Math1010print.aux

\relax 
\providecommand\zref@newlabel[2]{}
\providecommand\hyper@newdestlabel[2]{}
\providecommand\AtEndDvi@Check{}
\AtEndDvi@Check
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\HyPL@Entry{0<</S/r>>}
\@writefile{toc}{\thispagestyle {empty}}
\@writefile{toc}{\contentsline {chapter}{Table of Contents}{iii}{Doc-Start}}
\@writefile{toc}{\contentsline {chapter}{Preface}{vii}{section*.2}}
\HyPL@Entry{8<</S/D>>}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}The Real Numbers}{1}{chapter.1}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{chapter:reals}{{1}{1}{The Real Numbers}{chapter.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}Some Basic Set Theory Notions}{1}{section.1.1}}
\newlabel{SetTheory}{{1.1}{1}{Some Basic Set Theory Notions}{section.1.1}{}}
\newlabel{setdef}{{1}{1}{Some Basic Set Theory Notions}{definitioncounter.1}{}}
\newlabel{idea:setdescription}{{1}{1}{Some Basic Set Theory Notions}{keyideacounter.1}{}}
\newlabel{notationforsetinclusion}{{2}{2}{Some Basic Set Theory Notions}{definitioncounter.2}{}}
\newlabel{subsetdef}{{3}{2}{Some Basic Set Theory Notions}{definitioncounter.3}{}}
\newlabel{intersectionunion}{{4}{2}{Some Basic Set Theory Notions}{definitioncounter.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces A Venn diagram for $C$, $S$, and $V$\relax }}{3}{figure.caption.3}}
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{venndiagram}{{1.1}{3}{A Venn diagram for $C$, $S$, and $V$\relax }{figure.caption.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces Venn diagrams for intersection and union\relax }}{3}{figure.caption.4}}
\newlabel{fig:intunion}{{1.2}{3}{Venn diagrams for intersection and union\relax }{figure.caption.4}{}}
\newlabel{emptysetdefn}{{5}{3}{Some Basic Set Theory Notions}{definitioncounter.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.1}Sets of Real Numbers}{3}{subsection.1.1.1}}
\newlabel{SetsofNumbers}{{1.1.1}{3}{Sets of Real Numbers}{subsection.1.1.1}{}}
\newlabel{realnumberdefn}{{6}{3}{Sets of Real Numbers}{definitioncounter.6}{}}
\newlabel{setsofnumbersboxonthispage}{{6}{4}{Sets of Real Numbers}{section*.5}{}}
\newlabel{def:numbersets}{{7}{4}{Sets of Real Numbers}{definitioncounter.7}{}}
\pgfsyspdfmark {pgfid9}{14288487}{11646860}
\newlabel{numbersetex}{{1}{4}{Sets of Real Numbers}{examplecounter.1}{}}
\pgfsyspdfmark {pgfid10}{19793511}{11483020}
\newlabel{enumbersetex}{{6}{5}{Sets of Real Numbers}{Item.16}{}}
\pgfsyspdfmark {pgfid11}{4736286}{9481178}
\@writefile{lof}{\contentsline {figure}{\numberline {1.3}{\ignorespaces The real number line with two numbers $a$ and $b$, where $a<b$.\relax }}{6}{figure.caption.6}}
\newlabel{fig:numline}{{1.3}{6}{The real number line with two numbers $a$ and $b$, where $a<b$.\relax }{figure.caption.6}{}}
\newlabel{densityofqandp}{{1}{6}{Sets of Real Numbers}{theoremcounter.1}{}}
\newlabel{trichotomy}{{8}{6}{Sets of Real Numbers}{definitioncounter.8}{}}
\newlabel{def:intervalnot}{{9}{7}{Sets of Real Numbers}{definitioncounter.9}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.4}{\ignorespaces Union and intersection of intervals\relax }}{8}{figure.caption.7}}
\newlabel{fig:venn1}{{1.4}{8}{Union and intersection of intervals\relax }{figure.caption.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.5}{\ignorespaces The set $(-\infty , -2] \cup [2, \infty )$\relax }}{8}{figure.caption.8}}
\newlabel{fig:eg1-1}{{1.5}{8}{The set $(-\infty , -2] \cup [2, \infty )$\relax }{figure.caption.8}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.6}{\ignorespaces The set $(-\infty , 3) \cup (3, \infty )$\relax }}{8}{figure.caption.9}}
\newlabel{fig:eg1-2}{{1.6}{8}{The set $(-\infty , 3) \cup (3, \infty )$\relax }{figure.caption.9}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.7}{\ignorespaces The set $(-\infty , -3) \cup (-3,3) \cup (3, \infty )$\relax }}{8}{figure.caption.10}}
\newlabel{fig:eg1-3}{{1.7}{8}{The set $(-\infty , -3) \cup (-3,3) \cup (3, \infty )$\relax }{figure.caption.10}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.8}{\ignorespaces The set $(-1,3] \cup \{ 5\}$\relax }}{8}{figure.caption.11}}
\newlabel{fig:eg1-4}{{1.8}{8}{The set $(-1,3] \cup \{ 5\}$\relax }{figure.caption.11}{}}
\pgfsyspdfmark {pgfid15}{14288487}{37827378}
\newlabel{unionex}{{2}{8}{Sets of Real Numbers}{examplecounter.2}{}}
\newlabel{eunionex}{{8}{8}{Sets of Real Numbers}{Item.24}{}}
\pgfsyspdfmark {pgfid16}{14550631}{13022002}
\@writefile{toc}{\contentsline {section}{\numberline {1.2}Real Number Arithmetic}{10}{section.1.2}}
\newlabel{RealNumberArithmetic}{{1.2}{10}{Real Number Arithmetic}{section.1.2}{}}
\newlabel{realnumberaddition}{{10}{10}{Real Number Arithmetic}{definitioncounter.10}{}}
\newlabel{realnumbermultiplication}{{11}{10}{Real Number Arithmetic}{definitioncounter.11}{}}
\newlabel{howtoaddonetwothree}{{11}{11}{Real Number Arithmetic}{definitioncounter.11}{}}
\newlabel{distributiveproperty}{{12}{11}{Real Number Arithmetic}{definitioncounter.12}{}}
\newlabel{propertiesofzero}{{2}{12}{Real Number Arithmetic}{theoremcounter.2}{}}
\newlabel{fractionarithmetic}{{2}{12}{Real Number Arithmetic}{keyideacounter.2}{}}
\newlabel{propertiesofnegatives}{{3}{13}{Real Number Arithmetic}{keyideacounter.3}{}}
\pgfsyspdfmark {pgfid23}{4474142}{20969645}
\newlabel{fractionreview}{{3}{13}{Real Number Arithmetic}{examplecounter.3}{}}
\pgfsyspdfmark {pgfid24}{9979166}{20805805}
\pgfsyspdfmark {pgfid25}{16189033}{22048794}
\pgfsyspdfmark {pgfid26}{6374688}{28265207}
\newlabel{efractionreview}{{3}{16}{Real Number Arithmetic}{AMS.49}{}}
\pgfsyspdfmark {pgfid27}{14550631}{13949204}
\newlabel{propertiesofintegerexponents}{{13}{17}{Real Number Arithmetic}{definitioncounter.13}{}}
\newlabel{orderofoperations}{{14}{18}{Real Number Arithmetic}{definitioncounter.14}{}}
\pgfsyspdfmark {pgfid30}{14288487}{24405492}
\newlabel{exponentreview}{{4}{18}{Real Number Arithmetic}{examplecounter.4}{}}
\pgfsyspdfmark {pgfid31}{19793511}{24241652}
\pgfsyspdfmark {pgfid32}{6374688}{23670301}
\newlabel{eexponentreview}{{14}{20}{Real Number Arithmetic}{Item.65}{}}
\pgfsyspdfmark {pgfid33}{14550631}{28110928}
\newlabel{principalnthrootdefn}{{15}{20}{Real Number Arithmetic}{definitioncounter.15}{}}
\newlabel{radicalprops}{{3}{21}{Real Number Arithmetic}{theoremcounter.3}{}}
\newlabel{donotdistributeexponents}{{3}{21}{Real Number Arithmetic}{section*.50}{}}
\newlabel{rationalexponentdefn}{{16}{21}{Real Number Arithmetic}{definitioncounter.16}{}}
\pgfsyspdfmark {pgfid37}{14288487}{29760000}
\newlabel{ex_combineop}{{5}{22}{Real Number Arithmetic}{examplecounter.5}{}}
\pgfsyspdfmark {pgfid38}{19793511}{29596160}
\pgfsyspdfmark {pgfid39}{6374688}{21721364}
\pgfsyspdfmark {pgfid40}{16189033}{30044957}
\newlabel{eex_combineop}{{16}{25}{Real Number Arithmetic}{Item.73}{}}
\pgfsyspdfmark {pgfid41}{4736286}{35514269}
\@writefile{toc}{\contentsline {section}{\numberline {1.3}The Cartesian Coordinate Plane}{27}{section.1.3}}
\newlabel{CartesianPlane}{{1.3}{27}{The Cartesian Coordinate Plane}{section.1.3}{}}
\newlabel{idea:cartfacts}{{4}{28}{The Cartesian Coordinate Plane}{keyideacounter.4}{}}
\pgfsyspdfmark {pgfid43}{14288487}{17317661}
\newlabel{ex_cartplot}{{6}{28}{The Cartesian Coordinate Plane}{examplecounter.6}{}}
\pgfsyspdfmark {pgfid44}{19793511}{17153821}
\@writefile{lof}{\contentsline {figure}{\numberline {1.9}{\ignorespaces The four quadrants of the Cartesian plane\relax }}{29}{figure.caption.51}}
\newlabel{quadrant}{{1.9}{29}{The four quadrants of the Cartesian plane\relax }{figure.caption.51}{}}
\newlabel{eex_cartplot}{{4}{29}{The Cartesian Coordinate Plane}{examplecounter.6}{}}
\pgfsyspdfmark {pgfid45}{4736286}{24600336}
\newlabel{symmetrydefn}{{17}{29}{The Cartesian Coordinate Plane}{definitioncounter.17}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.10}{\ignorespaces The three types of symmetry in the plane\relax }}{30}{figure.caption.52}}
\newlabel{fig:plansymm}{{1.10}{30}{The three types of symmetry in the plane\relax }{figure.caption.52}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.11}{\ignorespaces The point $P(-2,3)$ and its three reflections\relax }}{30}{figure.caption.53}}
\newlabel{fig:pointsymm}{{1.11}{30}{The point $P(-2,3)$ and its three reflections\relax }{figure.caption.53}{}}
\pgfsyspdfmark {pgfid47}{14288487}{44110993}
\newlabel{ex_planesymm}{{7}{30}{The Cartesian Coordinate Plane}{examplecounter.7}{}}
\newlabel{eex_planesymm}{{17}{30}{The Cartesian Coordinate Plane}{Item.112}{}}
\pgfsyspdfmark {pgfid48}{15533671}{27883184}
\newlabel{idea:planereflect}{{5}{30}{The Cartesian Coordinate Plane}{keyideacounter.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.1}Distance in the Plane}{30}{subsection.1.3.1}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.12}{\ignorespaces Distance between $P$ and $Q$\relax }}{31}{figure.caption.58}}
\newlabel{fig:cartdist}{{1.12}{31}{Distance between $P$ and $Q$\relax }{figure.caption.58}{}}
\newlabel{distanceformula}{{6}{31}{Distance in the Plane}{keyideacounter.6}{}}
\pgfsyspdfmark {pgfid51}{4474142}{20706477}
\newlabel{ex_twoptdist}{{8}{31}{Distance in the Plane}{examplecounter.8}{}}
\newlabel{eex_twoptdist}{{6}{31}{Distance in the Plane}{examplecounter.8}{}}
\pgfsyspdfmark {pgfid52}{12045407}{11056407}
\pgfsyspdfmark {pgfid53}{4474142}{9787470}
\newlabel{ex_pointwithdist}{{9}{31}{Distance in the Plane}{examplecounter.9}{}}
\pgfsyspdfmark {pgfid54}{9979166}{9623630}
\@writefile{lof}{\contentsline {figure}{\numberline {1.13}{\ignorespaces Diagram for Example \ref  {ex_pointwithdist}\relax }}{32}{figure.caption.59}}
\newlabel{fig:pointdist}{{1.13}{32}{Diagram for Example \ref {ex_pointwithdist}\relax }{figure.caption.59}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.14}{\ignorespaces The midpoint of a line segment\relax }}{32}{figure.caption.60}}
\newlabel{fig:midptseg}{{1.14}{32}{The midpoint of a line segment\relax }{figure.caption.60}{}}
\newlabel{eex_pointwithdist}{{6}{32}{Distance in the Plane}{figure.caption.58}{}}
\pgfsyspdfmark {pgfid55}{28442471}{32516757}
\newlabel{midpointformula}{{7}{32}{Distance in the Plane}{keyideacounter.7}{}}
\pgfsyspdfmark {pgfid57}{14288487}{13886578}
\newlabel{ex_findmidpoint}{{10}{32}{Distance in the Plane}{examplecounter.10}{}}
\pgfsyspdfmark {pgfid58}{19793511}{13722738}
\newlabel{eex_findmidpoint}{{7}{33}{Distance in the Plane}{examplecounter.10}{}}
\pgfsyspdfmark {pgfid59}{11282709}{46707506}
\newlabel{inversemidpoint}{{7}{33}{Distance in the Plane}{section*.61}{}}
\pgfsyspdfmark {pgfid60}{4474142}{43725618}
\newlabel{inversemidpointex1}{{11}{33}{Distance in the Plane}{examplecounter.11}{}}
\newlabel{einversemidpointex1}{{7}{33}{Distance in the Plane}{examplecounter.11}{}}
\pgfsyspdfmark {pgfid61}{16446678}{33545172}
\newlabel{cartexerciseone}{{1}{34}{Distance in the Plane}{Item.113}{}}
\newlabel{distanceothercases}{{16}{34}{Distance in the Plane}{Item.128}{}}
\newlabel{verifymidpointformula}{{17}{34}{Distance in the Plane}{Item.132}{}}
\newlabel{orderedtripleexercise}{{21}{34}{Distance in the Plane}{Item.138}{}}
\@writefile{toc}{\contentsline {section}{\numberline {1.4}Complex Numbers}{35}{section.1.4}}
\newlabel{CmpNums}{{1.4}{35}{Complex Numbers}{section.1.4}{}}
\newlabel{idefn}{{18}{35}{Complex Numbers}{definitioncounter.18}{}}
\newlabel{complexdefn}{{19}{35}{Complex Numbers}{definitioncounter.19}{}}
\pgfsyspdfmark {pgfid64}{14288487}{46871346}
\newlabel{complexzeroex1}{{12}{36}{Complex Numbers}{examplecounter.12}{}}
\pgfsyspdfmark {pgfid65}{19793511}{46707506}
\newlabel{ecomplexzeroex1}{{19}{37}{Complex Numbers}{Item.152}{}}
\pgfsyspdfmark {pgfid66}{4736286}{37200816}
\newlabel{conjugateprops}{{4}{37}{Complex Numbers}{theoremcounter.4}{}}
\pgfsyspdfmark {pgfid68}{14288487}{33584968}
\newlabel{complexsolnsreviewex}{{13}{38}{Complex Numbers}{examplecounter.13}{}}
\pgfsyspdfmark {pgfid69}{19793511}{33421128}
\newlabel{ecomplexsolnsreviewex}{{4}{39}{Complex Numbers}{AMS.101}{}}
\pgfsyspdfmark {pgfid70}{4736286}{9949135}
\newlabel{discriminanttheoremcomplexversion}{{5}{40}{Complex Numbers}{theoremcounter.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.1}{\ignorespaces The graph of the relation $R=\{(-2,1), (4,3), (0,-3)\}$\relax }}{43}{figure.caption.102}}
\newlabel{fig:relgraph}{{2.1}{43}{The graph of the relation $R=\{(-2,1), (4,3), (0,-3)\}$\relax }{figure.caption.102}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.2}{\ignorespaces The graph of $A$\relax }}{43}{figure.caption.103}}
\newlabel{fig:relgraphA}{{2.2}{43}{The graph of $A$\relax }{figure.caption.103}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.3}{\ignorespaces The graph of $HLS_1$\relax }}{43}{figure.caption.104}}
\newlabel{fig:relgraphHLS}{{2.3}{43}{The graph of $HLS_1$\relax }{figure.caption.104}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {2}Relations and Functions}{43}{chapter.2}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{RelationsandFunctions}{{2}{43}{Relations and Functions}{chapter.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.1}Relations}{43}{section.2.1}}
\newlabel{Relations}{{2.1}{43}{Relations}{section.2.1}{}}
\newlabel{relatonsdefn}{{20}{43}{Relations}{definitioncounter.20}{}}
\pgfsyspdfmark {pgfid73}{4474142}{27089982}
\newlabel{relationgraphingexample}{{14}{43}{Relations}{examplecounter.14}{}}
\pgfsyspdfmark {pgfid74}{9979166}{26926142}
\newlabel{opencircleintroduction}{{3}{43}{Relations}{Item.197}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.5}{\ignorespaces Getting the right graph for $HLS_2$\relax }}{44}{figure.caption.106}}
\newlabel{HLS2opencircle}{{2.5}{44}{Getting the right graph for $HLS_2$\relax }{figure.caption.106}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.6}{\ignorespaces The graph of $V$\relax }}{44}{figure.caption.107}}
\newlabel{fig:relgraphV}{{2.6}{44}{The graph of $V$\relax }{figure.caption.107}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.7}{\ignorespaces The graph of $H$\relax }}{44}{figure.caption.108}}
\newlabel{fig:relgraphH}{{2.7}{44}{The graph of $H$\relax }{figure.caption.108}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.4}{\ignorespaces The graph of $R$\relax }}{44}{figure.caption.105}}
\newlabel{erelationgraphingexample}{{20}{44}{Relations}{figure.caption.105}{}}
\pgfsyspdfmark {pgfid75}{14550631}{12183677}
\newlabel{idea:horvertline}{{8}{45}{Relations}{keyideacounter.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.1}Graphs of Equations}{45}{subsection.2.1.1}}
\newlabel{GraphsofEquations}{{2.1.1}{45}{Graphs of Equations}{subsection.2.1.1}{}}
\newlabel{fgp}{{9}{45}{Graphs of Equations}{keyideacounter.9}{}}
\pgfsyspdfmark {pgfid78}{4474142}{16481875}
\newlabel{ex_checkgraph}{{15}{45}{Graphs of Equations}{examplecounter.15}{}}
\newlabel{eex_checkgraph}{{9}{45}{Graphs of Equations}{examplecounter.15}{}}
\pgfsyspdfmark {pgfid79}{19191989}{7797934}
\@writefile{lof}{\contentsline {figure}{\numberline {2.8}{\ignorespaces Points on the curve $x^2+y^3=1$\relax }}{46}{figure.caption.109}}
\newlabel{fig:plotpoints}{{2.8}{46}{Points on the curve $x^2+y^3=1$\relax }{figure.caption.109}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.9}{\ignorespaces The completed graph of $x^2+y^3=1$\relax }}{46}{figure.caption.110}}
\newlabel{fig:completecurve}{{2.9}{46}{The completed graph of $x^2+y^3=1$\relax }{figure.caption.110}{}}
\pgfsyspdfmark {pgfid80}{14288487}{46871346}
\newlabel{firstequgraph}{{16}{46}{Graphs of Equations}{examplecounter.16}{}}
\newlabel{efirstequgraph}{{9}{46}{Graphs of Equations}{examplecounter.16}{}}
\pgfsyspdfmark {pgfid81}{14550631}{22535353}
\newlabel{interceptsdefn}{{21}{46}{Graphs of Equations}{definitioncounter.21}{}}
\newlabel{idea:findintercepts}{{10}{47}{Graphs of Equations}{keyideacounter.10}{}}
\newlabel{symmetrytestequations}{{11}{47}{Graphs of Equations}{keyideacounter.11}{}}
\pgfsyspdfmark {pgfid85}{4474142}{8045854}
\newlabel{secondequgraph}{{17}{47}{Graphs of Equations}{examplecounter.17}{}}
\pgfsyspdfmark {pgfid86}{9979166}{7882014}
\@writefile{lof}{\contentsline {figure}{\numberline {2.10}{\ignorespaces Plotting the data so far\relax }}{48}{figure.caption.113}}
\newlabel{fig:interceptplot}{{2.10}{48}{Plotting the data so far\relax }{figure.caption.113}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.11}{\ignorespaces The final result\relax }}{48}{figure.caption.114}}
\newlabel{fig:finalplot}{{2.11}{48}{The final result\relax }{figure.caption.114}{}}
\newlabel{esecondequgraph}{{11}{48}{Graphs of Equations}{AMS.112}{}}
\pgfsyspdfmark {pgfid87}{15533671}{11806758}
\@writefile{lof}{\contentsline {figure}{\numberline {2.12}{\ignorespaces The graph of $R_{1}$\relax }}{53}{figure.caption.115}}
\newlabel{fig:relfunct1}{{2.12}{53}{The graph of $R_{1}$\relax }{figure.caption.115}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.13}{\ignorespaces The graph of $R_{2}$\relax }}{53}{figure.caption.116}}
\newlabel{fig:relfunct2}{{2.13}{53}{The graph of $R_{2}$\relax }{figure.caption.116}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.2}Introduction to Functions}{53}{section.2.2}}
\newlabel{IntrotoFunctions}{{2.2}{53}{Introduction to Functions}{section.2.2}{}}
\newlabel{functiondefn}{{22}{53}{Introduction to Functions}{definitioncounter.22}{}}
\pgfsyspdfmark {pgfid89}{4474142}{35230495}
\newlabel{ex_functrel}{{18}{53}{Introduction to Functions}{examplecounter.18}{}}
\newlabel{eex_functrel}{{22}{53}{Introduction to Functions}{Item.261}{}}
\pgfsyspdfmark {pgfid90}{15928392}{23312839}
\newlabel{VLT}{{6}{53}{Introduction to Functions}{theoremcounter.6}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.14}{\ignorespaces $S_{1}$ and the line $x=1$\relax }}{54}{figure.caption.117}}
\newlabel{fig:vlteg2}{{2.14}{54}{$S_{1}$ and the line $x=1$\relax }{figure.caption.117}{}}
\pgfsyspdfmark {pgfid92}{14288487}{46871346}
\newlabel{ex_vlt}{{19}{54}{Introduction to Functions}{examplecounter.19}{}}
\newlabel{eex_vlt}{{6}{54}{Introduction to Functions}{examplecounter.19}{}}
\pgfsyspdfmark {pgfid93}{26826257}{29654212}
\pgfsyspdfmark {pgfid94}{14288487}{24313037}
\newlabel{ex_vlt2}{{20}{54}{Introduction to Functions}{examplecounter.20}{}}
\pgfsyspdfmark {pgfid95}{19793511}{24149197}
\@writefile{lof}{\contentsline {figure}{\numberline {2.15}{\ignorespaces The graph of $G$ for Example \ref  {firstdomainexample}\relax }}{55}{figure.caption.118}}
\newlabel{fig:domaineg}{{2.15}{55}{The graph of $G$ for Example \ref {firstdomainexample}\relax }{figure.caption.118}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.16}{\ignorespaces Projecting the graph onto the $x$-axis in Example \ref  {firstdomainexample}\relax }}{55}{figure.caption.119}}
\newlabel{fig:domainegprojx}{{2.16}{55}{Projecting the graph onto the $x$-axis in Example \ref {firstdomainexample}\relax }{figure.caption.119}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.17}{\ignorespaces The domain of $G$ in Example \ref  {firstdomainexample}\relax }}{55}{figure.caption.120}}
\newlabel{fig:domainegdom}{{2.17}{55}{The domain of $G$ in Example \ref {firstdomainexample}\relax }{figure.caption.120}{}}
\newlabel{eex_vlt2}{{6}{55}{Introduction to Functions}{examplecounter.20}{}}
\pgfsyspdfmark {pgfid96}{8445832}{43561778}
\newlabel{domainrangedefn}{{23}{55}{Introduction to Functions}{definitioncounter.23}{}}
\pgfsyspdfmark {pgfid98}{4474142}{28835602}
\newlabel{firstdomainexample}{{21}{55}{Introduction to Functions}{examplecounter.21}{}}
\newlabel{efirstdomainexample}{{23}{55}{Introduction to Functions}{examplecounter.21}{}}
\pgfsyspdfmark {pgfid99}{16511576}{10050265}
\@writefile{lof}{\contentsline {figure}{\numberline {2.18}{\ignorespaces Projecting the graph onto the $y$-axis in Example \ref  {firstdomainexample}\relax }}{56}{figure.caption.125}}
\newlabel{fig:domainegprojy}{{2.18}{56}{Projecting the graph onto the $y$-axis in Example \ref {firstdomainexample}\relax }{figure.caption.125}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.19}{\ignorespaces The range of $G$ in Example \ref  {firstdomainexample}\relax }}{56}{figure.caption.126}}
\newlabel{fig:domainegran}{{2.19}{56}{The range of $G$ in Example \ref {firstdomainexample}\relax }{figure.caption.126}{}}
\pgfsyspdfmark {pgfid100}{14288487}{46871346}
\newlabel{introfunctionlastexample}{{22}{56}{Introduction to Functions}{examplecounter.22}{}}
\newlabel{eintrofunctionlastexample}{{23}{56}{Introduction to Functions}{AMS.124}{}}
\pgfsyspdfmark {pgfid101}{19881937}{13716278}
\@writefile{lof}{\contentsline {figure}{\numberline {2.20}{\ignorespaces Graphical depiction of a function\relax }}{60}{figure.caption.127}}
\newlabel{fig:functpic}{{2.20}{60}{Graphical depiction of a function\relax }{figure.caption.127}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.3}Function Notation}{60}{section.2.3}}
\newlabel{FunctionNotation}{{2.3}{60}{Function Notation}{section.2.3}{}}
\newlabel{functionargument}{{2.3}{60}{Function Notation}{section.2.3}{}}
\pgfsyspdfmark {pgfid102}{14288487}{19267079}
\newlabel{ex_functionformula}{{23}{60}{Function Notation}{examplecounter.23}{}}
\newlabel{eex_functionformula}{{2.3}{60}{Function Notation}{Item.324}{}}
\pgfsyspdfmark {pgfid103}{35376164}{8486858}
\pgfsyspdfmark {pgfid104}{4474142}{46871346}
\newlabel{funcnotatex1}{{24}{61}{Function Notation}{examplecounter.24}{}}
\newlabel{efuncnotatex1}{{2.3}{61}{Function Notation}{Item.334}{}}
\pgfsyspdfmark {pgfid105}{4736286}{4736286}
\pgfsyspdfmark {pgfid106}{14288487}{17926951}
\newlabel{ex_implieddom}{{25}{62}{Function Notation}{examplecounter.25}{}}
\pgfsyspdfmark {pgfid107}{19793511}{17763111}
\pgfsyspdfmark {pgfid108}{6374688}{22651180}
\newlabel{eex_implieddom}{{2.3}{64}{Function Notation}{Item.346}{}}
\pgfsyspdfmark {pgfid109}{14550631}{41833665}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.3.1}Modelling with Functions}{64}{subsection.2.3.1}}
\newlabel{modeling}{{2.3.1}{64}{Modelling with Functions}{subsection.2.3.1}{}}
\pgfsyspdfmark {pgfid110}{4474142}{36162196}
\newlabel{heightofrocketmodel}{{26}{65}{Modelling with Functions}{examplecounter.26}{}}
\newlabel{eheightofrocketmodel}{{2.3.1}{65}{Modelling with Functions}{Item.350}{}}
\pgfsyspdfmark {pgfid111}{4736286}{4736286}
\newlabel{piecewisefunction}{{2.3.1}{66}{Modelling with Functions}{section*.132}{}}
\newlabel{Sasquatchfunc1}{{72}{69}{Modelling with Functions}{Item.434}{}}
\newlabel{piecewiseordering}{{73}{69}{Modelling with Functions}{Item.435}{}}
\newlabel{50vs51}{{73b}{69}{Modelling with Functions}{Item.437}{}}
\newlabel{piecewiseshipping}{{74}{69}{Modelling with Functions}{Item.439}{}}
\newlabel{piecewisemobile}{{75}{69}{Modelling with Functions}{Item.442}{}}
\newlabel{greatestinteger}{{76}{69}{Modelling with Functions}{Item.446}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.4}Function Arithmetic}{70}{section.2.4}}
\newlabel{FunctionArithmetic}{{2.4}{70}{Function Arithmetic}{section.2.4}{}}
\newlabel{def:functarith}{{24}{70}{Function Arithmetic}{definitioncounter.24}{}}
\pgfsyspdfmark {pgfid113}{14288487}{15115633}
\newlabel{funcarithex}{{27}{70}{Function Arithmetic}{examplecounter.27}{}}
\pgfsyspdfmark {pgfid114}{19793511}{14951793}
\newlabel{quotdomainex}{{4}{70}{Function Arithmetic}{Item.459}{}}
\pgfsyspdfmark {pgfid115}{6374688}{18041704}
