-y-ticks 9;constrained TRUE;numpoints
100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function
\TEXUX{$\sqrt{x+1}-2$};linecolor "black";linestyle 1;pointstyle
"point";linethickness 3;lineAttributes "Solid";var1range
"-2.006,20.006";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";function
\TEXUX{$\sqrt{0.09-\left( x+1\right) ^{2}}-\allowbreak 2$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "-1.3,-0.7";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$-\sqrt{0.09-\left( x+1\right) ^{2}}-\allowbreak 2$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "-1.3,-0.7";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$\sqrt{0.04-\left( x+1\right) ^{2}}-\allowbreak 2$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "-1.2,-0.8";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$-\sqrt{0.04-\left( x+1\right) ^{2}}-\allowbreak 2$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "-1.2,-0.8";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$\sqrt{0.01-\left( x+1\right) ^{2}}-\allowbreak 2$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "-1.1,-0.9";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$-\sqrt{0.01-\left( x+1\right) ^{2}}-\allowbreak 2$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "-1.1,-0.9";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file
"T";tempfilename 'NCTLKL06.wmf';tempfile-properties "XPR";}}

\item $f\left( x\right) =10x-x^{2}+11$ \ where the domain is the closed
interval $\left[ 3,8\right] $.

\begin{tabular}[t]{lll}
domain: \ $\left[ 3,8\right] $ \ \ \ \ \ \ \  & increasing: on $\left(
3,5\right) $ & continuous on $\left[ 3,8\right] $ \\ 
range: $\left[ 27,36\right] $ & decreasing: on $\left( 5,8\right) $ & not
one-to-one \\ 
no intercepts & relative maximum: $\left( 5,36\right) $ \ \ \ \ \ \  & 
end-behavior: \\ 
no asymptotes & absolute maximum: $\left( 5,36\right) $ & $%
\lim\limits_{x\rightarrow -\infty }f\left( x\right) =$ undefined\ \  \\ 
& no relative minimum & $\lim\limits_{x\rightarrow \infty }f\left( x\right)
= $ undefined \\ 
& absolute minimum: $\left( 8,27\right) $ & 
\end{tabular}%
\FRAME{dtbpF}{2.7475in}{2.7475in}{0pt}{}{}{Plot}{\special{language
"Scientific Word";type "MAPLEPLOT";width 2.7475in;height 2.7475in;depth
0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine
"MuPAD";xmin "3.25";xmax "7.82";xviewmin "-2.006";xviewmax "10";yviewmin
"-5.00245";yviewmax "40";viewset"XY";rangeset"X";plottype 4;plotticks
1;num-x-ticks 12;num-y-ticks 9;numpoints 100;plotstyle "patch";axesstyle
"normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name
\TEXUX{$y$};function \TEXUX{$10x-x^{2}+11$};linecolor "black";linestyle
1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range
"3.25,7.82";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$32-3.5\sqrt{0.09-\left( x-3\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "2.7,3.3";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$32+3.5\sqrt{0.09-\left( x-3\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "2.7,3.3";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$3.5\sqrt{0.09-\left( x-8\right) ^{2}}+27$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "7.7,8.3";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$-3.5\sqrt{0.09-\left( x-8\right) ^{2}}+27$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "7.7,8.3";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$32-3.5\sqrt{0.04-\left( x-3\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "2.8,3.2";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$32+3.5\sqrt{0.04-\left( x-3\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "2.8,3.2";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$32-3.5\sqrt{0.01-\left( x-3\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "2.9,3.1";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$32+3.5\sqrt{0.01-\left( x-3\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "2.9,3.1";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$\left( \left( 3,32\right) ,\left( 3.2,32\right) \right) $};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "3.25,7.82";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";function \TEXUX{$\left( \left(
3,32.2\right) ,\left( 3.2,32.2\right) \right) $};linecolor "black";linestyle
1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range
"3.25,7.82";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";function
\TEXUX{$3.5\sqrt{0.04-\left( x-8\right) ^{2}}+27$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "7.8,8.2";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$-3.5\sqrt{0.04-\left( x-8\right) ^{2}}+27$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "7.8,8.2";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$3.5\sqrt{0.01-\left( x-8\right) ^{2}}+27$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "7.9,8.1";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$-3.5\sqrt{0.01-\left( x-8\right) ^{2}}+27$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "7.9,8.1";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$\left( \left( 8,27\right) ,\left( 8.2,27.2\right) \right)
$};linecolor "black";linestyle 1;pointstyle "point";linethickness
3;lineAttributes "Solid";var1range "3.25,7.82";num-x-gridlines
100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file
"T";tempfilename 'NCTLKL07.wmf';tempfile-properties "XPR";}}

\item $f\left( x\right) =x^{3}-3x^{2}$ on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$