\newlabel{efuncarithex}{{24}{72}{Function Arithmetic}{AMS.140}{}}
\pgfsyspdfmark {pgfid116}{14550631}{22173093}
\newlabel{differencequotient}{{25}{72}{Function Arithmetic}{definitioncounter.25}{}}
\pgfsyspdfmark {pgfid118}{4474142}{46871346}
\newlabel{differencequotientex}{{28}{73}{Function Arithmetic}{examplecounter.28}{}}
\pgfsyspdfmark {pgfid119}{9979166}{46707506}
\newlabel{edifferencequotientex}{{25}{74}{Function Arithmetic}{AMS.148}{}}
\pgfsyspdfmark {pgfid120}{14550631}{3163422}
\newlabel{diffquotgeompromise}{{25}{75}{Function Arithmetic}{section*.149}{}}
\newlabel{pricerevenuecostprofit}{{12}{75}{Function Arithmetic}{keyideacounter.12}{}}
\pgfsyspdfmark {pgfid122}{14288487}{46871346}
\newlabel{costrevenueprofitex1}{{29}{76}{Function Arithmetic}{examplecounter.29}{}}
\pgfsyspdfmark {pgfid123}{19793511}{46707506}
\newlabel{fixedcostex}{{1}{76}{Function Arithmetic}{Item.477}{}}
\pgfsyspdfmark {pgfid124}{16189033}{23430004}
\newlabel{ecostrevenueprofitex1}{{7}{77}{Function Arithmetic}{Item.483}{}}
\pgfsyspdfmark {pgfid125}{15169248}{34911026}
\@writefile{lof}{\contentsline {figure}{\numberline {2.21}{\ignorespaces Graphing the function $f(x) = x^2-x-6$\relax }}{80}{figure.caption.150}}
\newlabel{fig:fgraph1}{{2.21}{80}{Graphing the function $f(x) = x^2-x-6$\relax }{figure.caption.150}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.5}Graphs of Functions}{80}{section.2.5}}
\newlabel{GraphsofFunctions}{{2.5}{80}{Graphs of Functions}{section.2.5}{}}
\newlabel{fgpff}{{13}{80}{Graphs of Functions}{keyideacounter.13}{}}
\pgfsyspdfmark {pgfid127}{14288487}{32671713}
\newlabel{ex_fungraph1}{{30}{80}{Graphs of Functions}{examplecounter.30}{}}
\newlabel{eex_fungraph1}{{13}{80}{Graphs of Functions}{examplecounter.30}{}}
\pgfsyspdfmark {pgfid128}{16075537}{16160119}
\pgfsyspdfmark {pgfid129}{14288487}{13869836}
\newlabel{ex_pwgraph}{{31}{80}{Graphs of Functions}{examplecounter.31}{}}
\pgfsyspdfmark {pgfid130}{19793511}{13705996}
\@writefile{lof}{\contentsline {figure}{\numberline {2.22}{\ignorespaces The graph of $f(x)$ from Example \ref  {ex_pwgraph}\relax }}{81}{figure.caption.151}}
\newlabel{fig:pwgraph}{{2.22}{81}{The graph of $f(x)$ from Example \ref {ex_pwgraph}\relax }{figure.caption.151}{}}
\newlabel{eex_pwgraph}{{13}{81}{Graphs of Functions}{examplecounter.31}{}}
\pgfsyspdfmark {pgfid131}{21474952}{34124594}
\newlabel{zerosofafunction}{{26}{81}{Graphs of Functions}{definitioncounter.26}{}}
\newlabel{idea:symmtest}{{14}{81}{Graphs of Functions}{keyideacounter.14}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.23}{\ignorespaces The graph of $f(x)$ in Example \ref  {ex_evenodd}\relax }}{82}{figure.caption.152}}
\newlabel{fig:evenodd1}{{2.23}{82}{The graph of $f(x)$ in Example \ref {ex_evenodd}\relax }{figure.caption.152}{}}
\pgfsyspdfmark {pgfid134}{14288487}{42286096}
\newlabel{ex_evenodd}{{32}{82}{Graphs of Functions}{examplecounter.32}{}}
\pgfsyspdfmark {pgfid135}{19793511}{42122256}
\@writefile{lof}{\contentsline {figure}{\numberline {2.24}{\ignorespaces The graph of $g(x)$ in Example \ref  {ex_evenodd}\relax }}{83}{figure.caption.153}}
\newlabel{fig:evenodd2}{{2.24}{83}{The graph of $g(x)$ in Example \ref {ex_evenodd}\relax }{figure.caption.153}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.25}{\ignorespaces The graph of $h(x)$ in Example \ref  {ex_evenodd}\relax }}{83}{figure.caption.154}}
\newlabel{fig:evenodd3}{{2.25}{83}{The graph of $h(x)$ in Example \ref {ex_evenodd}\relax }{figure.caption.154}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.26}{\ignorespaces The graph of $i(x)$ in Example \ref  {ex_evenodd}\relax }}{83}{figure.caption.155}}
\newlabel{fig:evenodd4}{{2.26}{83}{The graph of $i(x)$ in Example \ref {ex_evenodd}\relax }{figure.caption.155}{}}
\pgfsyspdfmark {pgfid136}{6374688}{28072628}
\@writefile{lof}{\contentsline {figure}{\numberline {2.27}{\ignorespaces The graph of $j(x)$ in Example \ref  {ex_evenodd}\relax }}{84}{figure.caption.156}}
\newlabel{fig:evenodd5}{{2.27}{84}{The graph of $j(x)$ in Example \ref {ex_evenodd}\relax }{figure.caption.156}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.28}{\ignorespaces The graph of $p(x)$ in Example \ref  {ex_evenodd}\relax }}{84}{figure.caption.157}}
\newlabel{fig:evenodd6}{{2.28}{84}{The graph of $p(x)$ in Example \ref {ex_evenodd}\relax }{figure.caption.157}{}}
\newlabel{eex_evenodd}{{14}{84}{Graphs of Functions}{Item.557}{}}
\pgfsyspdfmark {pgfid137}{14550631}{10685198}
\@writefile{lof}{\contentsline {figure}{\numberline {2.29}{\ignorespaces The graph $y=f(x)$\relax }}{85}{figure.caption.158}}
\newlabel{fig:genbehav}{{2.29}{85}{The graph $y=f(x)$\relax }{figure.caption.158}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.5.1}General Function Behaviour}{85}{subsection.2.5.1}}
\newlabel{genfuncbehavior}{{2.5.1}{85}{General Function Behaviour}{subsection.2.5.1}{}}
\newlabel{incdeccnstdefn}{{27}{85}{General Function Behaviour}{definitioncounter.27}{}}
\newlabel{maxmindefn}{{28}{86}{General Function Behaviour}{definitioncounter.28}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.30}{\ignorespaces The graph for Example \ref  {tame}\relax }}{87}{figure.caption.159}}
\newlabel{fig:tamegraph}{{2.30}{87}{The graph for Example \ref {tame}\relax }{figure.caption.159}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.31}{\ignorespaces Solving $f(x)=1$ in Example \ref  {tame}.9\relax }}{87}{figure.caption.160}}
\newlabel{fig:tamey1}{{2.31}{87}{Solving $f(x)=1$ in Example \ref {tame}.9\relax }{figure.caption.160}{}}
\pgfsyspdfmark {pgfid140}{4474142}{46871346}
\newlabel{tame}{{33}{87}{General Function Behaviour}{examplecounter.33}{}}
\pgfsyspdfmark {pgfid141}{9979166}{46707506}
\@writefile{lof}{\contentsline {figure}{\numberline {2.32}{\ignorespaces The local maximum and minimum of $f(x) = \genfrac  {}{}{}0{15x}{x^2+3}$ in Example \ref  {ex_maxmincomp}\relax }}{88}{figure.caption.161}}
\newlabel{fig:maxmincomp1}{{2.32}{88}{The local maximum and minimum of $f(x) = \dfrac {15x}{x^2+3}$ in Example \ref {ex_maxmincomp}\relax }{figure.caption.161}{}}
\newlabel{etame}{{28}{88}{General Function Behaviour}{Item.589}{}}
\pgfsyspdfmark {pgfid142}{14550631}{22915141}
\pgfsyspdfmark {pgfid143}{14288487}{17281674}
\newlabel{ex_maxmincomp}{{34}{88}{General Function Behaviour}{examplecounter.34}{}}
\newlabel{eex_maxmincomp}{{28}{88}{General Function Behaviour}{examplecounter.34}{}}
\pgfsyspdfmark {pgfid144}{32340393}{4736286}
\@writefile{lof}{\contentsline {figure}{\numberline {2.33}{\ignorespaces Minimizing $d(x)$ in Example \ref  {distancefunctionex}\relax }}{89}{figure.caption.162}}
\newlabel{fig:distfunctex}{{2.33}{89}{Minimizing $d(x)$ in Example \ref {distancefunctionex}\relax }{figure.caption.162}{}}
\pgfsyspdfmark {pgfid145}{4474142}{46871346}
\newlabel{distancefunctionex}{{35}{89}{General Function Behaviour}{examplecounter.35}{}}
\newlabel{edistancefunctionex}{{28}{89}{General Function Behaviour}{examplecounter.35}{}}
\pgfsyspdfmark {pgfid146}{21401123}{27650143}
\@writefile{lof}{\contentsline {figure}{\numberline {2.35}{\ignorespaces The graph of a function $f$\relax }}{94}{figure.caption.164}}
\newlabel{fig:tran1}{{2.35}{94}{The graph of a function $f$\relax }{figure.caption.164}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2.6}Transformations}{94}{section.2.6}}
\newlabel{Transformations}{{2.6}{94}{Transformations}{section.2.6}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.34}{\ignorespaces Shifting the graph of $f$ up by 2 units\relax }}{94}{figure.caption.163}}
\newlabel{fig:shiftupf}{{2.34}{94}{Shifting the graph of $f$ up by 2 units\relax }{figure.caption.163}{}}
\newlabel{vshifts}{{7}{95}{Transformations}{theoremcounter.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.36}{\ignorespaces Shifting the graph of $f$ left by 2 units\relax }}{96}{figure.caption.165}}
\newlabel{fig:shiftleftf}{{2.36}{96}{Shifting the graph of $f$ left by 2 units\relax }{figure.caption.165}{}}
\newlabel{hshifts}{{8}{96}{Transformations}{theoremcounter.8}{}}
\pgfsyspdfmark {pgfid149}{14288487}{10133769}
\newlabel{transformationex1}{{36}{96}{Transformations}{examplecounter.36}{}}
\pgfsyspdfmark {pgfid150}{19793511}{9969929}
\@writefile{lof}{\contentsline {figure}{\numberline {2.37}{\ignorespaces The graph $y=f(x)=\sqrt  {x}$\relax }}{97}{figure.caption.166}}
\newlabel{fig:origf}{{2.37}{97}{The graph $y=f(x)=\sqrt {x}$\relax }{figure.caption.166}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.38}{\ignorespaces Graphing $g(x)=\sqrt  {x}-1$\relax }}{97}{figure.caption.167}}
\newlabel{fig:transeg1-1}{{2.38}{97}{Graphing $g(x)=\sqrt {x}-1$\relax }{figure.caption.167}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.39}{\ignorespaces Graphing $j(x)=\sqrt  {x}-1$\relax }}{97}{figure.caption.168}}
\newlabel{fig:transeg1-2}{{2.39}{97}{Graphing $j(x)=\sqrt {x}-1$\relax }{figure.caption.168}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.40}{\ignorespaces Graphing $m_1(x)=\sqrt  {x+3}$\relax }}{97}{figure.caption.169}}
\newlabel{fig:transeg1-3}{{2.40}{97}{Graphing $m_1(x)=\sqrt {x+3}$\relax }{figure.caption.169}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.41}{\ignorespaces Graphing $m(x)=\sqrt  {x+3}-2$\relax }}{97}{figure.caption.170}}
\newlabel{fig:transeg1-4}{{2.41}{97}{Graphing $m(x)=\sqrt {x+3}-2$\relax }{figure.caption.170}{}}
\newlabel{etransformationex1}{{8}{97}{Transformations}{Item.704}{}}
\pgfsyspdfmark {pgfid151}{20153359}{4736286}
\newlabel{reflections}{{9}{98}{Transformations}{theoremcounter.9}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.42}{\ignorespaces Reflecting the graph of $f$ across the $x$-axis\relax }}{98}{figure.caption.171}}
\newlabel{fig:reflectx}{{2.42}{98}{Reflecting the graph of $f$ across the $x$-axis\relax }{figure.caption.171}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.44}{\ignorespaces The graph $y=f(x)$ from Example \ref  {transformationex1}\relax }}{99}{figure.caption.173}}
\newlabel{fig:sqrtagain}{{2.44}{99}{The graph $y=f(x)$ from Example \ref {transformationex1}\relax }{figure.caption.173}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.45}{\ignorespaces Reflecting $y=f(x)$ across the $y$-axis to obtain the graph of $g(x)=\sqrt  {-x}$\relax }}{99}{figure.caption.174}}
\newlabel{fig:reflecteg1}{{2.45}{99}{Reflecting $y=f(x)$ across the $y$-axis to obtain the graph of $g(x)=\sqrt {-x}$\relax }{figure.caption.174}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.46}{\ignorespaces The intermediate function $j_1(x)=f(x+3)$\relax }}{99}{figure.caption.175}}
\newlabel{fig:reflecteg2}{{2.46}{99}{The intermediate function $j_1(x)=f(x+3)$\relax }{figure.caption.175}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.47}{\ignorespaces Reflecting $y=j_1(x)$ across the $y$-axis to obtain the graph of $j(x)=\sqrt  {3-x}$\relax }}{99}{figure.caption.176}}
\newlabel{fig:reflecteg3}{{2.47}{99}{Reflecting $y=j_1(x)$ across the $y$-axis to obtain the graph of $j(x)=\sqrt {3-x}$\relax }{figure.caption.176}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.43}{\ignorespaces Reflecting the graph of $f$ across the $y$-axis\relax }}{99}{figure.caption.172}}
\newlabel{fig:reflecty}{{2.43}{99}{Reflecting the graph of $f$ across the $y$-axis\relax }{figure.caption.172}{}}
\pgfsyspdfmark {pgfid153}{4474142}{35800067}
\newlabel{ex_reflecteg}{{37}{99}{Transformations}{examplecounter.37}{}}
\pgfsyspdfmark {pgfid154}{9979166}{35636227}
\@writefile{lof}{\contentsline {figure}{\numberline {2.48}{\ignorespaces Reflecting $y=f(x)$ across the $x$-axis to obtain the graph of $m_1(x)=-\sqrt  {x}$\relax }}{100}{figure.caption.177}}
\newlabel{fig:reflecteg4}{{2.48}{100}{Reflecting $y=f(x)$ across the $x$-axis to obtain the graph of $m_1(x)=-\sqrt {x}$\relax }{figure.caption.177}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.49}{\ignorespaces Shifting $y=m_1(x)$ up by three units to obtain the graph of $m(x)=3-\sqrt  {x}$\relax }}{100}{figure.caption.178}}
\newlabel{fig:reflecteg5}{{2.49}{100}{Shifting $y=m_1(x)$ up by three units to obtain the graph of $m(x)=3-\sqrt {x}$\relax }{figure.caption.178}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.50}{\ignorespaces Graphing $g(x) = 2f(x)$\relax }}{100}{figure.caption.179}}
\newlabel{fig:scalef}{{2.50}{100}{Graphing $g(x) = 2f(x)$\relax }{figure.caption.179}{}}
\newlabel{eex_reflecteg}{{9}{100}{Transformations}{Item.710}{}}
\pgfsyspdfmark {pgfid155}{14550631}{23514859}
\@writefile{lof}{\contentsline {figure}{\numberline {2.51}{\ignorespaces Vertical scaling by $\frac  {1}{2}$\relax }}{101}{figure.caption.180}}
\newlabel{fig:scalefhalf}{{2.51}{101}{Vertical scaling by $\frac {1}{2}$\relax }{figure.caption.180}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.52}{\ignorespaces The effect of horizontal scaling on a graph\relax }}{101}{figure.caption.181}}
\newlabel{fig:graphscale}{{2.52}{101}{The effect of horizontal scaling on a graph\relax }{figure.caption.181}{}}
\newlabel{vscalings}{{10}{101}{Transformations}{theoremcounter.10}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.53}{\ignorespaces The graph $y=\sqrt  {x}$\relax }}{102}{figure.caption.182}}
\newlabel{fig:sqrtoncemore}{{2.53}{102}{The graph $y=\sqrt {x}$\relax }{figure.caption.182}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.54}{\ignorespaces The graph $y=g(x) = 3\sqrt  {x}$\relax }}{102}{figure.caption.183}}
\newlabel{fig:scaleeg1}{{2.54}{102}{The graph $y=g(x) = 3\sqrt {x}$\relax }{figure.caption.183}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.55}{\ignorespaces The graph $y=j(x) = \sqrt  {9x}$\relax }}{102}{figure.caption.184}}
\newlabel{fig:scaleeg2}{{2.55}{102}{The graph $y=j(x) = \sqrt {9x}$\relax }{figure.caption.184}{}}
\newlabel{hscalings}{{11}{102}{Transformations}{theoremcounter.11}{}}
\pgfsyspdfmark {pgfid158}{14288487}{13363342}
\newlabel{transformationex3}{{38}{102}{Transformations}{examplecounter.38}{}}
\pgfsyspdfmark {pgfid159}{19793511}{13199502}
\@writefile{lof}{\contentsline {figure}{\numberline {2.56}{\ignorespaces The graph $y=m_1(x) = f(x+\frac  {3}{2})=\sqrt  {x+\frac  {3}{2}}$\relax }}{103}{figure.caption.185}}
\newlabel{fig:scaleeg3}{{2.56}{103}{The graph $y=m_1(x) = f(x+\frac {3}{2})=\sqrt {x+\frac {3}{2}}$\relax }{figure.caption.185}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.57}{\ignorespaces The graph $y=m_2(x) = m_1(\frac  {1}{2}x) = \sqrt  {\frac  {1}{2}x+\frac  {3}{2}}$\relax }}{103}{figure.caption.186}}
\newlabel{fig:scaleeg4}{{2.57}{103}{The graph $y=m_2(x) = m_1(\frac {1}{2}x) = \sqrt {\frac {1}{2}x+\frac {3}{2}}$\relax }{figure.caption.186}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.58}{\ignorespaces The graph $y=m_3(x) = -m_2(x)=-\sqrt  {\frac  {1}{2}x+\frac  {3}{2}}$\relax }}{103}{figure.caption.187}}
\newlabel{fig:scaleeg5}{{2.58}{103}{The graph $y=m_3(x) = -m_2(x)=-\sqrt {\frac {1}{2}x+\frac {3}{2}}$\relax }{figure.caption.187}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.59}{\ignorespaces The graph $y=m(x) = m_3(x)+1=-\sqrt  {\frac  {1}{2}x+\frac  {3}{2}}$\relax }}{103}{figure.caption.188}}
\newlabel{fig:scaleeg6}{{2.59}{103}{The graph $y=m(x) = m_3(x)+1=-\sqrt {\frac {1}{2}x+\frac {3}{2}}$\relax }{figure.caption.188}{}}
\newlabel{etransformationex3}{{11}{103}{Transformations}{Item.716}{}}
\pgfsyspdfmark {pgfid160}{4736286}{11072629}
\newlabel{transformationsthm}{{12}{104}{Transformations}{theoremcounter.12}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.61}{\ignorespaces The graph $y=f(x)$ for Example \ref  {transformationeg4}\relax }}{105}{figure.caption.190}}
\newlabel{fig:gentransorig}{{2.61}{105}{The graph $y=f(x)$ for Example \ref {transformationeg4}\relax }{figure.caption.190}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.62}{\ignorespaces Tracking the $x$ coordinates of transformed points\relax }}{105}{figure.caption.191}}
\newlabel{fig:transplotx}{{2.62}{105}{Tracking the $x$ coordinates of transformed points\relax }{figure.caption.191}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.63}{\ignorespaces Getting the corresponding $y$ coordinates\relax }}{105}{figure.caption.192}}
\newlabel{fig:transploty}{{2.63}{105}{Getting the corresponding $y$ coordinates\relax }{figure.caption.192}{}}
\pgfsyspdfmark {pgfid162}{4474142}{46871346}
\newlabel{transformationeg4}{{39}{105}{Transformations}{examplecounter.39}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.60}{\ignorespaces Determining the graph of $g(x) = -\frac  {3}{2}f(-2x+1) +2$\relax }}{105}{figure.caption.189}}
\newlabel{fig:gentransresult}{{2.60}{105}{Determining the graph of $g(x) = -\frac {3}{2}f(-2x+1) +2$\relax }{figure.caption.189}{}}
\newlabel{etransformationeg4}{{12}{105}{Transformations}{figure.caption.189}{}}
\pgfsyspdfmark {pgfid163}{6759112}{4736286}
\@writefile{lof}{\contentsline {figure}{\numberline {2.64}{\ignorespaces The sequence of transformations in Example \ref  {graphingcalctrans}\relax }}{106}{figure.caption.193}}
\newlabel{fig:gcalctr}{{2.64}{106}{The sequence of transformations in Example \ref {graphingcalctrans}\relax }{figure.caption.193}{}}
\pgfsyspdfmark {pgfid164}{14288487}{44118834}
\newlabel{graphingcalctrans}{{40}{106}{Transformations}{examplecounter.40}{}}
\newlabel{egraphingcalctrans}{{12}{106}{Transformations}{Item.724}{}}
\pgfsyspdfmark {pgfid165}{15533671}{26783537}
\@writefile{lof}{\contentsline {figure}{\numberline {3.1}{\ignorespaces The line between two points $P$ and $Q$\relax }}{111}{figure.caption.194}}
\newlabel{fig:linfun1}{{3.1}{111}{The line between two points $P$ and $Q$\relax }{figure.caption.194}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {3}Linear and Quadratic Functions}{111}{chapter.3}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{LinearQuadratic}{{3}{111}{Linear and Quadratic Functions}{chapter.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.1}Linear Functions}{111}{section.3.1}}
\newlabel{LinearFunctions}{{3.1}{111}{Linear Functions}{section.3.1}{}}
\newlabel{slope}{{29}{111}{Linear Functions}{definitioncounter.29}{}}
\pgfsyspdfmark {pgfid167}{4474142}{22620789}
\newlabel{slopeex}{{41}{111}{Linear Functions}{examplecounter.41}{}}
\pgfsyspdfmark {pgfid168}{9979166}{22456949}
\newlabel{eslopeex}{{29}{112}{Linear Functions}{Item.807}{}}
\pgfsyspdfmark {pgfid169}{14550631}{7528384}
\@writefile{lof}{\contentsline {figure}{\numberline {3.2}{\ignorespaces Slope as ``rise over run''\relax }}{113}{figure.caption.195}}
\newlabel{fig:riserun}{{3.2}{113}{Slope as ``rise over run''\relax }{figure.caption.195}{}}
\pgfsyspdfmark {pgfid170}{4474142}{34678814}
\newlabel{ex_tempchange}{{42}{113}{Linear Functions}{examplecounter.42}{}}
\newlabel{eex_tempchange}{{29}{113}{Linear Functions}{Item.813}{}}
\pgfsyspdfmark {pgfid171}{4736286}{13398401}
\@writefile{lof}{\contentsline {figure}{\numberline {3.3}{\ignorespaces Deriving the point-slope formula\relax }}{114}{figure.caption.196}}
\newlabel{fig:linfun2}{{3.3}{114}{Deriving the point-slope formula\relax }{figure.caption.196}{}}
\newlabel{pointslope}{{15}{114}{Linear Functions}{keyideacounter.15}{}}
\pgfsyspdfmark {pgfid173}{14288487}{34055409}
\newlabel{ex_pointslope}{{43}{114}{Linear Functions}{examplecounter.43}{}}
\newlabel{eex_pointslope}{{15}{114}{Linear Functions}{examplecounter.43}{}}
\pgfsyspdfmark {pgfid174}{31256295}{17721553}
\newlabel{slopeintercept}{{16}{114}{Linear Functions}{keyideacounter.16}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.4}{\ignorespaces The graph of $f(x)=3$\relax }}{115}{figure.caption.197}}
\newlabel{fig:linfun3}{{3.4}{115}{The graph of $f(x)=3$\relax }{figure.caption.197}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.5}{\ignorespaces The graph of $f(x)=3x-1$\relax }}{115}{figure.caption.198}}
\newlabel{fig:linfun4}{{3.5}{115}{The graph of $f(x)=3x-1$\relax }{figure.caption.198}{}}
\newlabel{linearfunction}{{30}{115}{Linear Functions}{definitioncounter.30}{}}
\newlabel{constantfunction}{{31}{115}{Linear Functions}{definitioncounter.31}{}}
\pgfsyspdfmark {pgfid178}{4474142}{18897211}
\newlabel{ex_linfungraph}{{44}{115}{Linear Functions}{examplecounter.44}{}}
\pgfsyspdfmark {pgfid179}{9979166}{18733371}
\@writefile{lof}{\contentsline {figure}{\numberline {3.6}{\ignorespaces The graph of $f(x)=\genfrac  {}{}{}0{3-2x}{4}$\relax }}{116}{figure.caption.199}}
\newlabel{fig:linfun5}{{3.6}{116}{The graph of $f(x)=\dfrac {3-2x}{4}$\relax }{figure.caption.199}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.7}{\ignorespaces The graph of $f(x)=\genfrac  {}{}{}0{x^2-4}{x-2}$\relax }}{116}{figure.caption.200}}
\newlabel{fig:linfun6}{{3.7}{116}{The graph of $f(x)=\dfrac {x^2-4}{x-2}$\relax }{figure.caption.200}{}}
\newlabel{eex_linfungraph}{{31}{116}{Linear Functions}{Item.821}{}}
\pgfsyspdfmark {pgfid180}{14550631}{29729182}
\pgfsyspdfmark {pgfid181}{14288487}{14583085}
\newlabel{PortaBoyCost}{{45}{116}{Linear Functions}{examplecounter.45}{}}
\pgfsyspdfmark {pgfid182}{19793511}{14419245}
\newlabel{ePortaBoyCost}{{31}{117}{Linear Functions}{Item.831}{}}
\pgfsyspdfmark {pgfid183}{4736286}{21870354}
\pgfsyspdfmark {pgfid184}{4474142}{18513314}
\newlabel{PortaBoyDemand}{{46}{117}{Linear Functions}{examplecounter.46}{}}
\pgfsyspdfmark {pgfid185}{9979166}{18349474}
\newlabel{ePortaBoyDemand}{{31}{118}{Linear Functions}{Item.841}{}}
\pgfsyspdfmark {pgfid186}{14550631}{7963813}
\@writefile{lof}{\contentsline {figure}{\numberline {3.8}{\ignorespaces The graph of $y=f(x)$ and its secant line through $(a,f(a))$ and $(b,f(b))$\relax }}{119}{figure.caption.202}}
\newlabel{fig:avsecant}{{3.8}{119}{The graph of $y=f(x)$ and its secant line through $(a,f(a))$ and $(b,f(b))$\relax }{figure.caption.202}{}}
\newlabel{arc}{{32}{119}{Linear Functions}{definitioncounter.32}{}}
\newlabel{instantaneousrateofchange}{{32}{119}{Linear Functions}{section*.201}{}}
\pgfsyspdfmark {pgfid188}{4474142}{20887073}
\newlabel{PortaBoyRevenue}{{47}{119}{Linear Functions}{examplecounter.47}{}}
\pgfsyspdfmark {pgfid189}{9979166}{20723233}
\newlabel{ePortaBoyRevenue}{{32}{120}{Linear Functions}{Item.849}{}}
\pgfsyspdfmark {pgfid190}{14550631}{24438789}
\newlabel{celsiustofahr}{{35}{121}{Linear Functions}{Item.884}{}}
\newlabel{exerredoportaboy}{{37}{122}{Linear Functions}{Item.889}{}}
\newlabel{parallellines}{{32}{123}{Linear Functions}{section*.203}{}}
\newlabel{perpendicularlines}{{32}{123}{Linear Functions}{section*.204}{}}
\newlabel{perpendicularlineproof}{{71}{123}{Linear Functions}{Item.930}{}}
\newlabel{inversemidpointex2}{{72}{123}{Linear Functions}{Item.931}{}}
\newlabel{identityexercise}{{73}{123}{Linear Functions}{Item.932}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.2}Absolute Value Functions}{124}{section.3.2}}
\newlabel{AbsoluteValueFunctions}{{3.2}{124}{Absolute Value Functions}{section.3.2}{}}
\newlabel{absolutevalue}{{33}{124}{Absolute Value Functions}{definitioncounter.33}{}}
\newlabel{absolutevalueprops}{{13}{124}{Absolute Value Functions}{theoremcounter.13}{}}
\pgfsyspdfmark {pgfid193}{4474142}{26436976}
\newlabel{absvalueeqnex}{{48}{125}{Absolute Value Functions}{examplecounter.48}{}}
\pgfsyspdfmark {pgfid194}{9979166}{26273136}
\newlabel{eabsvalueeqnex}{{13}{126}{Absolute Value Functions}{Item.946}{}}
\pgfsyspdfmark {pgfid195}{14550631}{8818255}
\@writefile{lof}{\contentsline {figure}{\numberline {3.9}{\ignorespaces Constructing the graph of $f(x)=\delimiter 69640972 x\delimiter 86418188 $\relax }}{127}{figure.caption.209}}
\newlabel{fig:absvalgraph}{{3.9}{127}{Constructing the graph of $f(x)=\lvert x\rvert $\relax }{figure.caption.209}{}}
\pgfsyspdfmark {pgfid196}{4474142}{46871346}
\newlabel{absvaluegraph1}{{49}{127}{Absolute Value Functions}{examplecounter.49}{}}
\pgfsyspdfmark {pgfid197}{9979166}{46707506}
\@writefile{lof}{\contentsline {figure}{\numberline {3.10}{\ignorespaces $g(x) = |x-3|$\relax }}{128}{figure.caption.210}}
\newlabel{fig:absvalgr2}{{3.10}{128}{$g(x) = |x-3|$\relax }{figure.caption.210}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.11}{\ignorespaces $h(x) = |x|-3$\relax }}{128}{figure.caption.211}}
\newlabel{fig:absvalgr3}{{3.11}{128}{$h(x) = |x|-3$\relax }{figure.caption.211}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.12}{\ignorespaces $i(x) = 4-2|3x+1|$\relax }}{128}{figure.caption.212}}
\newlabel{fig:absvalgr4}{{3.12}{128}{$i(x) = 4-2|3x+1|$\relax }{figure.caption.212}{}}
\newlabel{eabsvaluegraph1}{{13}{128}{Absolute Value Functions}{Item.954}{}}
\pgfsyspdfmark {pgfid198}{14550631}{7907126}
\@writefile{lof}{\contentsline {figure}{\numberline {3.13}{\ignorespaces $f(x) = \delimiter 69640972 x\delimiter 86418188 $ with three labelled points\relax }}{129}{figure.caption.213}}
\newlabel{fig:absvalgr5}{{3.13}{129}{$f(x) = \lvert x\rvert $ with three labelled points\relax }{figure.caption.213}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.14}{\ignorespaces $g(x) = |x-3| = f(x-3)$\relax }}{129}{figure.caption.214}}
\newlabel{fig:absvalgr6}{{3.14}{129}{$g(x) = |x-3| = f(x-3)$\relax }{figure.caption.214}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.15}{\ignorespaces $h(x)=\delimiter 69640972 x\delimiter 86418188 -3 = f(x)-3$\relax }}{129}{figure.caption.215}}
\newlabel{fig:absvalgr7}{{3.15}{129}{$h(x)=\lvert x\rvert -3 = f(x)-3$\relax }{figure.caption.215}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.16}{\ignorespaces $i(x) = 4-2|3x+1| = -2f(3x+1)+4$\relax }}{129}{figure.caption.216}}
\newlabel{fig:absvalgr8}{{3.16}{129}{$i(x) = 4-2|3x+1| = -2f(3x+1)+4$\relax }{figure.caption.216}{}}
\pgfsyspdfmark {pgfid199}{4474142}{46871346}
\newlabel{ex_absvaltrans}{{50}{129}{Absolute Value Functions}{examplecounter.50}{}}
\newlabel{eex_absvaltrans}{{13}{129}{Absolute Value Functions}{Item.960}{}}
\pgfsyspdfmark {pgfid200}{4736286}{8799985}
\@writefile{lof}{\contentsline {figure}{\numberline {3.17}{\ignorespaces $f(x) = \genfrac  {}{}{}0{\delimiter 69640972 x\delimiter 86418188 }{x}$\relax }}{130}{figure.caption.217}}
\newlabel{fig:absvalgr9}{{3.17}{130}{$f(x) = \dfrac {\lvert x\rvert }{x}$\relax }{figure.caption.217}{}}
\pgfsyspdfmark {pgfid201}{14288487}{46871346}
\newlabel{ex_absgraphhard}{{51}{130}{Absolute Value Functions}{examplecounter.51}{}}
\pgfsyspdfmark {pgfid202}{19793511}{46707506}
\@writefile{lof}{\contentsline {figure}{\numberline {3.18}{\ignorespaces $g(x) = \delimiter 69640972 x+2\delimiter 86418188 -\delimiter 69640972 x-3\delimiter 86418188 +1$\relax }}{131}{figure.caption.218}}
\newlabel{fig:absvalgrlast}{{3.18}{131}{$g(x) = \lvert x+2\rvert -\lvert x-3\rvert +1$\relax }{figure.caption.218}{}}
\newlabel{eex_absgraphhard}{{13}{131}{Absolute Value Functions}{figure.caption.217}{}}
\pgfsyspdfmark {pgfid203}{10217032}{21470780}
\newlabel{triangleinequalityreals}{{36}{132}{Absolute Value Functions}{Item.1000}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.19}{\ignorespaces The graph of the basic quadratic function $f(x)=x^2$\relax }}{133}{figure.caption.219}}
\newlabel{fig:parabstd}{{3.19}{133}{The graph of the basic quadratic function $f(x)=x^2$\relax }{figure.caption.219}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.20}{\ignorespaces The graph $y=x^2$ with points labelled\relax }}{133}{figure.caption.220}}
\newlabel{fig:parabgr1}{{3.20}{133}{The graph $y=x^2$ with points labelled\relax }{figure.caption.220}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.21}{\ignorespaces $g(x)=f(x+2)-3 = (x+2)^2-3$\relax }}{133}{figure.caption.221}}
\newlabel{fig:parabgr2}{{3.21}{133}{$g(x)=f(x+2)-3 = (x+2)^2-3$\relax }{figure.caption.221}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.3}Quadratic Functions}{133}{section.3.3}}
\newlabel{QuadraticFunctions}{{3.3}{133}{Quadratic Functions}{section.3.3}{}}
\newlabel{quadraticfunction}{{34}{133}{Quadratic Functions}{definitioncounter.34}{}}
\pgfsyspdfmark {pgfid205}{4474142}{28282562}
\newlabel{parabolaex1}{{52}{133}{Quadratic Functions}{examplecounter.52}{}}
\pgfsyspdfmark {pgfid206}{9979166}{28118722}
\@writefile{lof}{\contentsline {figure}{\numberline {3.22}{\ignorespaces $h(x) = -2f(x-3)+1 = -2(x-3)^2+1$\relax }}{134}{figure.caption.222}}
\newlabel{fig:parabgr3}{{3.22}{134}{$h(x) = -2f(x-3)+1 = -2(x-3)^2+1$\relax }{figure.caption.222}{}}
\newlabel{eparabolaex1}{{34}{134}{Quadratic Functions}{figure.caption.221}{}}
\pgfsyspdfmark {pgfid207}{14550631}{31279943}
\newlabel{standardgeneralformofparabolas}{{35}{134}{Quadratic Functions}{definitioncounter.35}{}}
\newlabel{standardformvertex}{{14}{134}{Quadratic Functions}{theoremcounter.14}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.23}{\ignorespaces The axis of symmetry of a parabola\relax }}{135}{figure.caption.223}}
\newlabel{fig:parabaxis}{{3.23}{135}{The axis of symmetry of a parabola\relax }{figure.caption.223}{}}
\pgfsyspdfmark {pgfid210}{4474142}{11892911}
\newlabel{parabolaex2}{{53}{135}{Quadratic Functions}{examplecounter.53}{}}
\pgfsyspdfmark {pgfid211}{9979166}{11729071}
\@writefile{lof}{\contentsline {figure}{\numberline {3.24}{\ignorespaces $f(x) = x^2-4x+3$\relax }}{136}{figure.caption.236}}
\newlabel{fig:parabgr4}{{3.24}{136}{$f(x) = x^2-4x+3$\relax }{figure.caption.236}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.25}{\ignorespaces $g(x) = 6-x-x^2$\relax }}{136}{figure.caption.237}}
\newlabel{fig:parabgr5}{{3.25}{136}{$g(x) = 6-x-x^2$\relax }{figure.caption.237}{}}
\pgfsyspdfmark {pgfid212}{16189033}{21208746}
\newlabel{eparabolaex2}{{14}{137}{Quadratic Functions}{figure.caption.237}{}}
\pgfsyspdfmark {pgfid213}{4736286}{42908880}
\newlabel{vertexofquadraticfunctions}{{15}{137}{Quadratic Functions}{theoremcounter.15}{}}
\newlabel{quadraticformula}{{16}{138}{Quadratic Functions}{theoremcounter.16}{}}
\newlabel{discriminant}{{36}{138}{Quadratic Functions}{definitioncounter.36}{}}
\newlabel{discriminanttrichotomy}{{17}{139}{Quadratic Functions}{theoremcounter.17}{}}
\pgfsyspdfmark {pgfid218}{4474142}{30475424}
\newlabel{PortaBoyProfit}{{54}{139}{Quadratic Functions}{examplecounter.54}{}}
\pgfsyspdfmark {pgfid219}{9979166}{30311584}
\@writefile{lof}{\contentsline {figure}{\numberline {3.26}{\ignorespaces The graph of the profit function $P(x)$\relax }}{140}{figure.caption.246}}
\newlabel{fig:parabprofit}{{3.26}{140}{The graph of the profit function $P(x)$\relax }{figure.caption.246}{}}
\newlabel{ePortaBoyProfit}{{17}{140}{Quadratic Functions}{Item.1018}{}}
\pgfsyspdfmark {pgfid220}{14550631}{6125350}
\@writefile{lof}{\contentsline {figure}{\numberline {3.27}{\ignorespaces A diagram of pasture dimensions\relax }}{141}{figure.caption.247}}
\newlabel{fig:parabalpaca}{{3.27}{141}{A diagram of pasture dimensions\relax }{figure.caption.247}{}}
\pgfsyspdfmark {pgfid221}{4474142}{46871346}
\newlabel{donniealpaca}{{55}{141}{Quadratic Functions}{examplecounter.55}{}}
\newlabel{edonniealpaca}{{17}{141}{Quadratic Functions}{examplecounter.55}{}}
\pgfsyspdfmark {pgfid222}{4736286}{13187905}
\pgfsyspdfmark {pgfid223}{4474142}{10171872}
\newlabel{ex_parababsgr}{{56}{141}{Quadratic Functions}{examplecounter.56}{}}
\pgfsyspdfmark {pgfid224}{9979166}{10008032}
\@writefile{lof}{\contentsline {figure}{\numberline {3.28}{\ignorespaces Obtaining the graph of $f(x)=\delimiter 69640972 x^2-x-6\delimiter 86418188 $\relax }}{142}{figure.caption.248}}
\newlabel{fig:parabgrlast}{{3.28}{142}{Obtaining the graph of $f(x)=\lvert x^2-x-6\rvert $\relax }{figure.caption.248}{}}
\newlabel{eex_parababsgr}{{17}{142}{Quadratic Functions}{figure.caption.247}{}}
\pgfsyspdfmark {pgfid225}{15533671}{31565414}
\newlabel{fixedperimetermaxareagarden}{{19}{143}{Quadratic Functions}{Item.1037}{}}
\newlabel{whatgoesup}{{25}{144}{Quadratic Functions}{Item.1043}{}}
\newlabel{parabolicbridgecable}{{26}{144}{Quadratic Functions}{Item.1050}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.29}{\ignorespaces Graphical interpretation of Example \ref  {ex_motineq}\relax }}{145}{figure.caption.249}}
\newlabel{fig:linearineq}{{3.29}{145}{Graphical interpretation of Example \ref {ex_motineq}\relax }{figure.caption.249}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.4}Inequalities with Absolute Value and Quadratic Functions}{145}{section.3.4}}
\newlabel{Inequalities}{{3.4}{145}{Inequalities with Absolute Value and Quadratic Functions}{section.3.4}{}}
\pgfsyspdfmark {pgfid226}{4474142}{41077102}
\newlabel{ex_motineq}{{57}{145}{Inequalities with Absolute Value and Quadratic Functions}{examplecounter.57}{}}
\newlabel{eex_motineq}{{3.4}{145}{Inequalities with Absolute Value and Quadratic Functions}{Item.1068}{}}
\pgfsyspdfmark {pgfid227}{4736286}{6931057}
\@writefile{lof}{\contentsline {figure}{\numberline {3.30}{\ignorespaces The graphs $y=f(x)$ and $y=g(x)$ for Example \ref  {ex_solvewithgraph}\relax }}{146}{figure.caption.250}}
\newlabel{fig:graphineq}{{3.30}{146}{The graphs $y=f(x)$ and $y=g(x)$ for Example \ref {ex_solvewithgraph}\relax }{figure.caption.250}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.31}{\ignorespaces The solution to $f(x)<g(x)$\relax }}{146}{figure.caption.251}}
\newlabel{fig:graphineq1}{{3.31}{146}{The solution to $f(x)<g(x)$\relax }{figure.caption.251}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.32}{\ignorespaces The solution to $f(x) \geq g(x)$\relax }}{146}{figure.caption.252}}
\newlabel{fig:graphineq2}{{3.32}{146}{The solution to $f(x) \geq g(x)$\relax }{figure.caption.252}{}}
\newlabel{idea:graphineq}{{17}{146}{Inequalities with Absolute Value and Quadratic Functions}{keyideacounter.17}{}}
\pgfsyspdfmark {pgfid229}{14288487}{33145287}
\newlabel{ex_solvewithgraph}{{58}{146}{Inequalities with Absolute Value and Quadratic Functions}{examplecounter.58}{}}
\newlabel{eex_solvewithgraph}{{17}{146}{Inequalities with Absolute Value and Quadratic Functions}{Item.1074}{}}
\pgfsyspdfmark {pgfid230}{14550631}{7327859}
\@writefile{lof}{\contentsline {figure}{\numberline {3.33}{\ignorespaces Solving $\delimiter 69640972 x\delimiter 86418188 <c$ graphically\relax }}{147}{figure.caption.253}}
\newlabel{fig:absvalineq}{{3.33}{147}{Solving $\lvert x\rvert <c$ graphically\relax }{figure.caption.253}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.34}{\ignorespaces Solving $\delimiter 69640972 x-1\delimiter 86418188 \geq 3$ in Example \ref  {ex_absvalineq}\relax }}{147}{figure.caption.254}}
\newlabel{fig:absvalineqeg1}{{3.34}{147}{Solving $\lvert x-1\rvert \geq 3$ in Example \ref {ex_absvalineq}\relax }{figure.caption.254}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.35}{\ignorespaces Solving $4 - 3|2x+1| > -2$ in Example \ref  {ex_absvalineq}\relax }}{147}{figure.caption.255}}
\newlabel{fig:absvalineqeg2}{{3.35}{147}{Solving $4 - 3|2x+1| > -2$ in Example \ref {ex_absvalineq}\relax }{figure.caption.255}{}}
\newlabel{absolutevalueineq}{{18}{147}{Inequalities with Absolute Value and Quadratic Functions}{theoremcounter.18}{}}
\pgfsyspdfmark {pgfid232}{4474142}{25483224}
\newlabel{ex_absvalineq}{{59}{147}{Inequalities with Absolute Value and Quadratic Functions}{examplecounter.59}{}}
\pgfsyspdfmark {pgfid233}{9979166}{25319384}
\@writefile{lof}{\contentsline {figure}{\numberline {3.36}{\ignorespaces Solving $2 < |x-1| \leq 5$ in Example \ref  {ex_absvalineq}\relax }}{148}{figure.caption.257}}
\newlabel{fig:absvalineqeg3}{{3.36}{148}{Solving $2 < |x-1| \leq 5$ in Example \ref {ex_absvalineq}\relax }{figure.caption.257}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.37}{\ignorespaces Solving $|x+1|\geq \genfrac  {}{}{}0{x+4}{2}$ in Example \ref  {ex_absvalineq}\relax }}{148}{figure.caption.258}}
\newlabel{fig:absvalineqeg4}{{3.37}{148}{Solving $|x+1|\geq \dfrac {x+4}{2}$ in Example \ref {ex_absvalineq}\relax }{figure.caption.258}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.38}{\ignorespaces $y=x^2-x-6$\relax }}{148}{figure.caption.259}}
\newlabel{fig:parabineq0}{{3.38}{148}{$y=x^2-x-6$\relax }{figure.caption.259}{}}
\newlabel{eex_absvalineq}{{18}{148}{Inequalities with Absolute Value and Quadratic Functions}{Item.1082}{}}
\pgfsyspdfmark {pgfid234}{14550631}{25405246}
\newlabel{firstsigndiagram}{{18}{148}{Inequalities with Absolute Value and Quadratic Functions}{section*.256}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.39}{\ignorespaces The sign diagram for $f(x)=2x^2+x-3$\relax }}{149}{figure.caption.260}}
\newlabel{fig:quadineqeg1}{{3.39}{149}{The sign diagram for $f(x)=2x^2+x-3$\relax }{figure.caption.260}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.40}{\ignorespaces Verifying the solution to $2x^2\leq 3-x$ graphically\relax }}{149}{figure.caption.261}}
\newlabel{fig:quadineqeg2}{{3.40}{149}{Verifying the solution to $2x^2\leq 3-x$ graphically\relax }{figure.caption.261}{}}
\newlabel{idea:quadineq}{{18}{149}{Inequalities with Absolute Value and Quadratic Functions}{keyideacounter.18}{}}
\pgfsyspdfmark {pgfid236}{4474142}{26658537}
\newlabel{ex_quadineq}{{60}{149}{Inequalities with Absolute Value and Quadratic Functions}{examplecounter.60}{}}
\pgfsyspdfmark {pgfid237}{9979166}{26494697}
\@writefile{lof}{\contentsline {figure}{\numberline {3.41}{\ignorespaces The sign diagram for $f(x)=x^2-2x-1$\relax }}{150}{figure.caption.262}}
\newlabel{fig:quadineqeg3}{{3.41}{150}{The sign diagram for $f(x)=x^2-2x-1$\relax }{figure.caption.262}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.42}{\ignorespaces Verifying the solution to $x^2-2x > 1$ graphically\relax }}{150}{figure.caption.263}}
\newlabel{fig:quadineqeg4}{{3.42}{150}{Verifying the solution to $x^2-2x > 1$ graphically\relax }{figure.caption.263}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.43}{\ignorespaces The sign diagram for $f(x)=x^2-2x+1$\relax }}{150}{figure.caption.264}}
\newlabel{fig:quadineqeg5}{{3.43}{150}{The sign diagram for $f(x)=x^2-2x+1$\relax }{figure.caption.264}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.44}{\ignorespaces Verifying the solution to $x^2+1\leq 2x$ graphically\relax }}{150}{figure.caption.265}}
\newlabel{fig:quadineqeg6}{{3.44}{150}{Verifying the solution to $x^2+1\leq 2x$ graphically\relax }{figure.caption.265}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.45}{\ignorespaces The sign diagram for $f(x)=x^2-3x$, where $x<1$\relax }}{150}{figure.caption.266}}
\newlabel{fig:quadineqeg7}{{3.45}{150}{The sign diagram for $f(x)=x^2-3x$, where $x<1$\relax }{figure.caption.266}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.46}{\ignorespaces The sign diagram for $g(x)=x^2-x-2$, where $x\geq 1$\relax }}{150}{figure.caption.267}}
\newlabel{fig:quadineqeg8}{{3.46}{150}{The sign diagram for $g(x)=x^2-x-2$, where $x\geq 1$\relax }{figure.caption.267}{}}
\pgfsyspdfmark {pgfid238}{16189033}{17981603}
\@writefile{lof}{\contentsline {figure}{\numberline {3.47}{\ignorespaces The overall sign diagram for Problem 4 in Example \ref  {ex_quadineq}\relax }}{151}{figure.caption.274}}
\newlabel{fig:quadineqeg9}{{3.47}{151}{The overall sign diagram for Problem 4 in Example \ref {ex_quadineq}\relax }{figure.caption.274}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.48}{\ignorespaces Verifying the inequality $2x-x^2\geq \delimiter 69640972 x-1\delimiter 86418188 -1$ graphically\relax }}{151}{figure.caption.275}}
\newlabel{fig:quadineqeglast}{{3.48}{151}{Verifying the inequality $2x-x^2\geq \lvert x-1\rvert -1$ graphically\relax }{figure.caption.275}{}}
\newlabel{eex_quadineq}{{18}{151}{Inequalities with Absolute Value and Quadratic Functions}{figure.caption.267}{}}
\pgfsyspdfmark {pgfid239}{4736286}{41211522}
\pgfsyspdfmark {pgfid240}{4474142}{34577699}
\newlabel{ex_tolerance}{{61}{151}{Inequalities with Absolute Value and Quadratic Functions}{examplecounter.61}{}}
\newlabel{eex_tolerance}{{18}{151}{Inequalities with Absolute Value and Quadratic Functions}{AMS.273}{}}
\pgfsyspdfmark {pgfid241}{24814757}{4736286}
\@writefile{lof}{\contentsline {figure}{\numberline {3.49}{\ignorespaces Graph of the relation $R$ in Example \ref  {ex_ineqrels}\relax }}{152}{figure.caption.276}}
\newlabel{fig:ineqrel1}{{3.49}{152}{Graph of the relation $R$ in Example \ref {ex_ineqrels}\relax }{figure.caption.276}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.50}{\ignorespaces Graph of the relation $S$ in Example \ref  {ex_ineqrels}\relax }}{152}{figure.caption.277}}
\newlabel{fig:ineqrel2}{{3.50}{152}{Graph of the relation $S$ in Example \ref {ex_ineqrels}\relax }{figure.caption.277}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.51}{\ignorespaces Graph of the relation $T$ in Example \ref  {ex_ineqrels}\relax }}{152}{figure.caption.278}}
\newlabel{fig:ineqrel3}{{3.51}{152}{Graph of the relation $T$ in Example \ref {ex_ineqrels}\relax }{figure.caption.278}{}}
\pgfsyspdfmark {pgfid242}{14288487}{44905266}
\newlabel{ex_ineqrels}{{62}{152}{Inequalities with Absolute Value and Quadratic Functions}{examplecounter.62}{}}
\newlabel{eex_ineqrels}{{18}{152}{Inequalities with Absolute Value and Quadratic Functions}{Item.1101}{}}
\pgfsyspdfmark {pgfid243}{14550631}{22983474}
\@writefile{toc}{\contentsline {chapter}{\numberline {4}Polynomial Functions}{155}{chapter.4}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Polynomials}{{4}{155}{Polynomial Functions}{chapter.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.1}Graphs of Polynomial Functions}{155}{section.4.1}}
\newlabel{GraphsofPolynomials}{{4.1}{155}{Graphs of Polynomial Functions}{section.4.1}{}}
\newlabel{polynomialfunction}{{37}{155}{Graphs of Polynomial Functions}{definitioncounter.37}{}}
\pgfsyspdfmark {pgfid245}{4474142}{22635505}
\newlabel{intropolyexample}{{63}{155}{Graphs of Polynomial Functions}{examplecounter.63}{}}
\pgfsyspdfmark {pgfid246}{9979166}{22471665}
\newlabel{eintropolyexample}{{37}{156}{Graphs of Polynomial Functions}{Item.1163}{}}
\pgfsyspdfmark {pgfid247}{14550631}{32079122}
\newlabel{degreeandallthat}{{38}{156}{Graphs of Polynomial Functions}{definitioncounter.38}{}}
\newlabel{indeterminantformone}{{38}{156}{Graphs of Polynomial Functions}{section*.279}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.1}{\ignorespaces Constructing the box in Example \ref  {boxnotopex}\relax }}{157}{figure.caption.280}}
\newlabel{fig:boxnotop}{{4.1}{157}{Constructing the box in Example \ref {boxnotopex}\relax }{figure.caption.280}{}}
\pgfsyspdfmark {pgfid249}{4474142}{46871346}
\newlabel{ex_degreeetc}{{64}{157}{Graphs of Polynomial Functions}{examplecounter.64}{}}
\newlabel{eex_degreeetc}{{38}{157}{Graphs of Polynomial Functions}{Item.1171}{}}
\pgfsyspdfmark {pgfid250}{4736286}{16030689}
\pgfsyspdfmark {pgfid251}{4474142}{13088387}
\newlabel{boxnotopex}{{65}{157}{Graphs of Polynomial Functions}{examplecounter.65}{}}
\pgfsyspdfmark {pgfid252}{9979166}{12924547}
\@writefile{lof}{\contentsline {figure}{\numberline {4.2}{\ignorespaces Optimizing the volume of the box in Example \ref  {boxnotopex}\relax }}{158}{figure.caption.281}}
\newlabel{fig:boxnotop2}{{4.2}{158}{Optimizing the volume of the box in Example \ref {boxnotopex}\relax }{figure.caption.281}{}}
\newlabel{eboxnotopex}{{38}{158}{Graphs of Polynomial Functions}{Item.1175}{}}
\pgfsyspdfmark {pgfid253}{14550631}{26170847}
\@writefile{lof}{\contentsline {figure}{\numberline {4.3}{\ignorespaces Graphing even powers of $x$\relax }}{159}{figure.caption.282}}
\newlabel{fig:graphevenpowers}{{4.3}{159}{Graphing even powers of $x$\relax }{figure.caption.282}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.4}{\ignorespaces Graphing odd powers of $x$\relax }}{159}{figure.caption.283}}
\newlabel{fig:graphoddpowers}{{4.4}{159}{Graphing odd powers of $x$\relax }{figure.caption.283}{}}
\newlabel{idea:endbehaveven}{{19}{159}{Graphs of Polynomial Functions}{keyideacounter.19}{}}
\newlabel{idea:endbehavodd}{{20}{159}{Graphs of Polynomial Functions}{keyideacounter.20}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.5}{\ignorespaces Pathologies not found on graphs of polynomials\relax }}{160}{figure.caption.284}}
\newlabel{cusppicture}{{4.5}{160}{Pathologies not found on graphs of polynomials\relax }{figure.caption.284}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.6}{\ignorespaces The graph of a polynomial\relax }}{160}{figure.caption.285}}
\newlabel{fig:polygraphgen}{{4.6}{160}{The graph of a polynomial\relax }{figure.caption.285}{}}
\newlabel{IVT}{{19}{160}{Graphs of Polynomial Functions}{theoremcounter.19}{}}
\pgfsyspdfmark {pgfid257}{14288487}{15182378}
\newlabel{ex_squareroot2}{{66}{160}{Graphs of Polynomial Functions}{examplecounter.66}{}}
\newlabel{eex_squareroot2}{{19}{160}{Graphs of Polynomial Functions}{examplecounter.66}{}}
\pgfsyspdfmark {pgfid258}{35377938}{9793767}
\@writefile{lof}{\contentsline {figure}{\numberline {4.7}{\ignorespaces The sign diagram of $f$ in Example \ref  {polygraphex}\relax }}{161}{figure.caption.286}}
\newlabel{fig:polysign1}{{4.7}{161}{The sign diagram of $f$ in Example \ref {polygraphex}\relax }{figure.caption.286}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.8}{\ignorespaces The graph $y=f(x)$ for Example \ref  {polygraphex}\relax }}{161}{figure.caption.287}}
\newlabel{fig:polygraph1}{{4.8}{161}{The graph $y=f(x)$ for Example \ref {polygraphex}\relax }{figure.caption.287}{}}
\newlabel{idea:polysigndiag}{{21}{161}{Graphs of Polynomial Functions}{keyideacounter.21}{}}
\pgfsyspdfmark {pgfid260}{4474142}{35090926}
\newlabel{polygraphex}{{67}{161}{Graphs of Polynomial Functions}{examplecounter.67}{}}
\newlabel{epolygraphex}{{21}{161}{Graphs of Polynomial Functions}{examplecounter.67}{}}
\pgfsyspdfmark {pgfid261}{5719326}{24618848}
\newlabel{EBPolynomials}{{20}{161}{Graphs of Polynomial Functions}{theoremcounter.20}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.9}{\ignorespaces Two views of the polynomials $f(x)$ and $g(x)$\relax }}{162}{figure.caption.288}}
\newlabel{fig:leadingterm}{{4.9}{162}{Two views of the polynomials $f(x)$ and $g(x)$\relax }{figure.caption.288}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.10}{\ignorespaces The graph $y=f(x)$ for Example \ref  {ex_multiplicity}\relax }}{163}{figure.caption.289}}
\newlabel{fig:polygraph2}{{4.10}{163}{The graph $y=f(x)$ for Example \ref {ex_multiplicity}\relax }{figure.caption.289}{}}
\newlabel{multiplicity}{{39}{163}{Graphs of Polynomial Functions}{definitioncounter.39}{}}
\newlabel{roleofmultiplicity}{{21}{163}{Graphs of Polynomial Functions}{theoremcounter.21}{}}
\pgfsyspdfmark {pgfid265}{4474142}{23001554}
\newlabel{ex_multiplicity}{{68}{163}{Graphs of Polynomial Functions}{examplecounter.68}{}}
\newlabel{eex_multiplicity}{{21}{163}{Graphs of Polynomial Functions}{examplecounter.68}{}}
\pgfsyspdfmark {pgfid266}{5719326}{6322642}