Since this is a cubic polynomial with a positive leading coefficient, the
end-behavior is $\lim\limits_{x\rightarrow -\infty }f\left( x\right)
=-\infty $ and $\lim\limits_{x\rightarrow \infty }f\left( x\right) =\infty $.%
\newline
We compute the intercepts: \ for the $y-$intercept$,$ we compute $f\left(
0\right) =0$. \ For the $x-$intercept, we solve the equation $x^{3}-3x^{2}=0$%
\begin{eqnarray*}
x^{3}-3x^{2} &=&0 \\
x^{2}\left( x-3\right) &=&0~~~\implies ~~~~x_{1}=0\text{ \ \ and }x_{2}=3
\end{eqnarray*}%
So the $x-$intercepts are $\left( 0,0\right) $ and $\left( 3,0\right) $.%
\newline
Since this is a polynomial, there will be no vertical or horizontal
asymptotes. \ \newline
%TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}%
%BeginExpansion
\begin{multicols}{2}%
%EndExpansion
To find out where this function is increasing/decreasing, we need to see the
sign of its derivative. \ We differentiate $f$%
\begin{equation*}
f^{\prime }\left( x\right) =3x^{2}-6x=3x\left( x-2\right)
\end{equation*}%
The derivative is a quadratic function, so its graph is a parabola. \ Its
zeroes are at $x=0$ and $x=2$. \ Since its leading coefficient is positive,
the parabola opens upward.\FRAME{dtbpFUX}{1.6155in}{1.6155in}{0pt}{\Qcb{$%
f^{\prime }\left( x\right) $}}{}{Plot}{\special{language "Scientific
Word";type "MAPLEPLOT";width 1.6155in;height 1.6155in;depth 0pt;display
"USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin
"-1.01";xmax "3.01";xviewmin "-1.01";xviewmax "3.01";yviewmin "-7";yviewmax
"10";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle
"patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name
\TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$3x\left( x-2\right)
$};linecolor "black";linestyle 1;pointstyle "point";linethickness
3;lineAttributes "Solid";var1range "-1.01,3.01";num-x-gridlines
100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file
"T";tempfilename 'NCTNBR0B.wmf';tempfile-properties "XPR";}}%
%TCIMACRO{\TeXButton{multicol end}{\end{multicols}}}%
%BeginExpansion
\end{multicols}%
%EndExpansion
We see that $f^{\prime }$ is positive on $\left( -\infty ,0\right) $ and on $%
\left( 2,\infty \right) $. \ This is where $f$ is increasing. \ $f^{\prime }$
is negative on $\left( 0,2\right) ,$ which means that $f$ is decreasing on $%
\left( 0,2\right) $.

At $x=0$, the derivative $f^{\prime }$ changes sign from positive to
negative. \ This implies that at $x=0,$ $f$ changes from increasing to
decreasing, and so there is a relative maximum there. \ At $x=2$, the
derivative $f^{\prime }$ changes sign from negative to positive. \ This
implies that at $x=2,$ $f$ changes behavior from decreasing to increasing,
indicating is a relative minimum there.

We evaluate the function at $x=0$ and $2$ to find the $y-$coordinates: \ $%
f\left( 0\right) =\allowbreak 0$ and $f\left( 2\right) =-4$. \ And so $f$
has a relative maximum: $\left( 0,0\right) $ and a relative minimum: $\left(
2,-4\right) $.

What about absolute extrema? \ If we recall that the two end-behaviors are $%
\infty $ and $-\infty $, that means that there will not be an absolute
minimum or maximum.

Because it is a polynomial function, $f$ is continuous on its entire domain, 
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$. \ Since we already found two $x-$intercepts, $f$ is clearly not
one-to-one. \ 

Based on all this, we can now graph the function and give its complete
analysis:

\begin{tabular}[t]{lll}
domain: $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ \ \ \ \ \ \ \ \ \  & increasing: on $\left( -\infty ,0\right) $ and on $%
\left( 2,\infty \right) $ & not one-to-one \\ 
range: $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ & decreasing on: \ $\left( 0,2\right) $ & continuous on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ \\ 
$y-$intercept: \ $\left( 0,0\right) $ & relative maximum: $\left( 0,0\right) 
$\  & end-behavior: \\ 
$x-$intercepts: \ $\left( 0,0\right) $ and $\left( 3,0\right) $ & no
absolute maximum & $\lim\limits_{x\rightarrow -\infty }f\left( x\right)
=-\infty $\ \ and \\ 
no asymptotes & relative minimum: \ $\left( 2,-4\right) $ & $%
\lim\limits_{x\rightarrow \infty }f\left( x\right) =\infty $ \\ 
& no absolute minimum &  \\ 
&  & 
\end{tabular}%
\FRAME{dtbpF}{3.1064in}{3.1064in}{0pt}{}{}{Plot}{\special{language
"Scientific Word";type "MAPLEPLOT";width 3.1064in;height 3.1064in;depth
0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine
"MuPAD";xmin "-20";xmax "20";xviewmin "-5";xviewmax "5";yviewmin
"-8";yviewmax "8";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks
11;num-y-ticks 17;numpoints 100;plotstyle "patch";axesstyle "normal";xis
\TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function
\TEXUX{$x^{3}-3x^{2}$};linecolor "black";linestyle 1;pointstyle
"point";linethickness 3;lineAttributes "Solid";var1range
"-20,20";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle
"Line";rangeset"X";function \TEXUX{$32-3.5\sqrt{0.09-\left( x-3\right)
^{2}}$};linecolor "black";linestyle 1;pointstyle "point";linethickness
3;lineAttributes "Solid";var1range "2.7,3.3";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$32+3.5\sqrt{0.09-\left( x-3\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "2.7,3.3";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$3.5\sqrt{0.09-\left( x-8\right) ^{2}}+27$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "7.7,8.3";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$-3.5\sqrt{0.09-\left( x-8\right) ^{2}}+27$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "7.7,8.3";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function
\TEXUX{$3x^{2}-6x$};linecolor "gray";linestyle 1;pointstyle
"point";linethickness 1;lineAttributes "Solid";var1range
"-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00c0c0c0]";curveStyle
"Line";valid_file "T";tempfilename 'NCTNHO0C.wmf';tempfile-properties "XPR";}%
}
\end{enumerate}

\vspace{2in}

\href{http://www.teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html%
}{For more documents like this, visit our page at\
http://www.teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail
questions or comments to mhidegkuti@ccc.edu.}

\end{document}