\newlabel{polygraphexercise}{{21}{164}{Graphs of Polynomial Functions}{Item.1188}{}}
\newlabel{LCDmaxprofit}{{21}{164}{Graphs of Polynomial Functions}{Item.1206}{}}
\newlabel{newportaboycost}{{32}{164}{Graphs of Polynomial Functions}{Item.1210}{}}
\newlabel{girthbox1}{{33a}{165}{Graphs of Polynomial Functions}{Item.1212}{}}
\newlabel{girthbox2}{{33b}{165}{Graphs of Polynomial Functions}{Item.1213}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.11}{\ignorespaces The graph $y=x^3+4x^2-5x-14$\relax }}{166}{figure.caption.290}}
\newlabel{fig:polyzeros1}{{4.11}{166}{The graph $y=x^3+4x^2-5x-14$\relax }{figure.caption.290}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.2}The Factor Theorem and the Remainder Theorem}{166}{section.4.2}}
\newlabel{Polydivision}{{4.2}{166}{The Factor Theorem and the Remainder Theorem}{section.4.2}{}}
\newlabel{polydiv}{{22}{166}{The Factor Theorem and the Remainder Theorem}{theoremcounter.22}{}}
\newlabel{remainderthm}{{23}{167}{The Factor Theorem and the Remainder Theorem}{theoremcounter.23}{}}
\newlabel{factorthm}{{24}{167}{The Factor Theorem and the Remainder Theorem}{theoremcounter.24}{}}
\pgfsyspdfmark {pgfid270}{4474142}{19446902}
\newlabel{ex_synthetic}{{69}{169}{The Factor Theorem and the Remainder Theorem}{examplecounter.69}{}}
\pgfsyspdfmark {pgfid271}{9979166}{19283062}
\newlabel{eex_synthetic}{{24}{170}{The Factor Theorem and the Remainder Theorem}{Item.1231}{}}
\pgfsyspdfmark {pgfid272}{17215057}{20733474}
\pgfsyspdfmark {pgfid273}{14288487}{18513015}
\newlabel{ex_polytheorems}{{70}{170}{The Factor Theorem and the Remainder Theorem}{examplecounter.70}{}}
\pgfsyspdfmark {pgfid274}{19793511}{18349175}
\newlabel{eex_polytheorems}{{24}{171}{The Factor Theorem and the Remainder Theorem}{Item.1235}{}}
\pgfsyspdfmark {pgfid275}{4736286}{31358674}
\pgfsyspdfmark {pgfid276}{4474142}{27042717}
\newlabel{ex_factormult}{{71}{171}{The Factor Theorem and the Remainder Theorem}{examplecounter.71}{}}
\newlabel{eex_factormult}{{24}{171}{The Factor Theorem and the Remainder Theorem}{examplecounter.71}{}}
\pgfsyspdfmark {pgfid277}{23862349}{12580115}
\newlabel{nzerosreal}{{25}{172}{The Factor Theorem and the Remainder Theorem}{theoremcounter.25}{}}
\newlabel{idea:zerofactor}{{22}{172}{The Factor Theorem and the Remainder Theorem}{keyideacounter.22}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.3}Real Zeros of Polynomials}{176}{section.4.3}}
\newlabel{RealZeros}{{4.3}{176}{Real Zeros of Polynomials}{section.4.3}{}}
\newlabel{RZT}{{26}{176}{Real Zeros of Polynomials}{theoremcounter.26}{}}
\pgfsyspdfmark {pgfid281}{14288487}{6472990}
\newlabel{RZTex}{{72}{176}{Real Zeros of Polynomials}{examplecounter.72}{}}
\pgfsyspdfmark {pgfid282}{19793511}{6309150}
\@writefile{lof}{\contentsline {figure}{\numberline {4.12}{\ignorespaces The graph $y=f(x)=2x^4+4x^3-x^2-6x-3$\relax }}{177}{figure.caption.291}}
\newlabel{fig:realzero1}{{4.12}{177}{The graph $y=f(x)=2x^4+4x^3-x^2-6x-3$\relax }{figure.caption.291}{}}
\newlabel{eRZTex}{{26}{177}{Real Zeros of Polynomials}{examplecounter.72}{}}
\pgfsyspdfmark {pgfid283}{4736286}{41988914}
\pgfsyspdfmark {pgfid284}{4474142}{37443522}
\newlabel{ex_rzerotech}{{73}{177}{Real Zeros of Polynomials}{examplecounter.73}{}}
\pgfsyspdfmark {pgfid285}{9979166}{37279682}
\@writefile{lof}{\contentsline {figure}{\numberline {4.13}{\ignorespaces Zooming in on the repeated zero in Example \ref  {ex_rzerotech}\relax }}{178}{figure.caption.292}}
\newlabel{fig:realzero2}{{4.13}{178}{Zooming in on the repeated zero in Example \ref {ex_rzerotech}\relax }{figure.caption.292}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.14}{\ignorespaces Two views of the graph  $y=f(x)=x^4+x^2-12$ in Example \ref  {usubex}\relax }}{178}{figure.caption.293}}
\newlabel{fig:realzero3}{{4.14}{178}{Two views of the graph\\ $y=f(x)=x^4+x^2-12$ in Example \ref {usubex}\relax }{figure.caption.293}{}}
\newlabel{eex_rzerotech}{{26}{178}{Real Zeros of Polynomials}{figure.caption.291}{}}
\pgfsyspdfmark {pgfid286}{14550631}{42132867}
\pgfsyspdfmark {pgfid287}{14288487}{30398514}
\newlabel{usubex}{{74}{178}{Real Zeros of Polynomials}{examplecounter.74}{}}
\pgfsyspdfmark {pgfid288}{19793511}{30234674}
\newlabel{eusubex}{{26}{179}{Real Zeros of Polynomials}{figure.caption.293}{}}
\pgfsyspdfmark {pgfid289}{4736286}{42979778}
\pgfsyspdfmark {pgfid290}{4474142}{32785342}
\newlabel{ex_zeronotech}{{75}{179}{Real Zeros of Polynomials}{examplecounter.75}{}}
\pgfsyspdfmark {pgfid291}{9979166}{32621502}
\pgfsyspdfmark {pgfid292}{6374688}{13728545}
\@writefile{lof}{\contentsline {figure}{\numberline {4.15}{\ignorespaces The graph $y=2x^4+4x^3-x^2-6x-3$\relax }}{180}{figure.caption.294}}
\newlabel{fig:realzero4}{{4.15}{180}{The graph $y=2x^4+4x^3-x^2-6x-3$\relax }{figure.caption.294}{}}
\newlabel{eex_zeronotech}{{26}{180}{Real Zeros of Polynomials}{Item.1296}{}}
\pgfsyspdfmark {pgfid293}{14550631}{18497082}
\@writefile{lof}{\contentsline {figure}{\numberline {4.16}{\ignorespaces The sign diagram for $p(x)$ in Example \ref  {polyeqineqexample}\relax }}{181}{figure.caption.295}}
\newlabel{fig:sdzeros1}{{4.16}{181}{The sign diagram for $p(x)$ in Example \ref {polyeqineqexample}\relax }{figure.caption.295}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.17}{\ignorespaces The polynomials $f(x)$ and $g(x)$ from Example \ref  {polyeqineqexample}, part 3\relax }}{181}{figure.caption.296}}
\newlabel{fig:realzeroeg1}{{4.17}{181}{The polynomials $f(x)$ and $g(x)$ from Example \ref {polyeqineqexample}, part 3\relax }{figure.caption.296}{}}
\pgfsyspdfmark {pgfid294}{4474142}{40224686}
\newlabel{polyeqineqexample}{{76}{181}{Real Zeros of Polynomials}{examplecounter.76}{}}
\pgfsyspdfmark {pgfid295}{9979166}{40060846}
\@writefile{lof}{\contentsline {figure}{\numberline {4.18}{\ignorespaces The sign diagram for $P(x)$ in Example \ref  {ex_zerosappl}\relax }}{182}{figure.caption.297}}
\newlabel{fig:sdzeros2}{{4.18}{182}{The sign diagram for $P(x)$ in Example \ref {ex_zerosappl}\relax }{figure.caption.297}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.19}{\ignorespaces Plotting the profit function $P(x)$ in Example \ref  {ex_zerosappl}\relax }}{182}{figure.caption.298}}
\newlabel{fig:applzeroeg}{{4.19}{182}{Plotting the profit function $P(x)$ in Example \ref {ex_zerosappl}\relax }{figure.caption.298}{}}
\newlabel{epolyeqineqexample}{{26}{182}{Real Zeros of Polynomials}{Item.1302}{}}
\pgfsyspdfmark {pgfid296}{14550631}{46707506}
\pgfsyspdfmark {pgfid297}{14288487}{43725618}
\newlabel{ex_zerosappl}{{77}{182}{Real Zeros of Polynomials}{examplecounter.77}{}}
\newlabel{eex_zerosappl}{{26}{182}{Real Zeros of Polynomials}{examplecounter.77}{}}
\pgfsyspdfmark {pgfid298}{20474897}{19968818}
\@writefile{toc}{\contentsline {section}{\numberline {4.4}Complex Zeros of Polynomials}{185}{section.4.4}}
\newlabel{ComplexZeros}{{4.4}{185}{Complex Zeros of Polynomials}{section.4.4}{}}
\newlabel{ftoa}{{27}{185}{Complex Zeros of Polynomials}{theoremcounter.27}{}}
\newlabel{complexfactorization}{{28}{186}{Complex Zeros of Polynomials}{theoremcounter.28}{}}
\pgfsyspdfmark {pgfid301}{14288487}{30785087}
\newlabel{ex_compfact}{{78}{186}{Complex Zeros of Polynomials}{examplecounter.78}{}}
\pgfsyspdfmark {pgfid302}{19793511}{30621247}
\newlabel{eex_compfact}{{28}{187}{Complex Zeros of Polynomials}{Item.1364}{}}
\pgfsyspdfmark {pgfid303}{4736286}{46707506}
\newlabel{conjugatepairsthm}{{29}{187}{Complex Zeros of Polynomials}{theoremcounter.29}{}}
\newlabel{realfactorization}{{30}{188}{Complex Zeros of Polynomials}{theoremcounter.30}{}}
\pgfsyspdfmark {pgfid306}{14288487}{38283353}
\newlabel{ex_compfact2}{{79}{188}{Complex Zeros of Polynomials}{examplecounter.79}{}}
\newlabel{eex_compfact2}{{30}{188}{Complex Zeros of Polynomials}{Item.1372}{}}
\pgfsyspdfmark {pgfid307}{14550631}{6712719}
\pgfsyspdfmark {pgfid308}{4474142}{46871346}
\newlabel{ex_complast}{{80}{189}{Complex Zeros of Polynomials}{examplecounter.80}{}}
\newlabel{eex_complast}{{30}{189}{Complex Zeros of Polynomials}{examplecounter.80}{}}
\pgfsyspdfmark {pgfid309}{24211592}{23661853}
\newlabel{complexepilogue}{{30}{189}{Complex Zeros of Polynomials}{section*.307}{}}
\newlabel{zbarexercise}{{28}{191}{Complex Zeros of Polynomials}{Item.1400}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {5}Rational Functions}{193}{chapter.5}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Rationals}{{5}{193}{Rational Functions}{chapter.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.1}Introduction to Rational Functions}{193}{section.5.1}}
\newlabel{IntroRational}{{5.1}{193}{Introduction to Rational Functions}{section.5.1}{}}
\newlabel{rationalfunction}{{40}{193}{Introduction to Rational Functions}{definitioncounter.40}{}}
\pgfsyspdfmark {pgfid311}{4474142}{26518833}
\newlabel{ratfuncex}{{81}{193}{Introduction to Rational Functions}{examplecounter.81}{}}
\pgfsyspdfmark {pgfid312}{9979166}{26354993}
\newlabel{eratfuncex}{{40}{194}{Introduction to Rational Functions}{Item.1408}{}}
\pgfsyspdfmark {pgfid313}{14550631}{9910198}
\@writefile{lof}{\contentsline {figure}{\numberline {5.1}{\ignorespaces The graph of $f(x) = \genfrac  {}{}{}0{2x-1}{x+1}$\relax }}{195}{figure.caption.308}}
\newlabel{fig:ratgr1}{{5.1}{195}{The graph of $f(x) = \dfrac {2x-1}{x+1}$\relax }{figure.caption.308}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces Values of $f(x)=\frac  {2x-1}{x+1}$ near $x=-1$\relax }}{195}{figure.caption.309}}
\newlabel{fig:rattab1}{{5.2}{195}{Values of $f(x)=\frac {2x-1}{x+1}$ near $x=-1$\relax }{figure.caption.309}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.3}{\ignorespaces Values of $f(x)=\genfrac  {}{}{}0{2x-1}{x+1}$ for large negative and positive values of $x$\relax }}{195}{figure.caption.310}}
\newlabel{fig:rattab2}{{5.3}{195}{Values of $f(x)=\dfrac {2x-1}{x+1}$ for large negative and positive values of $x$\relax }{figure.caption.310}{}}
\newlabel{va}{{41}{195}{Introduction to Rational Functions}{definitioncounter.41}{}}
\newlabel{ha}{{42}{195}{Introduction to Rational Functions}{definitioncounter.42}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.4}{\ignorespaces Values of $h(x)=\frac  {2x^2-1}{x^2-1}-\frac  {3x-2}{x^2-1}$ near $x=1$\relax }}{196}{figure.caption.311}}
\newlabel{fig:rattab3}{{5.4}{196}{Values of $h(x)=\frac {2x^2-1}{x^2-1}-\frac {3x-2}{x^2-1}$ near $x=1$\relax }{figure.caption.311}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.5}{\ignorespaces The graph $y=h(x)$ showing asymptotes and the `hole'\relax }}{196}{figure.caption.312}}
\newlabel{fig:ratgr3}{{5.5}{196}{The graph $y=h(x)$ showing asymptotes and the `hole'\relax }{figure.caption.312}{}}
\newlabel{vavshole}{{31}{196}{Introduction to Rational Functions}{theoremcounter.31}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.6}{\ignorespaces The graph $y=f(x)$ in Example \ref  {vavsholeexample}\relax }}{197}{figure.caption.313}}
\newlabel{fig:ratgr4}{{5.6}{197}{The graph $y=f(x)$ in Example \ref {vavsholeexample}\relax }{figure.caption.313}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.7}{\ignorespaces The graph $y=g(x)$ in Example \ref  {vavsholeexample}\relax }}{197}{figure.caption.314}}
\newlabel{fig:ratgr5}{{5.7}{197}{The graph $y=g(x)$ in Example \ref {vavsholeexample}\relax }{figure.caption.314}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.8}{\ignorespaces The graph $y=g(x)$ in Example \ref  {vavsholeexample}\relax }}{197}{figure.caption.315}}
\newlabel{fig:ratgr6}{{5.8}{197}{The graph $y=g(x)$ in Example \ref {vavsholeexample}\relax }{figure.caption.315}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.9}{\ignorespaces The graph $y=r(x)$ in Example \ref  {vavsholeexample}\relax }}{197}{figure.caption.316}}
\newlabel{fig:ratgr7}{{5.9}{197}{The graph $y=r(x)$ in Example \ref {vavsholeexample}\relax }{figure.caption.316}{}}
\pgfsyspdfmark {pgfid317}{4474142}{46871346}
\newlabel{vavsholeexample}{{82}{197}{Introduction to Rational Functions}{examplecounter.82}{}}
\newlabel{evavsholeexample}{{31}{197}{Introduction to Rational Functions}{Item.1416}{}}
\pgfsyspdfmark {pgfid318}{4736286}{8363215}
\@writefile{lof}{\contentsline {figure}{\numberline {5.10}{\ignorespaces The behaviour of $P$ as $t\to 5^-$\relax }}{198}{figure.caption.317}}
\newlabel{fig:rattab4}{{5.10}{198}{The behaviour of $P$ as $t\to 5^-$\relax }{figure.caption.317}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.11}{\ignorespaces The graph of $P(t)$, for $0\leq t<5$\relax }}{198}{figure.caption.318}}
\newlabel{fig:doomsday}{{5.11}{198}{The graph of $P(t)$, for $0\leq t<5$\relax }{figure.caption.318}{}}
\pgfsyspdfmark {pgfid319}{14288487}{46871346}
\newlabel{ex_doomsday}{{83}{198}{Introduction to Rational Functions}{examplecounter.83}{}}
\newlabel{eex_doomsday}{{31}{198}{Introduction to Rational Functions}{Item.1422}{}}
\pgfsyspdfmark {pgfid320}{14550631}{8325562}
\newlabel{hathm}{{32}{199}{Introduction to Rational Functions}{theoremcounter.32}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.12}{\ignorespaces Graphs of the three functions in Example \ref  {haexample}\relax }}{200}{figure.caption.319}}
\newlabel{fig:ratgr8}{{5.12}{200}{Graphs of the three functions in Example \ref {haexample}\relax }{figure.caption.319}{}}
\pgfsyspdfmark {pgfid322}{14288487}{46871346}
\newlabel{haexample}{{84}{200}{Introduction to Rational Functions}{examplecounter.84}{}}
\newlabel{ehaexample}{{32}{200}{Introduction to Rational Functions}{Item.1428}{}}
\pgfsyspdfmark {pgfid323}{14550631}{24162496}
\pgfsyspdfmark {pgfid324}{14288487}{21365990}
\newlabel{ex_fluspread}{{85}{200}{Introduction to Rational Functions}{examplecounter.85}{}}
\pgfsyspdfmark {pgfid325}{19793511}{21202150}
\@writefile{lof}{\contentsline {figure}{\numberline {5.13}{\ignorespaces The long-term behaviour of $N(t)$\relax }}{201}{figure.caption.320}}
\newlabel{fig:rattab5}{{5.13}{201}{The long-term behaviour of $N(t)$\relax }{figure.caption.320}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.14}{\ignorespaces The graph $y=\genfrac  {}{}{}0{x^2-4}{x+1}$ as $x\to -\infty $\relax }}{201}{figure.caption.321}}
\newlabel{fig:slantas1}{{5.14}{201}{The graph $y=\dfrac {x^2-4}{x+1}$ as $x\to -\infty $\relax }{figure.caption.321}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.15}{\ignorespaces The graph $y=\genfrac  {}{}{}0{x^2-4}{x+1}$ as $x\to +\infty $\relax }}{201}{figure.caption.322}}
\newlabel{fig:slantas2}{{5.15}{201}{The graph $y=\dfrac {x^2-4}{x+1}$ as $x\to +\infty $\relax }{figure.caption.322}{}}
\newlabel{eex_fluspread}{{32}{201}{Introduction to Rational Functions}{figure.caption.319}{}}
\pgfsyspdfmark {pgfid326}{4736286}{41154364}
\newlabel{sa}{{43}{201}{Introduction to Rational Functions}{definitioncounter.43}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.16}{\ignorespaces The graph $y=f(x)$ in Example \ref  {saexample}\relax }}{202}{figure.caption.323}}
\newlabel{fig:ratgr9}{{5.16}{202}{The graph $y=f(x)$ in Example \ref {saexample}\relax }{figure.caption.323}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.17}{\ignorespaces The graph $y=g(x)$ in Example \ref  {saexample}\relax }}{202}{figure.caption.324}}
\newlabel{fig:ratgr10}{{5.17}{202}{The graph $y=g(x)$ in Example \ref {saexample}\relax }{figure.caption.324}{}}
\newlabel{sathm}{{33}{202}{Introduction to Rational Functions}{theoremcounter.33}{}}
\pgfsyspdfmark {pgfid329}{14288487}{31905201}
\newlabel{saexample}{{86}{202}{Introduction to Rational Functions}{examplecounter.86}{}}
\pgfsyspdfmark {pgfid330}{19793511}{31741361}
\newlabel{sacancel}{{2}{202}{Introduction to Rational Functions}{Item.1436}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.18}{\ignorespaces The graph $y=h(x)$ in Example \ref  {saexample}\relax }}{203}{figure.caption.325}}
\newlabel{fig:ratgr11}{{5.18}{203}{The graph $y=h(x)$ in Example \ref {saexample}\relax }{figure.caption.325}{}}
\newlabel{esaexample}{{33}{203}{Introduction to Rational Functions}{figure.caption.324}{}}
\pgfsyspdfmark {pgfid331}{4736286}{40126999}
\@writefile{lof}{\contentsline {figure}{\numberline {5.19}{\ignorespaces The graph $y=h(x)$ from Example \ref  {ratfuncex}\relax }}{205}{figure.caption.326}}
\newlabel{fig:ratgraph1}{{5.19}{205}{The graph $y=h(x)$ from Example \ref {ratfuncex}\relax }{figure.caption.326}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.20}{\ignorespaces The sign diagram for the function $h(x)$ from Example \ref  {ratfuncex}\relax }}{205}{figure.caption.327}}
\newlabel{fig:ratsd1}{{5.20}{205}{The sign diagram for the function $h(x)$ from Example \ref {ratfuncex}\relax }{figure.caption.327}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.2}Graphs of Rational Functions}{205}{section.5.2}}
\newlabel{RationalGraphs}{{5.2}{205}{Graphs of Rational Functions}{section.5.2}{}}
\newlabel{rationalsigndiagram}{{23}{206}{Graphs of Rational Functions}{keyideacounter.23}{}}
\newlabel{idea:ratgraph}{{24}{206}{Graphs of Rational Functions}{keyideacounter.24}{}}
\pgfsyspdfmark {pgfid334}{14288487}{11578486}
\newlabel{ex_ratgraph1}{{87}{206}{Graphs of Rational Functions}{examplecounter.87}{}}
\pgfsyspdfmark {pgfid335}{19793511}{11414646}
\pgfsyspdfmark {pgfid336}{6374688}{33130524}
\@writefile{lof}{\contentsline {figure}{\numberline {5.21}{\ignorespaces The graph $y=\frac  {3x}{x^2-4}$ near its vertical asymptotes\relax }}{208}{figure.caption.328}}
\newlabel{fig:ratgraph2}{{5.21}{208}{The graph $y=\frac {3x}{x^2-4}$ near its vertical asymptotes\relax }{figure.caption.328}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.22}{\ignorespaces The end behavhiour of the graph $y=\frac  {3x}{x^2-4}$\relax }}{208}{figure.caption.329}}
\newlabel{fig:ratgraph3}{{5.22}{208}{The end behavhiour of the graph $y=\frac {3x}{x^2-4}$\relax }{figure.caption.329}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.23}{\ignorespaces The sign diagram for $f(x)=\frac  {3x}{x^2-4}$\relax }}{208}{figure.caption.330}}
\newlabel{fig:ratsd2}{{5.23}{208}{The sign diagram for $f(x)=\frac {3x}{x^2-4}$\relax }{figure.caption.330}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.24}{\ignorespaces The complete graph $y=\frac  {3x}{x^2-4}$ for Example \ref  {ex_ratgraph1}\relax }}{208}{figure.caption.331}}
\newlabel{fig:ratgraph4}{{5.24}{208}{The complete graph $y=\frac {3x}{x^2-4}$ for Example \ref {ex_ratgraph1}\relax }{figure.caption.331}{}}
\pgfsyspdfmark {pgfid337}{16189033}{33942805}
\newlabel{eex_ratgraph1}{{24}{209}{Graphs of Rational Functions}{figure.caption.331}{}}
\pgfsyspdfmark {pgfid338}{4736286}{42969346}
\pgfsyspdfmark {pgfid339}{4474142}{34532777}
\newlabel{ex_ratgraph2}{{88}{209}{Graphs of Rational Functions}{examplecounter.88}{}}
\pgfsyspdfmark {pgfid340}{9979166}{34368937}
\@writefile{lof}{\contentsline {figure}{\numberline {5.25}{\ignorespaces The graph $y=\frac  {2x^2-3x-5}{x^2-x-6}$ near the vertical asymptotes\relax }}{210}{figure.caption.332}}
\newlabel{fig:ratgraph5}{{5.25}{210}{The graph $y=\frac {2x^2-3x-5}{x^2-x-6}$ near the vertical asymptotes\relax }{figure.caption.332}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.26}{\ignorespaces The end behaviour of $y=\frac  {2x^2-3x-5}{x^2-x-6}$\relax }}{210}{figure.caption.333}}
\newlabel{fig:ratgraph6}{{5.26}{210}{The end behaviour of $y=\frac {2x^2-3x-5}{x^2-x-6}$\relax }{figure.caption.333}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.27}{\ignorespaces The sign diagram for $g(x)=\frac  {2x^2-3x-5}{x^2-x-6}$\relax }}{210}{figure.caption.334}}
\newlabel{fig:ratsd3}{{5.27}{210}{The sign diagram for $g(x)=\frac {2x^2-3x-5}{x^2-x-6}$\relax }{figure.caption.334}{}}
\pgfsyspdfmark {pgfid341}{16189033}{30411572}
\@writefile{lof}{\contentsline {figure}{\numberline {5.28}{\ignorespaces The complete graph $y=\frac  {2x^2-3x-5}{x^2-x-6}$ for Example \ref  {ex_ratgraph2}\relax }}{211}{figure.caption.335}}
\newlabel{fig:ratgraph7}{{5.28}{211}{The complete graph $y=\frac {2x^2-3x-5}{x^2-x-6}$ for Example \ref {ex_ratgraph2}\relax }{figure.caption.335}{}}
\newlabel{eex_ratgraph2}{{24}{211}{Graphs of Rational Functions}{figure.caption.334}{}}
\pgfsyspdfmark {pgfid342}{4736286}{40317094}
\pgfsyspdfmark {pgfid343}{4474142}{37491381}
\newlabel{ex_ratgraph3}{{89}{211}{Graphs of Rational Functions}{examplecounter.89}{}}
\pgfsyspdfmark {pgfid344}{9979166}{37327541}
\@writefile{lof}{\contentsline {figure}{\numberline {5.29}{\ignorespaces The behaviour of $y=h(x)$ near the hole and vertical asymptote\relax }}{212}{figure.caption.336}}
\newlabel{fig:ratgraph8}{{5.29}{212}{The behaviour of $y=h(x)$ near the hole and vertical asymptote\relax }{figure.caption.336}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.30}{\ignorespaces End behaviour for $y=h(x)$\relax }}{212}{figure.caption.337}}
\newlabel{fig:ratgraph9}{{5.30}{212}{End behaviour for $y=h(x)$\relax }{figure.caption.337}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.31}{\ignorespaces The sign diagram for  $h(x)=\frac  {2x^3+5x^2+4x+1}{x^2+3x+2}$\relax }}{212}{figure.caption.338}}
\newlabel{fig:ratsd4}{{5.31}{212}{The sign diagram for\\ $h(x)=\frac {2x^3+5x^2+4x+1}{x^2+3x+2}$\relax }{figure.caption.338}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.32}{\ignorespaces The graph $y=h(x)$ for Example \ref  {ex_ratgraph3}\relax }}{212}{figure.caption.339}}
\newlabel{fig:ratgraph10}{{5.32}{212}{The graph $y=h(x)$ for Example \ref {ex_ratgraph3}\relax }{figure.caption.339}{}}
\newlabel{eex_ratgraph3}{{24}{212}{Graphs of Rational Functions}{Item.1497}{}}
\pgfsyspdfmark {pgfid345}{33307643}{6741671}
\@writefile{lof}{\contentsline {figure}{\numberline {5.33}{\ignorespaces Comparing $y=r(x)$ to $y=x^2-1$\relax }}{213}{figure.caption.340}}
\newlabel{fig:ratgraph11}{{5.33}{213}{Comparing $y=r(x)$ to $y=x^2-1$\relax }{figure.caption.340}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.34}{\ignorespaces The limitations of our Precalculus methods\relax }}{213}{figure.caption.341}}
\newlabel{fig:ratgraph12}{{5.34}{213}{The limitations of our Precalculus methods\relax }{figure.caption.341}{}}
\pgfsyspdfmark {pgfid346}{4474142}{44905266}
\newlabel{calcisneededhere}{{90}{213}{Graphs of Rational Functions}{examplecounter.90}{}}
\newlabel{ecalcisneededhere}{{24}{213}{Graphs of Rational Functions}{Item.1503}{}}
\pgfsyspdfmark {pgfid347}{5719326}{10883732}
\@writefile{toc}{\contentsline {section}{\numberline {5.3}Rational Inequalities and Applications}{215}{section.5.3}}
\newlabel{RationalIneq}{{5.3}{215}{Rational Inequalities and Applications}{section.5.3}{}}
\pgfsyspdfmark {pgfid348}{4474142}{41897977}
\newlabel{ex_ratineq}{{91}{215}{Rational Inequalities and Applications}{examplecounter.91}{}}
\pgfsyspdfmark {pgfid349}{9979166}{41734137}
\@writefile{lof}{\contentsline {figure}{\numberline {5.35}{\ignorespaces The sign diagram for the inequality in Example \ref  {ex_ratineq}\relax }}{216}{figure.caption.346}}
\newlabel{fig:ratineq1}{{5.35}{216}{The sign diagram for the inequality in Example \ref {ex_ratineq}\relax }{figure.caption.346}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.36}{\ignorespaces The initial plot of $f(x)$ and $g(x)$\relax }}{216}{figure.caption.347}}
\newlabel{fig:ratineq2}{{5.36}{216}{The initial plot of $f(x)$ and $g(x)$\relax }{figure.caption.347}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.37}{\ignorespaces Zooming in to find the intersection points\relax }}{216}{figure.caption.348}}
\newlabel{fig:ratineq3}{{5.37}{216}{Zooming in to find the intersection points\relax }{figure.caption.348}{}}
\newlabel{eex_ratineq}{{5.3}{216}{Rational Inequalities and Applications}{Item.1536}{}}
\pgfsyspdfmark {pgfid350}{14550631}{26132404}
\pgfsyspdfmark {pgfid351}{14288487}{23137066}
\newlabel{upstreamdownstreamex}{{92}{216}{Rational Inequalities and Applications}{examplecounter.92}{}}
\pgfsyspdfmark {pgfid352}{19793511}{22973226}
\pgfsyspdfmark {pgfid353}{5719326}{24789202}
\newlabel{eupstreamdownstreamex}{{5.3}{218}{Rational Inequalities and Applications}{figure.caption.348}{}}
\pgfsyspdfmark {pgfid354}{15533671}{41988914}
\pgfsyspdfmark {pgfid355}{14288487}{37495374}
\newlabel{workex}{{93}{218}{Rational Inequalities and Applications}{examplecounter.93}{}}
\pgfsyspdfmark {pgfid356}{19793511}{37331534}
\@writefile{lof}{\contentsline {figure}{\numberline {5.38}{\ignorespaces The sign digram for $r(x)$\relax }}{219}{figure.caption.349}}
\newlabel{fig:ratineqsd}{{5.38}{219}{The sign digram for $r(x)$\relax }{figure.caption.349}{}}
\newlabel{eworkex}{{5.3}{219}{Rational Inequalities and Applications}{examplecounter.93}{}}
\pgfsyspdfmark {pgfid357}{16500232}{45134642}
\pgfsyspdfmark {pgfid358}{4474142}{36487326}
\newlabel{ex_ratcost}{{94}{219}{Rational Inequalities and Applications}{examplecounter.94}{}}
\pgfsyspdfmark {pgfid359}{9979166}{36323486}
\@writefile{lof}{\contentsline {figure}{\numberline {5.39}{\ignorespaces The box in Example \ref  {boxnotopfixedvolume}\relax }}{220}{figure.caption.350}}
\newlabel{fig:ratineq4}{{5.39}{220}{The box in Example \ref {boxnotopfixedvolume}\relax }{figure.caption.350}{}}
\newlabel{eex_ratcost}{{5.3}{220}{Rational Inequalities and Applications}{Item.1542}{}}
\pgfsyspdfmark {pgfid360}{14550631}{29756451}
\pgfsyspdfmark {pgfid361}{14288487}{27489153}
\newlabel{boxnotopfixedvolume}{{95}{220}{Rational Inequalities and Applications}{examplecounter.95}{}}
\pgfsyspdfmark {pgfid362}{19793511}{27325313}
\@writefile{lof}{\contentsline {figure}{\numberline {5.40}{\ignorespaces The sign digram for $h(x)$\relax }}{221}{figure.caption.351}}
\newlabel{fig:ratineqsd2}{{5.40}{221}{The sign digram for $h(x)$\relax }{figure.caption.351}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.41}{\ignorespaces Minimizing the surface area in Example \ref  {boxnotopfixedvolume}\relax }}{221}{figure.caption.352}}
\newlabel{fig:ratineqlast}{{5.41}{221}{Minimizing the surface area in Example \ref {boxnotopfixedvolume}\relax }{figure.caption.352}{}}
\pgfsyspdfmark {pgfid363}{6374688}{25877005}
\newlabel{eboxnotopfixedvolume}{{5.3}{222}{Rational Inequalities and Applications}{figure.caption.352}{}}
\pgfsyspdfmark {pgfid364}{14550631}{44632518}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.1}Variation}{222}{subsection.5.3.1}}
\newlabel{Variation}{{5.3.1}{222}{Variation}{subsection.5.3.1}{}}
\newlabel{variation}{{44}{222}{Variation}{definitioncounter.44}{}}
\pgfsyspdfmark {pgfid366}{14288487}{23652595}
\newlabel{ex_ratvariation}{{96}{222}{Variation}{examplecounter.96}{}}
\pgfsyspdfmark {pgfid367}{19793511}{23488755}
\newlabel{gravitylaw}{{5}{222}{Variation}{Item.1557}{}}
\newlabel{eex_ratvariation}{{44}{223}{Variation}{Item.1562}{}}
\pgfsyspdfmark {pgfid368}{4736286}{33226446}
\newlabel{fixedareaminperimetergarden}{{29}{225}{Variation}{Item.1591}{}}
\newlabel{BMIfirst}{{40a}{225}{Variation}{Item.1607}{}}
\newlabel{BMIsecond}{{40b}{225}{Variation}{Item.1608}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.1}{\ignorespaces Composition of functions\relax }}{227}{figure.caption.353}}
\newlabel{fig:funccomp1}{{6.1}{227}{Composition of functions\relax }{figure.caption.353}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {6}Function Composition and Inverses}{227}{chapter.6}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Further}{{6}{227}{Function Composition and Inverses}{chapter.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.1}Function Composition}{227}{section.6.1}}
\newlabel{FunctionComposition}{{6.1}{227}{Function Composition}{section.6.1}{}}
\newlabel{functioncompositiondefn}{{45}{227}{Function Composition}{definitioncounter.45}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.2}{\ignorespaces The sign diagram of $r(x)=x^2-4x+3$\relax }}{228}{figure.caption.354}}
\newlabel{fig:funcompsd1}{{6.2}{228}{The sign diagram of $r(x)=x^2-4x+3$\relax }{figure.caption.354}{}}
\pgfsyspdfmark {pgfid370}{14288487}{45743604}
\newlabel{functioncompex0}{{97}{228}{Function Composition}{examplecounter.97}{}}
\newlabel{efunctioncompex0}{{45}{228}{Function Composition}{Item.1616}{}}
\pgfsyspdfmark {pgfid371}{14550631}{29046538}
\pgfsyspdfmark {pgfid372}{14288487}{28082636}
\newlabel{functioncompex1}{{98}{228}{Function Composition}{examplecounter.98}{}}
\pgfsyspdfmark {pgfid373}{19793511}{27918796}
\newlabel{efunctioncompex1}{{45}{229}{Function Composition}{AMS.356}{}}
\pgfsyspdfmark {pgfid374}{11020376}{28918375}
\pgfsyspdfmark {pgfid375}{4474142}{25808574}
\newlabel{functioncompex2}{{99}{229}{Function Composition}{examplecounter.99}{}}
\pgfsyspdfmark {pgfid376}{9979166}{25644734}
\@writefile{lof}{\contentsline {figure}{\numberline {6.3}{\ignorespaces The sign diagram of  $r(x)=\genfrac  {}{}{}0{5x+3}{x+1}$\relax }}{230}{figure.caption.361}}
\newlabel{fig:funcompsd2}{{6.3}{230}{The sign diagram of \\$r(x)=\dfrac {5x+3}{x+1}$\relax }{figure.caption.361}{}}
\newlabel{efunctioncompex2}{{45}{230}{Function Composition}{AMS.360}{}}
\pgfsyspdfmark {pgfid377}{27224113}{4736286}
\pgfsyspdfmark {pgfid378}{4474142}{46871346}
\newlabel{functioncompex3}{{100}{231}{Function Composition}{examplecounter.100}{}}
\newlabel{efunctioncompex3}{{45}{231}{Function Composition}{examplecounter.100}{}}
\pgfsyspdfmark {pgfid379}{22223113}{4736286}
\pgfsyspdfmark {pgfid380}{14288487}{44789545}
\newlabel{functioncompex4}{{101}{232}{Function Composition}{examplecounter.101}{}}
\pgfsyspdfmark {pgfid381}{19793511}{44625705}
\newlabel{efunctioncompex4}{{45}{233}{Function Composition}{examplecounter.101}{}}
\pgfsyspdfmark {pgfid382}{8990257}{27523524}
\newlabel{functioncompprops}{{34}{233}{Function Composition}{theoremcounter.34}{}}
\pgfsyspdfmark {pgfid384}{14288487}{33097659}
\newlabel{ex_blowshere}{{102}{234}{Function Composition}{examplecounter.102}{}}
\newlabel{eex_blowshere}{{34}{234}{Function Composition}{examplecounter.102}{}}
\pgfsyspdfmark {pgfid385}{23694097}{22621759}
\pgfsyspdfmark {pgfid386}{14288487}{19617617}
\newlabel{ex_decomp}{{103}{234}{Function Composition}{examplecounter.103}{}}
\pgfsyspdfmark {pgfid387}{19793511}{19453777}
\newlabel{eex_decomp}{{34}{235}{Function Composition}{Item.1622}{}}
\pgfsyspdfmark {pgfid388}{4736286}{24217933}
\@writefile{lof}{\contentsline {figure}{\numberline {6.4}{\ignorespaces The relationship between a function and its inverse\relax }}{238}{figure.caption.362}}
\newlabel{fig:inverse1}{{6.4}{238}{The relationship between a function and its inverse\relax }{figure.caption.362}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.2}Inverse Functions}{238}{section.6.2}}
\newlabel{InverseFunctions}{{6.2}{238}{Inverse Functions}{section.6.2}{}}
\newlabel{inversefunctiondefn}{{46}{238}{Inverse Functions}{definitioncounter.46}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.5}{\ignorespaces Reflecting $y=f(x)$ across $y=x$ to obtain $y=g(x)$\relax }}{239}{figure.caption.364}}
\newlabel{fig:inverse2}{{6.5}{239}{Reflecting $y=f(x)$ across $y=x$ to obtain $y=g(x)$\relax }{figure.caption.364}{}}
\newlabel{inversefunctionprops}{{35}{239}{Inverse Functions}{theoremcounter.35}{}}
\newlabel{inversefunctionuniqueness}{{35}{239}{Inverse Functions}{section*.363}{}}
\newlabel{inverseuniquegraph}{{36}{239}{Inverse Functions}{theoremcounter.36}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.6}{\ignorespaces The function $f(x)=x^2$ is not invertible\relax }}{240}{figure.caption.365}}
\newlabel{fig:inverse3}{{6.6}{240}{The function $f(x)=x^2$ is not invertible\relax }{figure.caption.365}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.7}{\ignorespaces Reflecting $y=x^2$ across the line $y=x$ does not produce a function\relax }}{240}{figure.caption.366}}
\newlabel{fig:inverse4}{{6.7}{240}{Reflecting $y=x^2$ across the line $y=x$ does not produce a function\relax }{figure.caption.366}{}}
\newlabel{onetoone}{{47}{240}{Inverse Functions}{definitioncounter.47}{}}
\newlabel{HLT}{{37}{240}{Inverse Functions}{theoremcounter.37}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.8}{\ignorespaces The function $f$ is one-to-one\relax }}{241}{figure.caption.367}}
\newlabel{fig:inverse5}{{6.8}{241}{The function $f$ is one-to-one\relax }{figure.caption.367}{}}
\newlabel{inversefunctionequivalency}{{38}{241}{Inverse Functions}{theoremcounter.38}{}}
\pgfsyspdfmark {pgfid395}{4474142}{37705812}
\newlabel{inversefunctiononetooneex}{{104}{241}{Inverse Functions}{examplecounter.104}{}}
\pgfsyspdfmark {pgfid396}{9979166}{37541972}
\@writefile{lof}{\contentsline {figure}{\numberline {6.9}{\ignorespaces The function $g$ is one-to-one\relax }}{242}{figure.caption.368}}
\newlabel{fig:inverse6}{{6.9}{242}{The function $g$ is one-to-one\relax }{figure.caption.368}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.10}{\ignorespaces The function $h$ is not one-to-one\relax }}{242}{figure.caption.369}}
\newlabel{fig:inverse7}{{6.10}{242}{The function $h$ is not one-to-one\relax }{figure.caption.369}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.11}{\ignorespaces The function $F$ is not one-to-one\relax }}{242}{figure.caption.370}}
\newlabel{fig:inverse8}{{6.11}{242}{The function $F$ is not one-to-one\relax }{figure.caption.370}{}}
\newlabel{einversefunctiononetooneex}{{38}{242}{Inverse Functions}{Item.1712}{}}
\pgfsyspdfmark {pgfid397}{15533671}{13334840}
\newlabel{inverseprocedure}{{25}{243}{Inverse Functions}{keyideacounter.25}{}}
\pgfsyspdfmark {pgfid399}{4474142}{36637913}
\newlabel{ex_inverse1}{{105}{243}{Inverse Functions}{examplecounter.105}{}}
\pgfsyspdfmark {pgfid400}{9979166}{36474073}
\@writefile{lof}{\contentsline {figure}{\numberline {6.12}{\ignorespaces The graphs of $f$ and $f^{-1}$ from Example \ref  {ex_inverse1}\relax }}{244}{figure.caption.379}}
\newlabel{fig:inverse9}{{6.12}{244}{The graphs of $f$ and $f^{-1}$ from Example \ref {ex_inverse1}\relax }{figure.caption.379}{}}
\pgfsyspdfmark {pgfid401}{16189033}{23966184}
\@writefile{lof}{\contentsline {figure}{\numberline {6.13}{\ignorespaces The graphs of $g$ and $g^{-1}$ from Example \ref  {ex_inverse1}\relax }}{245}{figure.caption.380}}
\newlabel{fig:inverse10}{{6.13}{245}{The graphs of $g$ and $g^{-1}$ from Example \ref {ex_inverse1}\relax }{figure.caption.380}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.14}{\ignorespaces Restricting the domain of $f(x)=x^2$\relax }}{245}{figure.caption.381}}
\newlabel{fig:inverse11}{{6.14}{245}{Restricting the domain of $f(x)=x^2$\relax }{figure.caption.381}{}}
\newlabel{eex_inverse1}{{25}{245}{Inverse Functions}{figure.caption.379}{}}
\pgfsyspdfmark {pgfid402}{4736286}{22484662}
\@writefile{lof}{\contentsline {figure}{\numberline {6.15}{\ignorespaces The restricted function $g$ and its inverse\relax }}{246}{figure.caption.382}}
\newlabel{fig:inverse12}{{6.15}{246}{The restricted function $g$ and its inverse\relax }{figure.caption.382}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.16}{\ignorespaces $y=x^2-2x+4$, for $x\leq 1$\relax }}{246}{figure.caption.383}}
\newlabel{fig:inverse13}{{6.16}{246}{$y=x^2-2x+4$, for $x\leq 1$\relax }{figure.caption.383}{}}
\pgfsyspdfmark {pgfid403}{14288487}{46871346}
\newlabel{inverserestrictionex}{{106}{246}{Inverse Functions}{examplecounter.106}{}}
\pgfsyspdfmark {pgfid404}{19793511}{46707506}
\@writefile{lof}{\contentsline {figure}{\numberline {6.17}{\ignorespaces The graphs of $j$ and $j^{-1}$ from Example \ref  {inverserestrictionex}\relax }}{247}{figure.caption.384}}
\newlabel{fig:inverse14}{{6.17}{247}{The graphs of $j$ and $j^{-1}$ from Example \ref {inverserestrictionex}\relax }{figure.caption.384}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.18}{\ignorespaces $y=\sqrt  {x+2}-1$\relax }}{247}{figure.caption.385}}
\newlabel{fig:inverse15}{{6.18}{247}{$y=\sqrt {x+2}-1$\relax }{figure.caption.385}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.19}{\ignorespaces The graphs of $k$ and $k^{-1}$ from Example \ref  {inverserestrictionex}\relax }}{247}{figure.caption.386}}
\newlabel{fig:inverse16}{{6.19}{247}{The graphs of $k$ and $k^{-1}$ from Example \ref {inverserestrictionex}\relax }{figure.caption.386}{}}
\pgfsyspdfmark {pgfid405}{6374688}{27851158}
\newlabel{einverserestrictionex}{{25}{247}{Inverse Functions}{Item.1723}{}}
\pgfsyspdfmark {pgfid406}{4736286}{3163422}
\pgfsyspdfmark {pgfid407}{14288487}{45691698}
\newlabel{demandfunctionofprice}{{107}{248}{Inverse Functions}{examplecounter.107}{}}
\newlabel{edemandfunctionofprice}{{25}{248}{Inverse Functions}{Item.1731}{}}
\pgfsyspdfmark {pgfid408}{14550631}{5909271}
\newlabel{genericinverselast}{{24}{249}{Inverse Functions}{Item.1755}{}}
\newlabel{fcircginverse}{{31}{249}{Inverse Functions}{Item.1765}{}}
\newlabel{whatconditions}{{33}{249}{Inverse Functions}{Item.1767}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.20}{\ignorespaces Graphs of the first three even root functions\relax }}{250}{figure.caption.387}}
\newlabel{fig:algfun1}{{6.20}{250}{Graphs of the first three even root functions\relax }{figure.caption.387}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.21}{\ignorespaces Graphs of the first three odd root functions\relax }}{250}{figure.caption.388}}
\newlabel{fig:algfun2}{{6.21}{250}{Graphs of the first three odd root functions\relax }{figure.caption.388}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.3}Algebraic Functions}{250}{section.6.3}}
\newlabel{AlgebraicFunctions}{{6.3}{250}{Algebraic Functions}{section.6.3}{}}
\newlabel{principalnthrootdefn2}{{48}{250}{Algebraic Functions}{definitioncounter.48}{}}
\newlabel{radicalprops2}{{39}{250}{Algebraic Functions}{theoremcounter.39}{}}
\newlabel{rationalexponentdefn2}{{49}{251}{Algebraic Functions}{definitioncounter.49}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.22}{\ignorespaces $f(x) = 3x\sqrt  [3]{2-x}$\relax }}{252}{figure.caption.389}}
\newlabel{fig:algfun3}{{6.22}{252}{$f(x) = 3x\sqrt [3]{2-x}$\relax }{figure.caption.389}{}}
\newlabel{algebraicsigndiagram}{{26}{252}{Algebraic Functions}{keyideacounter.26}{}}
\pgfsyspdfmark {pgfid413}{14288487}{31038997}
\newlabel{ex_algfun1}{{108}{252}{Algebraic Functions}{examplecounter.108}{}}
\pgfsyspdfmark {pgfid414}{19793511}{30875157}
\@writefile{lof}{\contentsline {figure}{\numberline {6.23}{\ignorespaces Sign diagram for $r(x)=2-\sqrt  [4]{x+3}$\relax }}{253}{figure.caption.390}}
\newlabel{fig:algfun4}{{6.23}{253}{Sign diagram for $r(x)=2-\sqrt [4]{x+3}$\relax }{figure.caption.390}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.24}{\ignorespaces $g(x)=\sqrt  {2-\sqrt  [4]{x+3}}$\relax }}{253}{figure.caption.391}}
\newlabel{fig:algfun5}{{6.24}{253}{$g(x)=\sqrt {2-\sqrt [4]{x+3}}$\relax }{figure.caption.391}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.25}{\ignorespaces $h(x)=\sqrt  [3]{\genfrac  {}{}{}0{8x}{x+1}}$\relax }}{253}{figure.caption.392}}
\newlabel{fig:algfun6}{{6.25}{253}{$h(x)=\sqrt [3]{\dfrac {8x}{x+1}}$\relax }{figure.caption.392}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.26}{\ignorespaces The sign diagram of $r(x)=x^2-1$\relax }}{253}{figure.caption.393}}
\newlabel{fig:algfun7}{{6.26}{253}{The sign diagram of $r(x)=x^2-1$\relax }{figure.caption.393}{}}
\pgfsyspdfmark {pgfid415}{6374688}{26120851}
\@writefile{lof}{\contentsline {figure}{\numberline {6.27}{\ignorespaces $k(x)=\genfrac  {}{}{}0{2x}{\sqrt  {x^2-1}}$\relax }}{254}{figure.caption.394}}
\newlabel{fig:algfun8}{{6.27}{254}{$k(x)=\dfrac {2x}{\sqrt {x^2-1}}$\relax }{figure.caption.394}{}}
\newlabel{eex_algfun1}{{26}{254}{Algebraic Functions}{figure.caption.393}{}}
\pgfsyspdfmark {pgfid416}{14550631}{38121030}
\pgfsyspdfmark {pgfid417}{14288487}{26384567}
\newlabel{ex_algfun2}{{109}{254}{Algebraic Functions}{examplecounter.109}{}}
\pgfsyspdfmark {pgfid418}{19793511}{26220727}
\@writefile{lof}{\contentsline {figure}{\numberline {6.28}{\ignorespaces Sign diagram and graph for Example \ref  {ex_algfun2}.1\relax }}{255}{figure.caption.395}}
\newlabel{fig:algfun9}{{6.28}{255}{Sign diagram and graph for Example \ref {ex_algfun2}.1\relax }{figure.caption.395}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.29}{\ignorespaces The graphs of $f$ and $g$ near $x=0$\relax }}{255}{figure.caption.396}}
\newlabel{fig:algfun10}{{6.29}{255}{The graphs of $f$ and $g$ near $x=0$\relax }{figure.caption.396}{}}
\pgfsyspdfmark {pgfid419}{6374688}{23582800}
\@writefile{lof}{\contentsline {figure}{\numberline {6.30}{\ignorespaces Sign diagram and graph for Example \ref  {ex_algfun2}.2\relax }}{256}{figure.caption.397}}
\newlabel{fig:algfun11}{{6.30}{256}{Sign diagram and graph for Example \ref {ex_algfun2}.2\relax }{figure.caption.397}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.31}{\ignorespaces Diagram for Example \ref  {SasquatchCable}\relax }}{256}{figure.caption.398}}
\newlabel{fig:algfun12}{{6.31}{256}{Diagram for Example \ref {SasquatchCable}\relax }{figure.caption.398}{}}
\newlabel{eex_algfun2}{{26}{256}{Algebraic Functions}{figure.caption.396}{}}
\pgfsyspdfmark {pgfid420}{14550631}{24238028}
\pgfsyspdfmark {pgfid421}{14288487}{21287952}
\newlabel{SasquatchCable}{{110}{256}{Algebraic Functions}{examplecounter.110}{}}
\pgfsyspdfmark {pgfid422}{19793511}{21124112}
\newlabel{eSasquatchCable}{{26}{257}{Algebraic Functions}{Item.1787}{}}
\pgfsyspdfmark {pgfid423}{4736286}{31673542}
\newlabel{furtherequineqexercises}{{26}{258}{Algebraic Functions}{Item.1803}{}}
\newlabel{heightintermsofr}{{37a}{258}{Algebraic Functions}{Item.1825}{}}
\newlabel{WindChillTemperature}{{38}{258}{Algebraic Functions}{Item.1828}{}}
\newlabel{WindChill0}{{39b}{259}{Algebraic Functions}{Item.1833}{}}
\newlabel{pendulumproblem}{{40}{259}{Algebraic Functions}{Item.1835}{}}
\newlabel{pursuitfurther}{{44}{259}{Algebraic Functions}{Item.1845}{}}
\newlabel{nthrootsareirrational}{{47}{259}{Algebraic Functions}{Item.1851}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.1}{\ignorespaces Plotting $f(x)=2^x$\relax }}{261}{figure.caption.399}}
\newlabel{fig:introexp1}{{7.1}{261}{Plotting $f(x)=2^x$\relax }{figure.caption.399}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {7}Exponential and Logarithmic Functions}{261}{chapter.7}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{ExpLogs}{{7}{261}{Exponential and Logarithmic Functions}{chapter.7}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7.1}Introduction to Exponential and Logarithmic Functions}{261}{section.7.1}}
\newlabel{IntroExpLogs}{{7.1}{261}{Introduction to Exponential and Logarithmic Functions}{section.7.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.2}{\ignorespaces Reflecting $y=2^x$ across the $y$-axis to obtain the graph $y=2^{-x}$\relax }}{262}{figure.caption.401}}
\newlabel{fig:introexp2}{{7.2}{262}{Reflecting $y=2^x$ across the $y$-axis to obtain the graph $y=2^{-x}$\relax }{figure.caption.401}{}}
\newlabel{indeterminantformtwo}{{7.1}{262}{Introduction to Exponential and Logarithmic Functions}{section*.400}{}}
\newlabel{expfcndefn}{{50}{262}{Introduction to Exponential and Logarithmic Functions}{definitioncounter.50}{}}
\newlabel{expfcnprops}{{40}{263}{Introduction to Exponential and Logarithmic Functions}{theoremcounter.40}{}}
\pgfsyspdfmark {pgfid426}{4474142}{17859128}
\newlabel{cardepreciationex}{{111}{263}{Introduction to Exponential and Logarithmic Functions}{examplecounter.111}{}}
\pgfsyspdfmark {pgfid427}{9979166}{17695288}
\@writefile{lof}{\contentsline {figure}{\numberline {7.3}{\ignorespaces The graph $y=V(x)$ in Example \ref  {cardepreciationex}\relax }}{264}{figure.caption.402}}
\newlabel{fig:introexp3}{{7.3}{264}{The graph $y=V(x)$ in Example \ref {cardepreciationex}\relax }{figure.caption.402}{}}
\newlabel{ecardepreciationex}{{40}{264}{Introduction to Exponential and Logarithmic Functions}{Item.1858}{}}
\pgfsyspdfmark {pgfid428}{14550631}{33544542}
\pgfsyspdfmark {pgfid429}{14288487}{26664579}
\newlabel{exptempex}{{112}{264}{Introduction to Exponential and Logarithmic Functions}{examplecounter.112}{}}
\pgfsyspdfmark {pgfid430}{19793511}{26500739}
\@writefile{lof}{\contentsline {figure}{\numberline {7.4}{\ignorespaces Graphing $T(t)$ in Example \ref  {exptempex}\relax }}{265}{figure.caption.403}}
\newlabel{fig:introexp4}{{7.4}{265}{Graphing $T(t)$ in Example \ref {exptempex}\relax }{figure.caption.403}{}}
\newlabel{eexptempex}{{40}{265}{Introduction to Exponential and Logarithmic Functions}{Item.1864}{}}
\pgfsyspdfmark {pgfid431}{4736286}{25749736}
\newlabel{logfcndefn}{{51}{265}{Introduction to Exponential and Logarithmic Functions}{definitioncounter.51}{}}
\newlabel{def:natlog}{{52}{265}{Introduction to Exponential and Logarithmic Functions}{definitioncounter.52}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.5}{\ignorespaces The logarithm is the inverse of the exponential function\relax }}{266}{figure.caption.404}}
\newlabel{fig:introexp5}{{7.5}{266}{The logarithm is the inverse of the exponential function\relax }{figure.caption.404}{}}
\newlabel{logfcnprops}{{41}{266}{Introduction to Exponential and Logarithmic Functions}{theoremcounter.41}{}}
\pgfsyspdfmark {pgfid435}{4474142}{39323533}
\newlabel{ex_logprops}{{113}{267}{Introduction to Exponential and Logarithmic Functions}{examplecounter.113}{}}
\newlabel{eex_logprops}{{52}{267}{Introduction to Exponential and Logarithmic Functions}{Item.1878}{}}
\pgfsyspdfmark {pgfid436}{4736286}{3949854}
\@writefile{lof}{\contentsline {figure}{\numberline {7.6}{\ignorespaces $y=f(x)=2\qopname  \relax o{log}(3-x)-1$\relax }}{268}{figure.caption.405}}
\newlabel{fig:introexp6}{{7.6}{268}{$y=f(x)=2\log (3-x)-1$\relax }{figure.caption.405}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.7}{\ignorespaces Sign diagram for $r(x) = \frac  {x}{x-1}$\relax }}{268}{figure.caption.406}}
\newlabel{fig:introexpsd1}{{7.7}{268}{Sign diagram for $r(x) = \frac {x}{x-1}$\relax }{figure.caption.406}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.8}{\ignorespaces $y=g(x)=\qopname  \relax o{ln}\left (\frac  {x}{x-1}\right )$\relax }}{268}{figure.caption.407}}
\newlabel{fig:introexp7}{{7.8}{268}{$y=g(x)=\ln \left (\frac {x}{x-1}\right )$\relax }{figure.caption.407}{}}
\pgfsyspdfmark {pgfid437}{14288487}{42488325}
\newlabel{ex_logdomain}{{114}{268}{Introduction to Exponential and Logarithmic Functions}{examplecounter.114}{}}
\newlabel{eex_logdomain}{{52}{268}{Introduction to Exponential and Logarithmic Functions}{Item.1882}{}}
\pgfsyspdfmark {pgfid438}{14550631}{8332875}
\@writefile{lof}{\contentsline {figure}{\numberline {7.9}{\ignorespaces Graphing $f(x)=2^{x-1}-3$ in Example \ref  {proceduralinverse}\relax }}{269}{figure.caption.408}}
\newlabel{fig:introexp8}{{7.9}{269}{Graphing $f(x)=2^{x-1}-3$ in Example \ref {proceduralinverse}\relax }{figure.caption.408}{}}
\pgfsyspdfmark {pgfid439}{4474142}{46871346}
\newlabel{proceduralinverse}{{115}{269}{Introduction to Exponential and Logarithmic Functions}{examplecounter.115}{}}
\pgfsyspdfmark {pgfid440}{9979166}{46707506}
\@writefile{lof}{\contentsline {figure}{\numberline {7.10}{\ignorespaces Graphing $f^{-1}(x)=\qopname  \relax o{log}_2(x+3)+1$ in Example \ref  {proceduralinverse}\relax }}{270}{figure.caption.413}}
\newlabel{fig:introexp9}{{7.10}{270}{Graphing $f^{-1}(x)=\log _2(x+3)+1$ in Example \ref {proceduralinverse}\relax }{figure.caption.413}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.11}{\ignorespaces The graphs of $f$ and $f^{-1}$ in Example \ref  {proceduralinverse}\relax }}{270}{figure.caption.414}}
\newlabel{fig:introexp10}{{7.11}{270}{The graphs of $f$ and $f^{-1}$ in Example \ref {proceduralinverse}\relax }{figure.caption.414}{}}
\newlabel{eproceduralinverse}{{52}{270}{Introduction to Exponential and Logarithmic Functions}{Item.1898}{}}
\pgfsyspdfmark {pgfid441}{14550631}{16593345}
\newlabel{graphexpfirst}{{58}{272}{Introduction to Exponential and Logarithmic Functions}{Item.1956}{}}
\newlabel{graphexplast}{{63}{272}{Introduction to Exponential and Logarithmic Functions}{Item.1961}{}}
\newlabel{graphlogfirst}{{64}{272}{Introduction to Exponential and Logarithmic Functions}{Item.1962}{}}
\newlabel{graphloglast}{{69}{272}{Introduction to Exponential and Logarithmic Functions}{Item.1967}{}}
\newlabel{logarithmicscales}{{52}{272}{Introduction to Exponential and Logarithmic Functions}{Item.1972}{}}
\newlabel{Richterexercise}{{75}{272}{Introduction to Exponential and Logarithmic Functions}{Item.1973}{}}
\newlabel{decibelexercise}{{76}{272}{Introduction to Exponential and Logarithmic Functions}{Item.1978}{}}
\newlabel{pHexercise}{{77}{273}{Introduction to Exponential and Logarithmic Functions}{Item.1982}{}}
\newlabel{IntroExpLogsExercises}{{52}{274}{Introduction to Exponential and Logarithmic Functions}{Item.1987}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7.2}Properties of Logarithms}{274}{section.7.2}}
\newlabel{LogProperties}{{7.2}{274}{Properties of Logarithms}{section.7.2}{}}
\newlabel{invpropslogs}{{42}{274}{Properties of Logarithms}{theoremcounter.42}{}}
\newlabel{explogsonetoone}{{43}{274}{Properties of Logarithms}{theoremcounter.43}{}}
\newlabel{algpropexpfcns}{{44}{275}{Properties of Logarithms}{theoremcounter.44}{}}
\newlabel{algproplogfcns}{{45}{275}{Properties of Logarithms}{theoremcounter.45}{}}
\pgfsyspdfmark {pgfid446}{14288487}{45376138}
\newlabel{expandlogex}{{116}{276}{Properties of Logarithms}{examplecounter.116}{}}
\pgfsyspdfmark {pgfid447}{19793511}{45212298}
\newlabel{eexpandlogex}{{45}{277}{Properties of Logarithms}{AMS.446}{}}
\pgfsyspdfmark {pgfid448}{4736286}{4736286}
\pgfsyspdfmark {pgfid449}{14288487}{28400107}
\newlabel{contractlogex}{{117}{278}{Properties of Logarithms}{examplecounter.117}{}}
\pgfsyspdfmark {pgfid450}{19793511}{28236267}
\newlabel{econtractlogex}{{45}{279}{Properties of Logarithms}{AMS.464}{}}
\pgfsyspdfmark {pgfid451}{4736286}{25086984}
\newlabel{changeofbase}{{46}{279}{Properties of Logarithms}{theoremcounter.46}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.12}{\ignorespaces $y=f(x)=2^x$ and $y=g(x)=e^{x\qopname  \relax o{ln}(2)}$\relax }}{280}{figure.caption.465}}
\newlabel{fig:logprop1}{{7.12}{280}{$y=f(x)=2^x$ and $y=g(x)=e^{x\ln (2)}$\relax }{figure.caption.465}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.13}{\ignorespaces $y=f(x)=2^x$ and $y=g(x)=e^{x\qopname  \relax o{ln}(2)}$\relax }}{280}{figure.caption.466}}
\newlabel{fig:logprop2}{{7.13}{280}{$y=f(x)=2^x$ and $y=g(x)=e^{x\ln (2)}$\relax }{figure.caption.466}{}}
\pgfsyspdfmark {pgfid453}{14288487}{30927912}
\newlabel{ex_basechange}{{118}{280}{Properties of Logarithms}{examplecounter.118}{}}
\newlabel{eex_basechange}{{46}{280}{Properties of Logarithms}{Item.2013}{}}
\pgfsyspdfmark {pgfid454}{15533671}{5205242}
\newlabel{HendersonHasselbalch}{{43}{282}{Properties of Logarithms}{Item.2059}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7.3}Exponential Equations and Inequalities}{283}{section.7.3}}
\newlabel{ExpEquations}{{7.3}{283}{Exponential Equations and Inequalities}{section.7.3}{}}
\newlabel{idea:expeqns}{{27}{283}{Exponential Equations and Inequalities}{keyideacounter.27}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.14}{\ignorespaces {\color  {gray} $y=f(x)=2^{3x}$} and  $y=g(x)=16^{1-x}$\relax }}{284}{figure.caption.471}}
\newlabel{fig:expeqn1}{{7.14}{284}{{\color {gray} $y=f(x)=2^{3x}$} and\\ $y=g(x)=16^{1-x}$\relax }{figure.caption.471}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.15}{\ignorespaces {\color  {gray} $y=f(x)=2000$} and  $y=g(x)=1000\cdot e^{-0.1x}$\relax }}{284}{figure.caption.472}}
\newlabel{fig:expeqn2}{{7.15}{284}{{\color {gray} $y=f(x)=2000$} and\\ $y=g(x)=1000\cdot e^{-0.1x}$\relax }{figure.caption.472}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.16}{\ignorespaces {\color  {gray} $y=f(x)=9\cdot 3^x$} and  $y=g(x)=7^{2x}$\relax }}{284}{figure.caption.473}}
\newlabel{fig:expeqn3}{{7.16}{284}{{\color {gray} $y=f(x)=9\cdot 3^x$} and\\ $y=g(x)=7^{2x}$\relax }{figure.caption.473}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.17}{\ignorespaces {\color  {red} $y=f(x)=75$} and  $y=g(x)=\frac  {100}{1+3e^{-2x}}$\relax }}{284}{figure.caption.474}}
\newlabel{fig:expeqn4}{{7.17}{284}{{\color {red} $y=f(x)=75$} and\\ $y=g(x)=\frac {100}{1+3e^{-2x}}$\relax }{figure.caption.474}{}}
\pgfsyspdfmark {pgfid456}{14288487}{46871346}
\newlabel{expeqnsex1}{{119}{284}{Exponential Equations and Inequalities}{examplecounter.119}{}}
\pgfsyspdfmark {pgfid457}{19793511}{46707506}
\@writefile{lof}{\contentsline {figure}{\numberline {7.18}{\ignorespaces {\color  {gray} $y=f(x)=25^x$} and  $y=g(x)=5^x+6$\relax }}{285}{figure.caption.475}}
\newlabel{fig:expeqn5}{{7.18}{285}{{\color {gray} $y=f(x)=25^x$} and\\ $y=g(x)=5^x+6$\relax }{figure.caption.475}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.19}{\ignorespaces {\color  {gray} $y=f(x)=\genfrac  {}{}{}0{e^x-e^{-x}}{2}$} and  $y=g(x)=5$\relax }}{285}{figure.caption.476}}
\newlabel{fig:expeqn6}{{7.19}{285}{{\color {gray} $y=f(x)=\dfrac {e^x-e^{-x}}{2}$} and\\ $y=g(x)=5$\relax }{figure.caption.476}{}}
\newlabel{eexpeqnsex1}{{27}{285}{Exponential Equations and Inequalities}{Item.2077}{}}
\pgfsyspdfmark {pgfid458}{4736286}{21313671}
\@writefile{lof}{\contentsline {figure}{\numberline {7.20}{\ignorespaces Solving $2^{x^2-3x}-16\geq 0$\relax }}{286}{figure.caption.479}}
\newlabel{fig:expineq1}{{7.20}{286}{Solving $2^{x^2-3x}-16\geq 0$\relax }{figure.caption.479}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.21}{\ignorespaces Solving $\genfrac  {}{}{}0{e^x}{e^x-4}\leq 3$\relax }}{286}{figure.caption.480}}
\newlabel{fig:expineq2}{{7.21}{286}{Solving $\dfrac {e^x}{e^x-4}\leq 3$\relax }{figure.caption.480}{}}
\pgfsyspdfmark {pgfid459}{14288487}{46871346}
\newlabel{expineq}{{120}{286}{Exponential Equations and Inequalities}{examplecounter.120}{}}
\pgfsyspdfmark {pgfid460}{19793511}{46707506}
\@writefile{lof}{\contentsline {figure}{\numberline {7.22}{\ignorespaces Solving $xe^{2x}<4x$\relax }}{287}{figure.caption.483}}
\newlabel{fig:expineq3}{{7.22}{287}{Solving $xe^{2x}<4x$\relax }{figure.caption.483}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.23}{\ignorespaces Solving $T(t)=100$ in Example \ref  {ex_newtoncool2}\relax }}{287}{figure.caption.484}}
\newlabel{fig:expnewton}{{7.23}{287}{Solving $T(t)=100$ in Example \ref {ex_newtoncool2}\relax }{figure.caption.484}{}}
\newlabel{eexpineq}{{27}{287}{Exponential Equations and Inequalities}{AMS.482}{}}
\pgfsyspdfmark {pgfid461}{4736286}{26076445}
\pgfsyspdfmark {pgfid462}{4474142}{24135965}
\newlabel{ex_newtoncool2}{{121}{287}{Exponential Equations and Inequalities}{examplecounter.121}{}}
\newlabel{eex_newtoncool2}{{27}{287}{Exponential Equations and Inequalities}{examplecounter.121}{}}
\pgfsyspdfmark {pgfid463}{12129032}{7627037}
\@writefile{lof}{\contentsline {figure}{\numberline {7.24}{\ignorespaces {\color  {gray} $y=f(x)=\frac  {5e^x}{e^x+1}$}  $y=g(x)=\qopname  \relax o{ln}\left (\frac  {x}{5-x}\right )$\relax }}{288}{figure.caption.487}}
\newlabel{fig:expinverse}{{7.24}{288}{{\color {gray} $y=f(x)=\frac {5e^x}{e^x+1}$}\\ $y=g(x)=\ln \left (\frac {x}{5-x}\right )$\relax }{figure.caption.487}{}}
\pgfsyspdfmark {pgfid464}{14288487}{46871346}
\newlabel{expfracinverse}{{122}{288}{Exponential Equations and Inequalities}{examplecounter.122}{}}
\newlabel{eexpfracinverse}{{27}{288}{Exponential Equations and Inequalities}{AMS.486}{}}
\pgfsyspdfmark {pgfid465}{15533671}{19825731}
\newlabel{expineqfirst}{{34}{289}{Exponential Equations and Inequalities}{Item.2117}{}}
\newlabel{expineqlast}{{39}{289}{Exponential Equations and Inequalities}{Item.2122}{}}
\newlabel{onetoonelogexercise}{{46}{289}{Exponential Equations and Inequalities}{Item.2129}{}}
\newlabel{hyperbolicsine}{{48}{289}{Exponential Equations and Inequalities}{Item.2131}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.25}{\ignorespaces {\color  {gray} $y=f(x)=\qopname  \relax o{log}_{117}(1-3x)$}   and $y=g(x)=\qopname  \relax o{log}_{117}(x^2-3)$\relax }}{291}{figure.caption.488}}
\newlabel{fig:logeqn1}{{7.25}{291}{{\color {gray} $y=f(x)=\log _{117}(1-3x)$} \\ and $y=g(x)=\log _{117}(x^2-3)$\relax }{figure.caption.488}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.26}{\ignorespaces {\color  {gray} $y=f(x)=2-\qopname  \relax o{ln}(x-3)$}   and $y=g(x)=1$\relax }}{291}{figure.caption.489}}
\newlabel{fig:logeqn2}{{7.26}{291}{{\color {gray} $y=f(x)=2-\ln (x-3)$} \\ and $y=g(x)=1$\relax }{figure.caption.489}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7.4}Logarithmic Equations and Inequalities}{291}{section.7.4}}
\newlabel{LogEquations}{{7.4}{291}{Logarithmic Equations and Inequalities}{section.7.4}{}}
\newlabel{idea:logeqn}{{28}{291}{Logarithmic Equations and Inequalities}{keyideacounter.28}{}}
\pgfsyspdfmark {pgfid467}{4474142}{26095555}
\newlabel{LogEqnsEx1}{{123}{291}{Logarithmic Equations and Inequalities}{examplecounter.123}{}}
\pgfsyspdfmark {pgfid468}{9979166}{25931715}
\@writefile{lof}{\contentsline {figure}{\numberline {7.27}{\ignorespaces {\color  {gray} $y=f(x)=\qopname  \relax o{log}_6(x+4)+\qopname  \relax o{log}_6(3-x)$} and $y=g(x)=1$\relax }}{292}{figure.caption.494}}
\newlabel{fig:logeqn3}{{7.27}{292}{{\color {gray} $y=f(x)=\log _6(x+4)+\log _6(3-x)$} and $y=g(x)=1$\relax }{figure.caption.494}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.28}{\ignorespaces {\color  {gray} $y=f(x)=\qopname  \relax o{log}_7(1-2x)$}   and $y=g(x)=1-\qopname  \relax o{log}_7(3-x)$\relax }}{292}{figure.caption.495}}
\newlabel{fig:logeqn4}{{7.28}{292}{{\color {gray} $y=f(x)=\log _7(1-2x)$} \\ and $y=g(x)=1-\log _7(3-x)$\relax }{figure.caption.495}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.29}{\ignorespaces {\color  {gray} $y=f(x)=\qopname  \relax o{log}_2(x+3)$}   and $y=g(x)=log_2(6-x)+3$\relax }}{292}{figure.caption.496}}
\newlabel{fig:logeqn5}{{7.29}{292}{{\color {gray} $y=f(x)=\log _2(x+3)$} \\ and $y=g(x)=log_2(6-x)+3$\relax }{figure.caption.496}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.30}{\ignorespaces {\color  {gray} $y=f(x)=1+2\qopname  \relax o{log}_4(x+1)$}   and $y=g(x)=2\qopname  \relax o{log}_2(x)$\relax }}{292}{figure.caption.497}}
\newlabel{fig:logeqn6}{{7.30}{292}{{\color {gray} $y=f(x)=1+2\log _4(x+1)$} \\ and $y=g(x)=2\log _2(x)$\relax }{figure.caption.497}{}}
\newlabel{eLogEqnsEx1}{{28}{292}{Logarithmic Equations and Inequalities}{AMS.493}{}}
\pgfsyspdfmark {pgfid469}{14550631}{4736286}
\@writefile{lof}{\contentsline {figure}{\numberline {7.31}{\ignorespaces Solving $\genfrac  {}{}{}0{1}{\qopname  \relax o{ln}(x)+1}\leq 1$\relax }}{293}{figure.caption.498}}
\newlabel{fig:logineq1}{{7.31}{293}{Solving $\dfrac {1}{\ln (x)+1}\leq 1$\relax }{figure.caption.498}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.32}{\ignorespaces Solving  $(\qopname  \relax o{log}_2(x))^2<2\qopname  \relax o{log}_2(x)+3$\relax }}{293}{figure.caption.499}}
\newlabel{fig:logineq2}{{7.32}{293}{Solving\\ $(\log _2(x))^2<2\log _2(x)+3$\relax }{figure.caption.499}{}}
\pgfsyspdfmark {pgfid470}{4474142}{38494949}
\newlabel{logineq}{{124}{293}{Logarithmic Equations and Inequalities}{examplecounter.124}{}}
\pgfsyspdfmark {pgfid471}{9979166}{38331109}
\@writefile{lof}{\contentsline {figure}{\numberline {7.33}{\ignorespaces Solving $x\qopname  \relax o{log}(x+1)\geq x$\relax }}{294}{figure.caption.500}}
\newlabel{fig:logineq3}{{7.33}{294}{Solving $x\log (x+1)\geq x$\relax }{figure.caption.500}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.34}{\ignorespaces The graphs of $y = f(x) = -\qopname  \relax o{log}(x)$, {\color  {gray} $y = 7.8$} and {\color  {gray} $y = 8.5$}\relax }}{294}{figure.caption.501}}
\newlabel{fig:pHlog}{{7.34}{294}{The graphs of $y = f(x) = -\log (x)$, {\color {gray} $y = 7.8$} and {\color {gray} $y = 8.5$}\relax }{figure.caption.501}{}}
\newlabel{elogineq}{{28}{294}{Logarithmic Equations and Inequalities}{Item.2158}{}}
\pgfsyspdfmark {pgfid472}{14550631}{30450927}
\pgfsyspdfmark {pgfid473}{14288487}{23523500}
\newlabel{ex_fishph}{{125}{294}{Logarithmic Equations and Inequalities}{examplecounter.125}{}}
\newlabel{eex_fishph}{{28}{294}{Logarithmic Equations and Inequalities}{examplecounter.125}{}}
\pgfsyspdfmark {pgfid474}{15533671}{8675136}
\@writefile{lof}{\contentsline {figure}{\numberline {7.35}{\ignorespaces {\color  {gray} $y=f(x)=\genfrac  {}{}{}0{\qopname  \relax o{log}(x)}{1-\qopname  \relax o{log}(x)}$}  and $y=g(x)=10^{\frac  {x}{x+1}}$\relax }}{295}{figure.caption.506}}
\newlabel{fig:loginverse}{{7.35}{295}{{\color {gray} $y=f(x)=\dfrac {\log (x)}{1-\log (x)}$}\\ and $y=g(x)=10^{\frac {x}{x+1}}$\relax }{figure.caption.506}{}}
\pgfsyspdfmark {pgfid475}{4474142}{46871346}
\newlabel{logfracinverse}{{126}{295}{Logarithmic Equations and Inequalities}{examplecounter.126}{}}
\newlabel{elogfracinverse}{{28}{295}{Logarithmic Equations and Inequalities}{AMS.505}{}}
\pgfsyspdfmark {pgfid476}{5719326}{27671446}
\newlabel{sixfourRichterequ}{{10}{296}{Logarithmic Equations and Inequalities}{Item.2168}{}}
\newlabel{sixfourpHequ}{{11}{296}{Logarithmic Equations and Inequalities}{Item.2169}{}}
\newlabel{sixfourdecibelequ}{{12}{296}{Logarithmic Equations and Inequalities}{Item.2170}{}}
\newlabel{sixfourdecibelineq}{{27}{296}{Logarithmic Equations and Inequalities}{Item.2185}{}}
\newlabel{sixfourRichterineq}{{28}{296}{Logarithmic Equations and Inequalities}{Item.2186}{}}
\newlabel{sixfourpHineq}{{29}{296}{Logarithmic Equations and Inequalities}{Item.2187}{}}
\newlabel{onetooneexpexercise}{{35}{296}{Logarithmic Equations and Inequalities}{Item.2193}{}}
\newlabel{inversehyptangent}{{39}{296}{Logarithmic Equations and Inequalities}{Item.2200}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7.5}Applications of Exponential and Logarithmic Functions}{298}{section.7.5}}
\newlabel{ExpLogApplications}{{7.5}{298}{Applications of Exponential and Logarithmic Functions}{section.7.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.5.1}Applications of Exponential Functions}{298}{subsection.7.5.1}}
\newlabel{expapp}{{7.5.1}{298}{Applications of Exponential Functions}{subsection.7.5.1}{}}
\newlabel{simpleinterest}{{29}{298}{Applications of Exponential Functions}{keyideacounter.29}{}}
\newlabel{compoundinterest}{{30}{299}{Applications of Exponential Functions}{keyideacounter.30}{}}
\pgfsyspdfmark {pgfid479}{4474142}{22228173}
\newlabel{compoundinterestex}{{127}{299}{Applications of Exponential Functions}{examplecounter.127}{}}
\pgfsyspdfmark {pgfid480}{9979166}{22064333}
\newlabel{ecompoundinterestex}{{30}{300}{Applications of Exponential Functions}{Item.2211}{}}
\pgfsyspdfmark {pgfid481}{14550631}{33432897}
\newlabel{whatise}{{47}{300}{Applications of Exponential Functions}{theoremcounter.47}{}}
\newlabel{continuouscompoundinterest}{{31}{301}{Applications of Exponential Functions}{keyideacounter.31}{}}
\newlabel{lawofuninhibitedgrowth}{{32}{301}{Applications of Exponential Functions}{keyideacounter.32}{}}
\pgfsyspdfmark {pgfid485}{4474142}{12766322}
\newlabel{ex_athero}{{128}{301}{Applications of Exponential Functions}{examplecounter.128}{}}
\pgfsyspdfmark {pgfid486}{9979166}{12602482}
\newlabel{eex_athero}{{32}{302}{Applications of Exponential Functions}{examplecounter.128}{}}
\pgfsyspdfmark {pgfid487}{19032551}{43326521}
\newlabel{radioactivedecay}{{33}{302}{Applications of Exponential Functions}{keyideacounter.33}{}}
\pgfsyspdfmark {pgfid489}{14288487}{21341824}
\newlabel{ex_iodine}{{129}{302}{Applications of Exponential Functions}{examplecounter.129}{}}
\newlabel{eex_iodine}{{33}{302}{Applications of Exponential Functions}{examplecounter.129}{}}
\pgfsyspdfmark {pgfid490}{14550631}{10626542}
\newlabel{newtonslawofcooling}{{34}{303}{Applications of Exponential Functions}{keyideacounter.34}{}}
\pgfsyspdfmark {pgfid492}{4474142}{19368668}
\newlabel{exptempex2}{{130}{303}{Applications of Exponential Functions}{examplecounter.130}{}}
\pgfsyspdfmark {pgfid493}{9979166}{19204828}
\newlabel{eexptempex2}{{34}{304}{Applications of Exponential Functions}{Item.2215}{}}
\pgfsyspdfmark {pgfid494}{14550631}{41366406}
\newlabel{logisticgrowth}{{35}{304}{Applications of Exponential Functions}{keyideacounter.35}{}}
\pgfsyspdfmark {pgfid496}{14288487}{10248174}
\newlabel{ex_rumour}{{131}{304}{Applications of Exponential Functions}{examplecounter.131}{}}
\pgfsyspdfmark {pgfid497}{19793511}{10084334}
\@writefile{lof}{\contentsline {figure}{\numberline {7.36}{\ignorespaces  $y=\genfrac  {}{}{}0{84}{1+2799e^{-x}}$  and {\color  {gray} $y=84$}\relax }}{305}{figure.caption.507}}
\newlabel{fig:expapp1}{{7.36}{305}{$y=\dfrac {84}{1+2799e^{-x}}$\\ and {\color {gray} $y=84$}\relax }{figure.caption.507}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.37}{\ignorespaces  $y=\genfrac  {}{}{}0{84}{1+2799e^{-x}}$  and {\color  {gray} $y=42$}\relax }}{305}{figure.caption.508}}
\newlabel{fig:expapp2}{{7.37}{305}{$y=\dfrac {84}{1+2799e^{-x}}$\\ and {\color {gray} $y=42$}\relax }{figure.caption.508}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.38}{\ignorespaces $y=\genfrac  {}{}{}0{84}{1+2799e^{-x}}$  for $0\leq x\leq 8$\relax }}{305}{figure.caption.509}}
\newlabel{fig:expapp3}{{7.38}{305}{$y=\dfrac {84}{1+2799e^{-x}}$\\ for $0\leq x\leq 8$\relax }{figure.caption.509}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7.39}{\ignorespaces $y=\genfrac  {}{}{}0{84}{1+2799e^{-x}}$  for $8\leq x\leq 16$\relax }}{305}{figure.caption.510}}
\newlabel{fig:expapp4}{{7.39}{305}{$y=\dfrac {84}{1+2799e^{-x}}$\\ for $8\leq x\leq 16$\relax }{figure.caption.510}{}}
\newlabel{eex_rumour}{{35}{305}{Applications of Exponential Functions}{Item.2223}{}}
\pgfsyspdfmark {pgfid498}{4736286}{13586444}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.5.2}Applications of Logarithms}{306}{subsection.7.5.2}}
\pgfsyspdfmark {pgfid499}{14288487}{41303058}
\newlabel{ex_password}{{132}{306}{Applications of Logarithms}{examplecounter.132}{}}
\newlabel{eex_password}{{7.5.2}{306}{Applications of Logarithms}{Item.2227}{}}
\pgfsyspdfmark {pgfid500}{14550631}{21148167}
\pgfsyspdfmark {pgfid501}{14288487}{15020551}
\newlabel{ex_bloodbuffer}{{133}{306}{Applications of Logarithms}{examplecounter.133}{}}
\newlabel{eex_bloodbuffer}{{7.5.2}{306}{Applications of Logarithms}{examplecounter.133}{}}
\pgfsyspdfmark {pgfid502}{24386257}{6992391}
\newlabel{basicinterestexfirst}{{1}{307}{Applications of Logarithms}{Item.2228}{}}
\newlabel{basicinterestexlast}{{6}{307}{Applications of Logarithms}{Item.2233}{}}
\newlabel{radioactivefirst}{{14}{307}{Applications of Logarithms}{Item.2248}{}}
\newlabel{radioactivelast}{{18}{308}{Applications of Logarithms}{Item.2252}{}}
\newlabel{PainesvillePopulationTwoPoint}{{26}{308}{Applications of Logarithms}{Item.2270}{}}
\newlabel{pursuitlog}{{32}{309}{Applications of Logarithms}{Item.2286}{}}
\newlabel{explogsappcircuitone}{{33}{309}{Applications of Logarithms}{Item.2287}{}}
\newlabel{catenary}{{35}{309}{Applications of Logarithms}{Item.2289}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.3}{\ignorespaces A ray with initial point $P$\relax }}{311}{figure.caption.514}}
\newlabel{fig:angles1}{{8.3}{311}{A ray with initial point $P$\relax }{figure.caption.514}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.4}{\ignorespaces Two ways to measure an angle\relax }}{311}{figure.caption.515}}
\newlabel{fig:angles4}{{8.4}{311}{Two ways to measure an angle\relax }{figure.caption.515}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.5}{\ignorespaces Labelling angles\relax }}{311}{figure.caption.516}}
\newlabel{fig:angles5}{{8.5}{311}{Labelling angles\relax }{figure.caption.516}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {8}Foundations of Trigonometry}{311}{chapter.8}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{IntroTrig}{{8}{311}{Foundations of Trigonometry}{chapter.8}{}}
\@writefile{toc}{\contentsline {section}{\numberline {8.1}Angles and their Measure}{311}{section.8.1}}
\newlabel{Angles}{{8.1}{311}{Angles and their Measure}{section.8.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.1}{\ignorespaces Typical angles\relax }}{311}{figure.caption.511}}
\newlabel{fig:angles2}{{8.1}{311}{Typical angles\relax }{figure.caption.511}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.2}{\ignorespaces Less typical angles\relax }}{311}{figure.caption.512}}
\newlabel{fig:angles3}{{8.2}{311}{Less typical angles\relax }{figure.caption.512}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.8}{\ignorespaces Supplementary and complementary angles\relax }}{312}{figure.caption.518}}
\newlabel{fig:angles8}{{8.8}{312}{Supplementary and complementary angles\relax }{figure.caption.518}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.9}{\ignorespaces The sign of an angle\relax }}{312}{figure.caption.519}}
\newlabel{fig:angles9}{{8.9}{312}{The sign of an angle\relax }{figure.caption.519}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.10}{\ignorespaces Angles can comprise more than one revolution\relax }}{312}{figure.caption.520}}
\newlabel{fig:angles10}{{8.10}{312}{Angles can comprise more than one revolution\relax }{figure.caption.520}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.6}{\ignorespaces Defining degree measure\relax }}{312}{figure.caption.513}}
\newlabel{fig:angles6}{{8.6}{312}{Defining degree measure\relax }{figure.caption.513}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.7}{\ignorespaces Measuring angles in degrees\relax }}{312}{figure.caption.517}}
\newlabel{fig:angles7}{{8.7}{312}{Measuring angles in degrees\relax }{figure.caption.517}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.11}{\ignorespaces Two coterminal angles, ${"α}= 120^{\circ }$ and ${"β}= -240^{\circ }$, in standard position.\relax }}{313}{figure.caption.521}}
\newlabel{fig:angles11}{{8.11}{313}{Two coterminal angles, $\alpha = 120^{\circ }$ and $\beta = -240^{\circ }$, in standard position.\relax }{figure.caption.521}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.12}{\ignorespaces ${"α}= 60^{\circ }$ in standard position\relax }}{313}{figure.caption.522}}
\newlabel{fig:angles12}{{8.12}{313}{$\alpha = 60^{\circ }$ in standard position\relax }{figure.caption.522}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.13}{\ignorespaces ${"β}= -225^{\circ }$ in standard position\relax }}{313}{figure.caption.523}}
\newlabel{fig:angles13}{{8.13}{313}{$\beta = -225^{\circ }$ in standard position\relax }{figure.caption.523}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.14}{\ignorespaces ${"γ}= 540^{\circ }$ in standard position\relax }}{313}{figure.caption.524}}
\newlabel{fig:angles14}{{8.14}{313}{$\gamma = 540^{\circ }$ in standard position\relax }{figure.caption.524}{}}
\pgfsyspdfmark {pgfid503}{4474142}{30003618}
\newlabel{orientedcoterminaldegree}{{134}{313}{Angles and their Measure}{examplecounter.134}{}}
\pgfsyspdfmark {pgfid504}{9979166}{29839778}
\@writefile{lof}{\contentsline {figure}{\numberline {8.15}{\ignorespaces ${"φ}= -750^{\circ }$ in standard position\relax }}{314}{figure.caption.525}}
\newlabel{fig:angles15}{{8.15}{314}{$\phi = -750^{\circ }$ in standard position\relax }{figure.caption.525}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.16}{\ignorespaces The radian measure of ${"θ}$ is $\genfrac  {}{}{}0{s}{r}$\relax }}{314}{figure.caption.526}}
\newlabel{fig:angles16}{{8.16}{314}{The radian measure of $\theta $ is $\dfrac {s}{r}$\relax }{figure.caption.526}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.17}{\ignorespaces An angle of $k$ radians subtends an arc of length $k\cdot r$\relax }}{314}{figure.caption.527}}
\newlabel{fig:angles17}{{8.17}{314}{An angle of $k$ radians subtends an arc of length $k\cdot r$\relax }{figure.caption.527}{}}
\newlabel{eorientedcoterminaldegree}{{8.1}{314}{Angles and their Measure}{figure.caption.524}{}}
\pgfsyspdfmark {pgfid505}{14550631}{42975056}
\newlabel{pidefn}{{53}{314}{Angles and their Measure}{definitioncounter.53}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.18}{\ignorespaces ${"α}= \frac  {{"π}}{6}$ in standard position.\relax }}{315}{figure.caption.528}}
\newlabel{fig:angles18}{{8.18}{315}{$\alpha = \frac {\pi }{6}$ in standard position.\relax }{figure.caption.528}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.19}{\ignorespaces ${"β}=-\frac  {4{"π}}{3}$ in standard position.\relax }}{315}{figure.caption.529}}
\newlabel{fig:angles19}{{8.19}{315}{$\beta =-\frac {4\pi }{3}$ in standard position.\relax }{figure.caption.529}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.20}{\ignorespaces ${"γ}= \frac  {9{"π}}{4}$ in standard position.\relax }}{315}{figure.caption.530}}
\newlabel{fig:angles20}{{8.20}{315}{$\gamma = \frac {9\pi }{4}$ in standard position.\relax }{figure.caption.530}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.21}{\ignorespaces ${"φ}= -\frac  {5{"π}}{2}$ in standard position.\relax }}{315}{figure.caption.531}}
\newlabel{fig:angles21}{{8.21}{315}{$\phi = -\frac {5\pi }{2}$ in standard position.\relax }{figure.caption.531}{}}
\pgfsyspdfmark {pgfid507}{4474142}{34688124}
\newlabel{orientedcoterminalradian}{{135}{315}{Angles and their Measure}{examplecounter.135}{}}
\pgfsyspdfmark {pgfid508}{9979166}{34524284}
\newlabel{eorientedcoterminalradian}{{53}{316}{Angles and their Measure}{Item.2305}{}}
\pgfsyspdfmark {pgfid509}{14550631}{46707506}
\newlabel{degreenradianconversion}{{36}{316}{Angles and their Measure}{keyideacounter.36}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.23}{\ignorespaces $t=\frac  {3{"π}}{4}$\relax }}{317}{figure.caption.533}}
\newlabel{fig:angles22}{{8.23}{317}{$t=\frac {3\pi }{4}$\relax }{figure.caption.533}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.24}{\ignorespaces $t=-2{"π}$\relax }}{317}{figure.caption.534}}
\newlabel{fig:angles23}{{8.24}{317}{$t=-2\pi $\relax }{figure.caption.534}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.25}{\ignorespaces $t=-2$\relax }}{317}{figure.caption.535}}
\newlabel{fig:angles24}{{8.25}{317}{$t=-2$\relax }{figure.caption.535}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.22}{\ignorespaces Identifying real numnbers with angles\relax }}{317}{figure.caption.532}}
\newlabel{wrappingfunction}{{8.22}{317}{Identifying real numnbers with angles\relax }{figure.caption.532}{}}
\pgfsyspdfmark {pgfid511}{4474142}{34612448}
\newlabel{realwrap}{{136}{317}{Angles and their Measure}{examplecounter.136}{}}
\newlabel{erealwrap}{{36}{317}{Angles and their Measure}{Item.2313}{}}
\pgfsyspdfmark {pgfid512}{4736286}{5289383}
\@writefile{lof}{\contentsline {figure}{\numberline {8.26}{\ignorespaces $t=117$\relax }}{318}{figure.caption.536}}
\newlabel{fig:angles25}{{8.26}{318}{$t=117$\relax }{figure.caption.536}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.27}{\ignorespaces Circular motion\relax }}{318}{figure.caption.537}}
\newlabel{fig:angles26}{{8.27}{318}{Circular motion\relax }{figure.caption.537}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {8.1.1}Applications of Radian Measure: Circular Motion}{318}{subsection.8.1.1}}
\newlabel{circularmotion}{{8.1.1}{318}{Applications of Radian Measure: Circular Motion}{subsection.8.1.1}{}}
\pgfsyspdfmark {pgfid513}{14288487}{21329652}
\newlabel{EarthRotationEx}{{137}{318}{Applications of Radian Measure: Circular Motion}{examplecounter.137}{}}
\newlabel{eEarthRotationEx}{{8.1.1}{318}{Applications of Radian Measure: Circular Motion}{examplecounter.137}{}}
\pgfsyspdfmark {pgfid514}{14550631}{6618813}
\newlabel{spinningyoyo}{{42}{320}{Applications of Radian Measure: Circular Motion}{Item.2355}{}}
\newlabel{yoyotrick}{{44}{321}{Applications of Radian Measure: Circular Motion}{Item.2357}{}}
\newlabel{harddrive}{{45}{321}{Applications of Radian Measure: Circular Motion}{Item.2358}{}}
\newlabel{giantwheelmotion}{{47}{321}{Applications of Radian Measure: Circular Motion}{Item.2360}{}}
\newlabel{circularsectorarea}{{48}{321}{Applications of Radian Measure: Circular Motion}{Item.2361}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.29}{\ignorespaces Defining $\qopname  \relax o{cos}({"θ})$ and $\qopname  \relax o{sin}({"θ})$\relax }}{322}{figure.caption.539}}
\newlabel{fig:circle1}{{8.29}{322}{Defining $\cos (\theta )$ and $\sin (\theta )$\relax }{figure.caption.539}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.30}{\ignorespaces Finding $\qopname  \relax o{cos}(270^\circ )$ and $\qopname  \relax o{sin}(270^\circ )$\relax }}{322}{figure.caption.540}}
\newlabel{fig:circle2}{{8.30}{322}{Finding $\cos (270^\circ )$ and $\sin (270^\circ )$\relax }{figure.caption.540}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.31}{\ignorespaces Finding $\qopname  \relax o{cos}(-{"π})$ and $\qopname  \relax o{sin}(-{"π})$\relax }}{322}{figure.caption.541}}
\newlabel{fig:circle3}{{8.31}{322}{Finding $\cos (-\pi )$ and $\sin (-\pi )$\relax }{figure.caption.541}{}}
\@writefile{toc}{\contentsline {section}{\numberline {8.2}The Unit Circle: Sine and Cosine}{322}{section.8.2}}
\newlabel{TheUnitCircle}{{8.2}{322}{The Unit Circle: Sine and Cosine}{section.8.2}{}}
\pgfsyspdfmark {pgfid515}{14288487}{34239811}
\newlabel{cosinesineviaunitcircle}{{138}{322}{The Unit Circle: Sine and Cosine}{examplecounter.138}{}}
\pgfsyspdfmark {pgfid516}{19793511}{34075971}
\@writefile{lof}{\contentsline {figure}{\numberline {8.28}{\ignorespaces Finding $\qopname  \relax o{cos}\left (\frac  {{"π}}{4}\right )$ and $\qopname  \relax o{sin}\left (\frac  {{"π}}{4}\right )$\relax }}{323}{figure.caption.538}}
\newlabel{fig:circle4}{{8.28}{323}{Finding $\cos \left (\frac {\pi }{4}\right )$ and $\sin \left (\frac {\pi }{4}\right )$\relax }{figure.caption.538}{}}
\pgfsyspdfmark {pgfid517}{6374688}{23292638}
\@writefile{lof}{\contentsline {figure}{\numberline {8.32}{\ignorespaces Finding $\qopname  \relax o{cos}\left (\frac  {{"π}}{6}\right )$ and $\qopname  \relax o{sin}\left (\frac  {{"π}}{6}\right )$\relax }}{323}{figure.caption.542}}
\newlabel{fig:circle5}{{8.32}{323}{Finding $\cos \left (\frac {\pi }{6}\right )$ and $\sin \left (\frac {\pi }{6}\right )$\relax }{figure.caption.542}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.33}{\ignorespaces Finding $\qopname  \relax o{cos}\left (\frac  {{"π}}{3}\right )$ and $\qopname  \relax o{sin}\left (\frac  {{"π}}{3}\right )$\relax }}{324}{figure.caption.543}}
\newlabel{fig:circle6}{{8.33}{324}{Finding $\cos \left (\frac {\pi }{3}\right )$ and $\sin \left (\frac {\pi }{3}\right )$\relax }{figure.caption.543}{}}
\newlabel{ecosinesineviaunitcircle}{{8.2}{324}{The Unit Circle: Sine and Cosine}{figure.caption.543}{}}
\pgfsyspdfmark {pgfid518}{14550631}{31562194}
\newlabel{cosinesinepythid}{{48}{324}{The Unit Circle: Sine and Cosine}{theoremcounter.48}{}}
\pgfsyspdfmark {pgfid520}{14288487}{9867954}
\newlabel{cosinesinepythidex}{{139}{324}{The Unit Circle: Sine and Cosine}{examplecounter.139}{}}
\pgfsyspdfmark {pgfid521}{19793511}{9704114}
\@writefile{lof}{\contentsline {figure}{\numberline {8.35}{\ignorespaces Reference angle ${"α}$ for a Quadrant I angle\relax }}{325}{figure.caption.545}}
\newlabel{fig:circle8}{{8.35}{325}{Reference angle $\alpha $ for a Quadrant I angle\relax }{figure.caption.545}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.36}{\ignorespaces Reference angle ${"α}$ for a Quadrant II angle\relax }}{325}{figure.caption.546}}
\newlabel{fig:circle9}{{8.36}{325}{Reference angle $\alpha $ for a Quadrant II angle\relax }{figure.caption.546}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.37}{\ignorespaces Reference angle ${"α}$ for a Quadrant III angle\relax }}{325}{figure.caption.547}}
\newlabel{fig:circle10}{{8.37}{325}{Reference angle $\alpha $ for a Quadrant III angle\relax }{figure.caption.547}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.38}{\ignorespaces Reference angle ${"α}$ for a Quadrant IV angle\relax }}{325}{figure.caption.548}}
\newlabel{fig:circle11}{{8.38}{325}{Reference angle $\alpha $ for a Quadrant IV angle\relax }{figure.caption.548}{}}
\newlabel{ecosinesinepythidex}{{48}{325}{The Unit Circle: Sine and Cosine}{Item.2385}{}}
\pgfsyspdfmark {pgfid522}{4736286}{34110549}
\@writefile{lof}{\contentsline {figure}{\numberline {8.34}{\ignorespaces Refelcting $P(x,y)$ across the $y$-axis to obtain a Quadrant I angle\relax }}{325}{figure.caption.544}}
\newlabel{fig:circle7}{{8.34}{325}{Refelcting $P(x,y)$ across the $y$-axis to obtain a Quadrant I angle\relax }{figure.caption.544}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.39}{\ignorespaces Finding $\qopname  \relax o{cos}\left (\frac  {5{"π}}{4}\right )$ and $\qopname  \relax o{sin}\left (\frac  {5{"π}}{4}\right )$\relax }}{326}{figure.caption.550}}
\newlabel{fig:circle12}{{8.39}{326}{Finding $\cos \left (\frac {5\pi }{4}\right )$ and $\sin \left (\frac {5\pi }{4}\right )$\relax }{figure.caption.550}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.40}{\ignorespaces Finding $\qopname  \relax o{cos}\left (\frac  {11 {"π}}{6}\right )$ and $\qopname  \relax o{sin}\left (\frac  {11 {"π}}{6}\right )$\relax }}{326}{figure.caption.551}}
\newlabel{fig:circle13}{{8.40}{326}{Finding $\cos \left (\frac {11 \pi }{6}\right )$ and $\sin \left (\frac {11 \pi }{6}\right )$\relax }{figure.caption.551}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.41}{\ignorespaces Finding $\qopname  \relax o{cos}\left (-\frac  {5 {"π}}{4}\right )$ and $\qopname  \relax o{sin}\left (-\frac  {5 {"π}}{4}\right )$\relax }}{326}{figure.caption.552}}
\newlabel{fig:circle14}{{8.41}{326}{Finding $\cos \left (-\frac {5 \pi }{4}\right )$ and $\sin \left (-\frac {5 \pi }{4}\right )$\relax }{figure.caption.552}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.42}{\ignorespaces Finding $\qopname  \relax o{cos}\left (\frac  {7 {"π}}{3}\right )$ and $\qopname  \relax o{sin}\left (\frac  {7 {"π}}{3}\right )$\relax }}{326}{figure.caption.553}}
\newlabel{fig:circle15}{{8.42}{326}{Finding $\cos \left (\frac {7 \pi }{3}\right )$ and $\sin \left (\frac {7 \pi }{3}\right )$\relax }{figure.caption.553}{}}
\newlabel{refanglethm}{{49}{326}{The Unit Circle: Sine and Cosine}{theoremcounter.49}{}}
\newlabel{CosineSineFacts}{{49}{326}{The Unit Circle: Sine and Cosine}{section*.549}{}}
\pgfsyspdfmark {pgfid524}{14288487}{30068351}
\newlabel{refangleex}{{140}{326}{The Unit Circle: Sine and Cosine}{examplecounter.140}{}}
\newlabel{erefangleex}{{49}{326}{The Unit Circle: Sine and Cosine}{Item.2393}{}}
\pgfsyspdfmark {pgfid525}{14550631}{4736286}
\@writefile{lof}{\contentsline {figure}{\numberline {8.43}{\ignorespaces Important Points on the Unit Circle\relax }}{327}{figure.caption.554}}
\newlabel{commonanglesunitcircle}{{8.43}{327}{Important Points on the Unit Circle\relax }{figure.caption.554}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.46}{\ignorespaces Sketching ${"α}$\relax }}{328}{figure.caption.557}}
\newlabel{fig:circle16}{{8.46}{328}{Sketching $\alpha $\relax }{figure.caption.557}{}}
\pgfsyspdfmark {pgfid526}{14288487}{44740117}
\newlabel{advancedrefangleex}{{141}{328}{The Unit Circle: Sine and Cosine}{examplecounter.141}{}}
\pgfsyspdfmark {pgfid527}{19793511}{44576277}
\@writefile{lof}{\contentsline {figure}{\numberline {8.44}{\ignorespaces Finding $\qopname  \relax o{cos}({"θ})$ and $\qopname  \relax o{sin}({"θ})$ in Example \ref  {advancedrefangleex}.2(a)\relax }}{328}{figure.caption.555}}
\newlabel{fig:circle17}{{8.44}{328}{Finding $\cos (\theta )$ and $\sin (\theta )$ in Example \ref {advancedrefangleex}.2(a)\relax }{figure.caption.555}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.45}{\ignorespaces Finding $\qopname  \relax o{cos}({"θ})$ and $\qopname  \relax o{sin}({"θ})$ in Example \ref  {advancedrefangleex}.2(b)\relax }}{329}{figure.caption.556}}
\newlabel{fig:circle18}{{8.45}{329}{Finding $\cos (\theta )$ and $\sin (\theta )$ in Example \ref {advancedrefangleex}.2(b)\relax }{figure.caption.556}{}}
\pgfsyspdfmark {pgfid528}{7816480}{33389217}
\@writefile{lof}{\contentsline {figure}{\numberline {8.47}{\ignorespaces Finding $\qopname  \relax o{cos}({"θ})$ and $\qopname  \relax o{sin}({"θ})$ in Example \ref  {advancedrefangleex}.2(c)\relax }}{329}{figure.caption.558}}
\newlabel{fig:circle19}{{8.47}{329}{Finding $\cos (\theta )$ and $\sin (\theta )$ in Example \ref {advancedrefangleex}.2(c)\relax }{figure.caption.558}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.48}{\ignorespaces Finding $\qopname  \relax o{cos}({"θ})$ and $\qopname  \relax o{sin}({"θ})$ in Example \ref  {advancedrefangleex}.2(a)\relax }}{330}{figure.caption.559}}
\newlabel{fig:circle20}{{8.48}{330}{Finding $\cos (\theta )$ and $\sin (\theta )$ in Example \ref {advancedrefangleex}.2(a)\relax }{figure.caption.559}{}}
\newlabel{eadvancedrefangleex}{{49}{330}{The Unit Circle: Sine and Cosine}{figure.caption.559}{}}
\pgfsyspdfmark {pgfid529}{14550631}{33674903}
\pgfsyspdfmark {pgfid530}{14288487}{30704564}
\newlabel{solveforangle}{{142}{330}{The Unit Circle: Sine and Cosine}{examplecounter.142}{}}
\pgfsyspdfmark {pgfid531}{19793511}{30540724}
\newlabel{cosineishalf}{{1}{330}{The Unit Circle: Sine and Cosine}{Item.2406}{}}
\newlabel{sineisnegativehalf}{{2}{330}{The Unit Circle: Sine and Cosine}{Item.2407}{}}
\newlabel{cosineiszero}{{3}{330}{The Unit Circle: Sine and Cosine}{Item.2408}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.49}{\ignorespaces Angles with $\qopname  \relax o{cos}({"θ})=\frac  {1}{2}$\relax }}{330}{figure.caption.560}}
\newlabel{fig:circle21}{{8.49}{330}{Angles with $\cos (\theta )=\frac {1}{2}$\relax }{figure.caption.560}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.50}{\ignorespaces Angles with $\qopname  \relax o{sin}({"θ})=-\frac  {1}{2}$\relax }}{331}{figure.caption.561}}
\newlabel{fig:circle22}{{8.50}{331}{Angles with $\sin (\theta )=-\frac {1}{2}$\relax }{figure.caption.561}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.51}{\ignorespaces Angles with $\qopname  \relax o{cos}({"θ})=0$\relax }}{331}{figure.caption.562}}
\newlabel{fig:circle23}{{8.51}{331}{Angles with $\cos (\theta )=0$\relax }{figure.caption.562}{}}
\newlabel{esolveforangle}{{49}{331}{The Unit Circle: Sine and Cosine}{figure.caption.562}{}}
\pgfsyspdfmark {pgfid532}{4736286}{4736286}
\@writefile{toc}{\contentsline {subsection}{\numberline {8.2.1}Beyond the Unit Circle}{332}{subsection.8.2.1}}
\newlabel{cosinesinebeyond}{{8.2.1}{332}{Beyond the Unit Circle}{subsection.8.2.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.52}{\ignorespaces Determining coordinates of $Q(x,y)$ in terms of $\qopname  \relax o{cos}({"θ})$ and $\qopname  \relax o{sin}({"θ})$\relax }}{332}{figure.caption.563}}
\newlabel{fig:circle24}{{8.52}{332}{Determining coordinates of $Q(x,y)$ in terms of $\cos (\theta )$ and $\sin (\theta )$\relax }{figure.caption.563}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.53}{\ignorespaces The terminal side of ${"θ}$ contains $Q(4,-2)$\relax }}{333}{figure.caption.564}}
\newlabel{fig:circle25}{{8.53}{333}{The terminal side of $\theta $ contains $Q(4,-2)$\relax }{figure.caption.564}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.54}{\ignorespaces A point on the Earth at $41.628^\circ $N\relax }}{333}{figure.caption.565}}
\newlabel{fig:circle26}{{8.54}{333}{A point on the Earth at $41.628^\circ $N\relax }{figure.caption.565}{}}
\newlabel{cosinesinecircle}{{50}{333}{Beyond the Unit Circle}{theoremcounter.50}{}}
\pgfsyspdfmark {pgfid534}{4474142}{35788010}
\newlabel{cosinesinecircleex}{{143}{333}{Beyond the Unit Circle}{examplecounter.143}{}}
\newlabel{ecosinesinecircleex}{{50}{333}{Beyond the Unit Circle}{Item.2415}{}}
\pgfsyspdfmark {pgfid535}{4736286}{12065374}
\@writefile{lof}{\contentsline {figure}{\numberline {8.56}{\ignorespaces Relating triangular and circular trigonometry\relax }}{334}{figure.caption.567}}
\newlabel{fig:tritrig1}{{8.56}{334}{Relating triangular and circular trigonometry\relax }{figure.caption.567}{}}
\newlabel{equationsforcircularmotion}{{51}{334}{Beyond the Unit Circle}{theoremcounter.51}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.55}{\ignorespaces Equations for circular motion\relax }}{334}{figure.caption.566}}
\newlabel{fig:circle27}{{8.55}{334}{Equations for circular motion\relax }{figure.caption.566}{}}
\pgfsyspdfmark {pgfid537}{14288487}{25682124}
\newlabel{Lakelandrotates}{{144}{334}{Beyond the Unit Circle}{examplecounter.144}{}}
\newlabel{eLakelandrotates}{{51}{334}{Beyond the Unit Circle}{examplecounter.144}{}}
\pgfsyspdfmark {pgfid538}{19669351}{19673132}
\@writefile{lof}{\contentsline {figure}{\numberline {8.57}{\ignorespaces The triangle for Example \ref  {righttriangleex}\relax }}{335}{figure.caption.568}}
\newlabel{fig:tritrig2}{{8.57}{335}{The triangle for Example \ref {righttriangleex}\relax }{figure.caption.568}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.58}{\ignorespaces The completed triangle for Example \ref  {righttriangleex}\relax }}{335}{figure.caption.569}}
\newlabel{fig:tritrig3}{{8.58}{335}{The completed triangle for Example \ref {righttriangleex}\relax }{figure.caption.569}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.59}{\ignorespaces Defining $\qopname  \relax o{cos}(t)$ and $\qopname  \relax o{sin}(t)$ as functions of a real variable\relax }}{335}{figure.caption.570}}
\newlabel{fig:circle28}{{8.59}{335}{Defining $\cos (t)$ and $\sin (t)$ as functions of a real variable\relax }{figure.caption.570}{}}
\newlabel{cosinesinetriangle}{{52}{335}{Beyond the Unit Circle}{theoremcounter.52}{}}
\pgfsyspdfmark {pgfid540}{4474142}{38764330}
\newlabel{righttriangleex}{{145}{335}{Beyond the Unit Circle}{examplecounter.145}{}}
\newlabel{erighttriangleex}{{52}{335}{Beyond the Unit Circle}{examplecounter.145}{}}
\pgfsyspdfmark {pgfid541}{5719326}{22812513}
\newlabel{cosinesinefunctiondomainrange}{{53}{336}{Beyond the Unit Circle}{theoremcounter.53}{}}
\newlabel{cosinesineequationsrealnumbers}{{53}{336}{Beyond the Unit Circle}{section*.571}{}}
\newlabel{cofunctionforeshadowing}{{74}{339}{Beyond the Unit Circle}{Item.2489}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.60}{\ignorespaces Explaining the tangent and secant functions\relax }}{340}{figure.caption.572}}
\newlabel{fig:cirfun1}{{8.60}{340}{Explaining the tangent and secant functions\relax }{figure.caption.572}{}}
\@writefile{toc}{\contentsline {section}{\numberline {8.3}The Six Circular Functions and Fundamental Identities}{340}{section.8.3}}
\newlabel{CircularFunctions}{{8.3}{340}{The Six Circular Functions and Fundamental Identities}{section.8.3}{}}
\newlabel{circularfunctions}{{54}{340}{The Six Circular Functions and Fundamental Identities}{definitioncounter.54}{}}
\newlabel{recipquotid}{{54}{341}{The Six Circular Functions and Fundamental Identities}{theoremcounter.54}{}}
\pgfsyspdfmark {pgfid545}{4474142}{25967545}
\newlabel{circularfunctionsex}{{146}{341}{The Six Circular Functions and Fundamental Identities}{examplecounter.146}{}}
\pgfsyspdfmark {pgfid546}{9979166}{25803705}
\newlabel{commontanmistake}{{6}{342}{The Six Circular Functions and Fundamental Identities}{Item.2502}{}}
\newlabel{ecircularfunctionsex}{{54}{342}{The Six Circular Functions and Fundamental Identities}{Item.2502}{}}
\pgfsyspdfmark {pgfid547}{14550631}{32366061}
\newlabel{genrefanglethm}{{55}{342}{The Six Circular Functions and Fundamental Identities}{theoremcounter.55}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.61}{\ignorespaces Solving $\qopname  \relax o{tan}({"θ})=\sqrt  {3}$\relax }}{343}{figure.caption.573}}
\newlabel{fig:cirfun2}{{8.61}{343}{Solving $\tan (\theta )=\sqrt {3}$\relax }{figure.caption.573}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.62}{\ignorespaces Solving $\qopname  \relax o{cot}({"θ})=-1$\relax }}{343}{figure.caption.574}}
\newlabel{fig:cirfun3}{{8.62}{343}{Solving $\cot (\theta )=-1$\relax }{figure.caption.574}{}}
\pgfsyspdfmark {pgfid549}{4474142}{46871346}
\newlabel{solveforangle2}{{147}{343}{The Six Circular Functions and Fundamental Identities}{examplecounter.147}{}}
\newlabel{cotangentisnegativeone}{{3}{343}{The Six Circular Functions and Fundamental Identities}{Item.2505}{}}
\newlabel{esolveforangle2}{{55}{343}{The Six Circular Functions and Fundamental Identities}{Item.2508}{}}
\pgfsyspdfmark {pgfid550}{4736286}{14665605}
\newlabel{pythids}{{56}{344}{The Six Circular Functions and Fundamental Identities}{theoremcounter.56}{}}
\pgfsyspdfmark {pgfid552}{14288487}{24062382}
\newlabel{idornotex1}{{148}{344}{The Six Circular Functions and Fundamental Identities}{examplecounter.148}{}}
\pgfsyspdfmark {pgfid553}{19793511}{23898542}
\newlabel{pythconjex}{{6}{344}{The Six Circular Functions and Fundamental Identities}{Item.2517}{}}
\pgfsyspdfmark {pgfid554}{6374688}{20455892}
\newlabel{eidornotex1}{{56}{346}{The Six Circular Functions and Fundamental Identities}{Item.2523}{}}
\pgfsyspdfmark {pgfid555}{14550631}{27555656}
\newlabel{PythagoreanConjugates}{{37}{346}{The Six Circular Functions and Fundamental Identities}{keyideacounter.37}{}}
\newlabel{IdentityHelp}{{38}{347}{The Six Circular Functions and Fundamental Identities}{keyideacounter.38}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.1}Beyond the Unit Circle}{348}{subsection.8.3.1}}
\newlabel{circularfunctionsbeyond}{{8.3.1}{348}{Beyond the Unit Circle}{subsection.8.3.1}{}}
\newlabel{circularfunctionscircle}{{57}{348}{Beyond the Unit Circle}{theoremcounter.57}{}}
\pgfsyspdfmark {pgfid559}{14288487}{27173023}
\newlabel{circularfunctionscircleex}{{149}{348}{Beyond the Unit Circle}{examplecounter.149}{}}
\newlabel{ecircularfunctionscircleex}{{57}{348}{Beyond the Unit Circle}{Item.2527}{}}
\pgfsyspdfmark {pgfid560}{14550631}{4736286}
\@writefile{lof}{\contentsline {figure}{\numberline {8.63}{\ignorespaces A right-angled triangle \relax }}{349}{figure.caption.576}}
\newlabel{fig:cirfun4}{{8.63}{349}{A right-angled triangle \relax }{figure.caption.576}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.64}{\ignorespaces The angle of inclination from the base line to the object is ${"θ}$\relax }}{349}{figure.caption.577}}
\newlabel{fig:cirfun5}{{8.64}{349}{The angle of inclination from the base line to the object is $\theta $\relax }{figure.caption.577}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.65}{\ignorespaces Finding the height of the Clocktower\relax }}{349}{figure.caption.578}}
\newlabel{fig:cirfun6}{{8.65}{349}{Finding the height of the Clocktower\relax }{figure.caption.578}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.66}{\ignorespaces Finding the height of a California Redwood\relax }}{349}{figure.caption.579}}
\newlabel{fig:cirfun7}{{8.66}{349}{Finding the height of a California Redwood\relax }{figure.caption.579}{}}
\newlabel{circularfunctionstriangle}{{58}{349}{Beyond the Unit Circle}{theoremcounter.58}{}}
\newlabel{angleofelevation}{{58}{349}{Beyond the Unit Circle}{section*.575}{}}
\pgfsyspdfmark {pgfid562}{4474142}{29894957}
\newlabel{circularfunctionstriangleex}{{150}{349}{Beyond the Unit Circle}{examplecounter.150}{}}
\pgfsyspdfmark {pgfid563}{9979166}{29731117}
\newlabel{ecircularfunctionstriangleex}{{58}{350}{Beyond the Unit Circle}{figure.caption.579}{}}
\pgfsyspdfmark {pgfid564}{14550631}{36190332}
\newlabel{extendedinterval}{{58}{351}{Beyond the Unit Circle}{section*.580}{}}
\newlabel{circularfunctionsdomainrange}{{59}{352}{Beyond the Unit Circle}{theoremcounter.59}{}}
\newlabel{angleofdepression}{{78}{355}{Beyond the Unit Circle}{Item.2609}{}}
\newlabel{sasquatchfire}{{78b}{355}{Beyond the Unit Circle}{Item.2611}{}}
\newlabel{cofunctionforeshadowingagain}{{134}{356}{Beyond the Unit Circle}{Item.2668}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.67}{\ignorespaces Establishing Theorem \ref  {evenodd}\relax }}{358}{figure.caption.581}}
\newlabel{fig:identities1}{{8.67}{358}{Establishing Theorem \ref {evenodd}\relax }{figure.caption.581}{}}
\@writefile{toc}{\contentsline {section}{\numberline {8.4}Trigonometric Identities}{358}{section.8.4}}
\newlabel{Identities}{{8.4}{358}{Trigonometric Identities}{section.8.4}{}}
\newlabel{evenodd}{{60}{358}{Trigonometric Identities}{theoremcounter.60}{}}
\newlabel{cosinesumdifference}{{61}{358}{Trigonometric Identities}{theoremcounter.61}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.68}{\ignorespaces Establishing Theorem \ref  {cosinesumdifference}\relax }}{359}{figure.caption.582}}
\newlabel{fig:identities2}{{8.68}{359}{Establishing Theorem \ref {cosinesumdifference}\relax }{figure.caption.582}{}}
\pgfsyspdfmark {pgfid568}{14288487}{45735034}
\newlabel{cosinesumdiffex}{{151}{360}{Trigonometric Identities}{examplecounter.151}{}}
\newlabel{ecosinesumdiffex}{{61}{360}{Trigonometric Identities}{Item.2681}{}}
\pgfsyspdfmark {pgfid569}{14550631}{24306429}
\newlabel{cofunctionidentities}{{62}{360}{Trigonometric Identities}{theoremcounter.62}{}}
\newlabel{sinesumdifference}{{63}{361}{Trigonometric Identities}{theoremcounter.63}{}}
\pgfsyspdfmark {pgfid572}{4474142}{26855907}
\newlabel{sinesumanddiffex}{{152}{361}{Trigonometric Identities}{examplecounter.152}{}}
\pgfsyspdfmark {pgfid573}{9979166}{26692067}
\newlabel{esinesumanddiffex}{{63}{362}{Trigonometric Identities}{Item.2687}{}}
\pgfsyspdfmark {pgfid574}{14550631}{6718528}
\newlabel{circularsumdifference}{{64}{363}{Trigonometric Identities}{theoremcounter.64}{}}
\newlabel{doubleangle}{{65}{363}{Trigonometric Identities}{theoremcounter.65}{}}
\pgfsyspdfmark {pgfid577}{14288487}{46871346}
\newlabel{doubleangleex}{{153}{364}{Trigonometric Identities}{examplecounter.153}{}}
\pgfsyspdfmark {pgfid578}{19793511}{46707506}
\newlabel{doubleanglesinewtan}{{3}{364}{Trigonometric Identities}{Item.2690}{}}
\newlabel{cosinepolynomial}{{4}{364}{Trigonometric Identities}{Item.2691}{}}
\newlabel{edoubleangleex}{{65}{365}{Trigonometric Identities}{Item.2695}{}}
\pgfsyspdfmark {pgfid579}{4736286}{30349053}
\newlabel{powerreduction}{{66}{365}{Trigonometric Identities}{theoremcounter.66}{}}
\pgfsyspdfmark {pgfid581}{4474142}{15350465}
\newlabel{powerreductionex}{{154}{365}{Trigonometric Identities}{examplecounter.154}{}}
\pgfsyspdfmark {pgfid582}{9979166}{15186625}
\newlabel{epowerreductionex}{{66}{366}{Trigonometric Identities}{examplecounter.154}{}}
\pgfsyspdfmark {pgfid583}{14550631}{39101716}
\newlabel{halfangle}{{67}{366}{Trigonometric Identities}{theoremcounter.67}{}}
\pgfsyspdfmark {pgfid585}{14288487}{14068860}
\newlabel{ex_halfangle}{{155}{366}{Trigonometric Identities}{examplecounter.155}{}}
\pgfsyspdfmark {pgfid586}{19793511}{13905020}
\newlabel{eex_halfangle}{{67}{367}{Trigonometric Identities}{Item.2701}{}}
\pgfsyspdfmark {pgfid587}{4736286}{3163422}
\newlabel{producttosum}{{68}{368}{Trigonometric Identities}{theoremcounter.68}{}}
\newlabel{sumtoproduct}{{69}{368}{Trigonometric Identities}{theoremcounter.69}{}}
\pgfsyspdfmark {pgfid590}{14288487}{18654143}
\newlabel{prodtosumtoprod}{{156}{368}{Trigonometric Identities}{examplecounter.156}{}}
\pgfsyspdfmark {pgfid591}{19793511}{18490303}
\newlabel{eprodtosumtoprod}{{69}{369}{Trigonometric Identities}{Item.2705}{}}
\pgfsyspdfmark {pgfid592}{4736286}{39269155}
\newlabel{cos75}{{7}{370}{Trigonometric Identities}{Item.2712}{}}
\newlabel{sin105}{{9}{370}{Trigonometric Identities}{Item.2714}{}}
\newlabel{cos7pi12}{{16}{370}{Trigonometric Identities}{Item.2721}{}}
\newlabel{sinpi12}{{18}{370}{Trigonometric Identities}{Item.2723}{}}
\newlabel{sine3theta}{{65}{371}{Trigonometric Identities}{Item.2788}{}}
\newlabel{cosine4theta}{{69}{371}{Trigonometric Identities}{Item.2792}{}}
\newlabel{preludetoarctrigsine}{{86}{372}{Trigonometric Identities}{Item.2809}{}}
\newlabel{preludetoarctrigtan}{{88}{372}{Trigonometric Identities}{Item.2814}{}}
\@writefile{toc}{\contentsline {section}{\numberline {8.5}Graphs of the Trigonometric Functions}{373}{section.8.5}}
\newlabel{TrigGraphs}{{8.5}{373}{Graphs of the Trigonometric Functions}{section.8.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {8.5.1}Graphs of the Cosine and Sine Functions}{373}{subsection.8.5.1}}
\newlabel{periodic}{{55}{373}{Graphs of the Cosine and Sine Functions}{definitioncounter.55}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.71}{\ignorespaces Graphing $y=\qopname  \relax o{cos}(x)$\relax }}{374}{figure.caption.585}}
\newlabel{fig:triggraph1}{{8.71}{374}{Graphing $y=\cos (x)$\relax }{figure.caption.585}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.72}{\ignorespaces Graphing $y=\qopname  \relax o{sin}(x)$\relax }}{374}{figure.caption.586}}
\newlabel{fig:triggraph2}{{8.72}{374}{Graphing $y=\sin (x)$\relax }{figure.caption.586}{}}
\newlabel{cosinesinefunctionprops}{{70}{374}{Graphs of the Cosine and Sine Functions}{theoremcounter.70}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.69}{\ignorespaces An accurately scaled graph of $y = \qopname  \relax o{cos}(x)$.\relax }}{374}{figure.caption.583}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.70}{\ignorespaces An accurately scaled graph of $y = \qopname  \relax o{sin}(x)$.\relax }}{374}{figure.caption.584}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.75}{\ignorespaces Reference points for $f(x)$ in Example \ref  {cosinesinegraphex1}\relax }}{375}{figure.caption.589}}
\newlabel{fig:triggraph3}{{8.75}{375}{Reference points for $f(x)$ in Example \ref {cosinesinegraphex1}\relax }{figure.caption.589}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.76}{\ignorespaces Reference points for $g(x)$ in Example \ref  {cosinesinegraphex1}\relax }}{375}{figure.caption.590}}
\newlabel{fig:triggraph4}{{8.76}{375}{Reference points for $g(x)$ in Example \ref {cosinesinegraphex1}\relax }{figure.caption.590}{}}
\pgfsyspdfmark {pgfid595}{4474142}{34311291}
\newlabel{cosinesinegraphex1}{{157}{375}{Graphs of the Cosine and Sine Functions}{examplecounter.157}{}}
\pgfsyspdfmark {pgfid596}{9979166}{34147451}
\@writefile{lof}{\contentsline {figure}{\numberline {8.73}{\ignorespaces Plotting one cycle of $y=f(x)$ in Example \ref  {cosinesinegraphex1}\relax }}{375}{figure.caption.587}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.77}{\ignorespaces Properties of sinusoids\relax }}{376}{figure.caption.591}}
\newlabel{fig:triggraph5}{{8.77}{376}{Properties of sinusoids\relax }{figure.caption.591}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.74}{\ignorespaces Plotting one cycle of $y=g(x)$ in Example \ref  {cosinesinegraphex1}\relax }}{376}{figure.caption.588}}
\newlabel{ecosinesinegraphex1}{{70}{376}{Graphs of the Cosine and Sine Functions}{figure.caption.590}{}}
\pgfsyspdfmark {pgfid597}{14550631}{29904637}
\newlabel{sinusoidform}{{71}{376}{Graphs of the Cosine and Sine Functions}{theoremcounter.71}{}}
\newlabel{phaseshiftissue}{{71}{377}{Graphs of the Cosine and Sine Functions}{section*.592}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.79}{\ignorespaces One cycle of $y=f(x)$ in Example \ref  {fitsinusoidtodata1}\relax }}{378}{figure.caption.595}}
\newlabel{fig:triggraph6}{{8.79}{378}{One cycle of $y=f(x)$ in Example \ref {fitsinusoidtodata1}\relax }{figure.caption.595}{}}
\pgfsyspdfmark {pgfid599}{14288487}{46871346}
\newlabel{fitsinusoidtodata1}{{158}{378}{Graphs of the Cosine and Sine Functions}{examplecounter.158}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.78}{\ignorespaces Extending the graph of $y=f(x)$\relax }}{378}{figure.caption.593}}
\newlabel{fig:triggraph7}{{8.78}{378}{Extending the graph of $y=f(x)$\relax }{figure.caption.593}{}}
\newlabel{efitsinusoidtodata1}{{71}{378}{Graphs of the Cosine and Sine Functions}{figure.caption.593}{}}
\pgfsyspdfmark {pgfid600}{14550631}{13592709}
\newlabel{expandedsinusoid}{{71}{378}{Graphs of the Cosine and Sine Functions}{section*.594}{}}
\pgfsyspdfmark {pgfid601}{4474142}{42048260}
\newlabel{expandedsinusoidex1}{{159}{379}{Graphs of the Cosine and Sine Functions}{examplecounter.159}{}}
\pgfsyspdfmark {pgfid602}{9979166}{41884420}
\newlabel{eexpandedsinusoidex1}{{71}{380}{Graphs of the Cosine and Sine Functions}{Item.2844}{}}
\pgfsyspdfmark {pgfid603}{14550631}{33026244}
\@writefile{toc}{\contentsline {subsection}{\numberline {8.5.2}Graphs of the Secant and Cosecant Functions}{380}{subsection.8.5.2}}
\newlabel{secantcosecantgraphsection}{{8.5.2}{380}{Graphs of the Secant and Cosecant Functions}{subsection.8.5.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.80}{\ignorespaces The `fundamental cycle' of $y = \qopname  \relax o{sec}(x)$.\relax }}{381}{figure.caption.596}}
\newlabel{fig:secgraph1}{{8.80}{381}{The `fundamental cycle' of $y = \sec (x)$.\relax }{figure.caption.596}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.81}{\ignorespaces The graph of $y=\qopname  \relax o{sec}{x}$\relax }}{381}{figure.caption.597}}
\newlabel{fig:secgraph2}{{8.81}{381}{The graph of $y=\sec {x}$\relax }{figure.caption.597}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.82}{\ignorespaces The `fundamental cycle' of $y = \qopname  \relax o{csc}(x)$.\relax }}{382}{figure.caption.598}}
\newlabel{fig:cscgraph1}{{8.82}{382}{The `fundamental cycle' of $y = \csc (x)$.\relax }{figure.caption.598}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.83}{\ignorespaces The graph of $y=\qopname  \relax o{csc}{x}$\relax }}{382}{figure.caption.599}}
\newlabel{fig:cscgraph2}{{8.83}{382}{The graph of $y=\csc {x}$\relax }{figure.caption.599}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.85}{\ignorespaces Reference points for $f(x)$ in Example \ref  {seccscgraphex}\relax }}{383}{figure.caption.601}}
\newlabel{fig:secgraph3}{{8.85}{383}{Reference points for $f(x)$ in Example \ref {seccscgraphex}\relax }{figure.caption.601}{}}
\newlabel{secantcosecantfunctionprops}{{72}{383}{Graphs of the Secant and Cosecant Functions}{theoremcounter.72}{}}
\pgfsyspdfmark {pgfid605}{4474142}{23460416}
\newlabel{seccscgraphex}{{160}{383}{Graphs of the Secant and Cosecant Functions}{examplecounter.160}{}}
\pgfsyspdfmark {pgfid606}{9979166}{23296576}
\@writefile{lof}{\contentsline {figure}{\numberline {8.87}{\ignorespaces Reference points for $g(x)$ in Example \ref  {seccscgraphex}\relax }}{384}{figure.caption.603}}
\newlabel{fig:cscgraph3}{{8.87}{384}{Reference points for $g(x)$ in Example \ref {seccscgraphex}\relax }{figure.caption.603}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.84}{\ignorespaces Plotting one cycle of $y=f(x)$ in Example \ref  {seccscgraphex}\relax }}{384}{figure.caption.600}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.86}{\ignorespaces Plotting one cycle of $y=g(x)$ in Example \ref  {seccscgraphex}\relax }}{384}{figure.caption.602}}
\newlabel{eseccscgraphex}{{8.5.2}{384}{Graphs of the Secant and Cosecant Functions}{figure.caption.602}{}}
\pgfsyspdfmark {pgfid607}{14550631}{16343968}
\@writefile{toc}{\contentsline {subsection}{\numberline {8.5.3}Graphs of the Tangent and Cotangent Functions}{384}{subsection.8.5.3}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.88}{\ignorespaces The graph of $y = \qopname  \relax o{tan}(x)$ over $[0,2{"π}]$\relax }}{385}{figure.caption.604}}
\newlabel{fig:tangraph1}{{8.88}{385}{The graph of $y = \tan (x)$ over $[0,2\pi ]$\relax }{figure.caption.604}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.89}{\ignorespaces The graph of $y = \qopname  \relax o{tan}(x)$\relax }}{385}{figure.caption.605}}
\newlabel{fig:tangraph2}{{8.89}{385}{The graph of $y = \tan (x)$\relax }{figure.caption.605}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.90}{\ignorespaces The graph of $y = \qopname  \relax o{cot}(x)$ over $[0,2{"π}]$\relax }}{386}{figure.caption.606}}
\newlabel{fig:cotgraph1}{{8.90}{386}{The graph of $y = \cot (x)$ over $[0,2\pi ]$\relax }{figure.caption.606}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.91}{\ignorespaces The graph of $y = \qopname  \relax o{cot}(x)$\relax }}{386}{figure.caption.607}}
\newlabel{fig:cotgraph2}{{8.91}{386}{The graph of $y = \cot (x)$\relax }{figure.caption.607}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.93}{\ignorespaces Reference points for $f(x)$ in Example \ref  {tancotgraphex}\relax }}{387}{figure.caption.609}}
\newlabel{fig:tangraph3}{{8.93}{387}{Reference points for $f(x)$ in Example \ref {tancotgraphex}\relax }{figure.caption.609}{}}
\newlabel{tangentcotangentfunctionprops}{{73}{387}{Graphs of the Tangent and Cotangent Functions}{theoremcounter.73}{}}
\pgfsyspdfmark {pgfid609}{4474142}{24763416}
\newlabel{tancotgraphex}{{161}{387}{Graphs of the Tangent and Cotangent Functions}{examplecounter.161}{}}
\pgfsyspdfmark {pgfid610}{9979166}{24599576}
\@writefile{lof}{\contentsline {figure}{\numberline {8.95}{\ignorespaces Reference points for $g(x)$ in Example \ref  {tancotgraphex}\relax }}{388}{figure.caption.611}}
\newlabel{fig:cotgraph3}{{8.95}{388}{Reference points for $g(x)$ in Example \ref {tancotgraphex}\relax }{figure.caption.611}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.92}{\ignorespaces Plotting one cycle of $y=f(x)$ in Example \ref  {tancotgraphex}\relax }}{388}{figure.caption.608}}
\@writefile{lof}{\contentsline {figure}{\numberline {8.94}{\ignorespaces Plotting one cycle of $y=g(x)$ in Example \ref  {tancotgraphex}\relax }}{388}{figure.caption.610}}
\newlabel{etancotgraphex}{{8.5.3}{388}{Graphs of the Tangent and Cotangent Functions}{figure.caption.610}{}}
\pgfsyspdfmark {pgfid611}{14550631}{13742231}
\newlabel{expandedsinusoidexerfirst}{{25}{389}{Graphs of the Tangent and Cotangent Functions}{Item.2877}{}}
\newlabel{expandedsinusoidexerlast}{{34}{389}{Graphs of the Tangent and Cotangent Functions}{Item.2886}{}}
\newlabel{sinusoidexercise2}{{36}{389}{Graphs of the Tangent and Cotangent Functions}{Item.2888}{}}
\newlabel{idengraphfirst}{{38}{390}{Graphs of the Tangent and Cotangent Functions}{Item.2890}{}}
\newlabel{idengraphlast}{{43}{390}{Graphs of the Tangent and Cotangent Functions}{Item.2895}{}}
\newlabel{exploregraphslast}{{50}{390}{Graphs of the Tangent and Cotangent Functions}{Item.2902}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.3}{\ignorespaces Reflecting $y=\qopname  \relax o{cos}(x)$ across $y=x$ yields $y=\qopname  \relax o{arccos}(x)$\relax }}{391}{figure.caption.614}}
\newlabel{fig:arctrig2}{{9.3}{391}{Reflecting $y=\cos (x)$ across $y=x$ yields $y=\arccos (x)$\relax }{figure.caption.614}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {9}Further Topics in Trigonometry}{391}{chapter.9}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{AppExt}{{9}{391}{Further Topics in Trigonometry}{chapter.9}{}}
\@writefile{toc}{\contentsline {section}{\numberline {9.1}Inverse Trigonometric Functions}{391}{section.9.1}}
\newlabel{ArcTrig}{{9.1}{391}{Inverse Trigonometric Functions}{section.9.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.1}{\ignorespaces Restricting the domain of $f(x) = \qopname  \relax o{cos}(x)$ to $[0,{"π}]$.\relax }}{391}{figure.caption.612}}
\newlabel{fig:arctrig1}{{9.1}{391}{Restricting the domain of $f(x) = \cos (x)$ to $[0,\pi ]$.\relax }{figure.caption.612}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.2}{\ignorespaces Restricting the domain of $f(x) = \qopname  \relax o{sin}(x)$ to $\left [-\frac  {{"π}}{2}, \frac  {{"π}}{2}\right ]$.\relax }}{391}{figure.caption.613}}
\newlabel{fig:arctrig3}{{9.2}{391}{Restricting the domain of $f(x) = \sin (x)$ to $\left [-\frac {\pi }{2}, \frac {\pi }{2}\right ]$.\relax }{figure.caption.613}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.4}{\ignorespaces Reflecting $y=\qopname  \relax o{sin}(x)$ across $y=x$ yields $y=\qopname  \relax o{arcsin}(x)$\relax }}{392}{figure.caption.615}}
\newlabel{fig:arctrig4}{{9.4}{392}{Reflecting $y=\sin (x)$ across $y=x$ yields $y=\arcsin (x)$\relax }{figure.caption.615}{}}
\newlabel{arccosinesinefunctionprops}{{74}{392}{Inverse Trigonometric Functions}{theoremcounter.74}{}}
\pgfsyspdfmark {pgfid613}{14288487}{25043436}
\newlabel{arccosinesineex}{{162}{392}{Inverse Trigonometric Functions}{examplecounter.162}{}}
\pgfsyspdfmark {pgfid614}{19793511}{24879596}
\newlabel{algarcsincos}{{2}{392}{Inverse Trigonometric Functions}{Item.2913}{}}
\newlabel{tanarccos}{{2a}{392}{Inverse Trigonometric Functions}{Item.2914}{}}
\newlabel{cosarcsin}{{2b}{392}{Inverse Trigonometric Functions}{Item.2915}{}}
\pgfsyspdfmark {pgfid615}{6374688}{19780544}
\@writefile{lof}{\contentsline {figure}{\numberline {9.5}{\ignorespaces Reflecting $y=\qopname  \relax o{tan}(x)$ across $y=x$ yields $y=\qopname  \relax o{arctan}(x)$\relax }}{394}{figure.caption.616}}
\newlabel{fig:arctrig5}{{9.5}{394}{Reflecting $y=\tan (x)$ across $y=x$ yields $y=\arctan (x)$\relax }{figure.caption.616}{}}
\newlabel{earccosinesineex}{{74}{394}{Inverse Trigonometric Functions}{Item.2927}{}}
\pgfsyspdfmark {pgfid616}{14550631}{28809176}
\@writefile{lof}{\contentsline {figure}{\numberline {9.6}{\ignorespaces Reflecting $y=\qopname  \relax o{cot}(x)$ across $y=x$ yields $y=\operatorname  {arccot}(x)$\relax }}{395}{figure.caption.617}}
\newlabel{fig:arctrig5a}{{9.6}{395}{Reflecting $y=\cot (x)$ across $y=x$ yields $y=\operatorname {arccot}(x)$\relax }{figure.caption.617}{}}
\newlabel{arctangentcotangentfunctionprops}{{75}{395}{Inverse Trigonometric Functions}{theoremcounter.75}{}}
\pgfsyspdfmark {pgfid618}{4474142}{22905290}
\newlabel{ex_arctan}{{163}{395}{Inverse Trigonometric Functions}{examplecounter.163}{}}
\pgfsyspdfmark {pgfid619}{9979166}{22741450}
\newlabel{eex_arctan}{{75}{396}{Inverse Trigonometric Functions}{Item.2943}{}}
\pgfsyspdfmark {pgfid620}{14550631}{3163422}
\@writefile{lof}{\contentsline {figure}{\numberline {9.7}{\ignorespaces The fundamental cycles of $f(x)=\qopname  \relax o{sec}(x)$ and $g(x)=\qopname  \relax o{csc}(x)$\relax }}{397}{figure.caption.618}}
\newlabel{fig:arctrig6}{{9.7}{397}{The fundamental cycles of $f(x)=\sec (x)$ and $g(x)=\csc (x)$\relax }{figure.caption.618}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {9.1.1}Inverses of Secant and Cosecant: Trigonometry Friendly Approach}{397}{subsection.9.1.1}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.8}{\ignorespaces The ``Trigonometry Friendly'' definition of $\operatorname  {arcsec}(x)$\relax }}{398}{figure.caption.619}}
\newlabel{fig:arctrig7}{{9.8}{398}{The ``Trigonometry Friendly'' definition of $\operatorname {arcsec}(x)$\relax }{figure.caption.619}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.9}{\ignorespaces The ``Trigonometry Friendly'' definition of $\operatorname  {arccsc}(x)$\relax }}{398}{figure.caption.620}}
\newlabel{fig:arctrig8}{{9.9}{398}{The ``Trigonometry Friendly'' definition of $\operatorname {arccsc}(x)$\relax }{figure.caption.620}{}}
\newlabel{arcsecantcosecantfunctionprops1}{{76}{398}{Inverses of Secant and Cosecant: Trigonometry Friendly Approach}{theoremcounter.76}{}}
\pgfsyspdfmark {pgfid622}{14288487}{21939821}
\newlabel{arcsecantcosecantex1}{{164}{398}{Inverses of Secant and Cosecant: Trigonometry Friendly Approach}{examplecounter.164}{}}
\pgfsyspdfmark {pgfid623}{19793511}{21775981}
\newlabel{earcsecantcosecantex1}{{76}{399}{Inverses of Secant and Cosecant: Trigonometry Friendly Approach}{Item.2959}{}}
\pgfsyspdfmark {pgfid624}{4736286}{5260226}
\@writefile{lof}{\contentsline {figure}{\numberline {9.10}{\ignorespaces The ``Calculus Friendly'' definition of $\operatorname  {arcsec}(x)$\relax }}{400}{figure.caption.621}}
\newlabel{fig:arctrig9}{{9.10}{400}{The ``Calculus Friendly'' definition of $\operatorname {arcsec}(x)$\relax }{figure.caption.621}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.11}{\ignorespaces The ``Calculus Friendly definition of $\operatorname  {arccsc}(x)$\relax }}{400}{figure.caption.622}}
\newlabel{fig:arctrig10}{{9.11}{400}{The ``Calculus Friendly definition of $\operatorname {arccsc}(x)$\relax }{figure.caption.622}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {9.1.2}Inverses of Secant and Cosecant: Calculus Friendly Approach}{400}{subsection.9.1.2}}
\newlabel{arcsecantcosecantfunctionprops2}{{77}{400}{Inverses of Secant and Cosecant: Calculus Friendly Approach}{theoremcounter.77}{}}
\pgfsyspdfmark {pgfid626}{14288487}{13672782}
\newlabel{arcsecantcosecantex2}{{165}{400}{Inverses of Secant and Cosecant: Calculus Friendly Approach}{examplecounter.165}{}}
\pgfsyspdfmark {pgfid627}{19793511}{13508942}
\pgfsyspdfmark {pgfid628}{6374688}{27363620}
\newlabel{earcsecantcosecantex2}{{77}{402}{Inverses of Secant and Cosecant: Calculus Friendly Approach}{Item.2975}{}}
\pgfsyspdfmark {pgfid629}{14550631}{28751386}
\@writefile{toc}{\contentsline {subsection}{\numberline {9.1.3}Calculators and the Inverse Circular Functions.}{402}{subsection.9.1.3}}
\newlabel{sectionarcstuffoncalc}{{9.1.3}{402}{Calculators and the Inverse Circular Functions}{subsection.9.1.3}{}}
\pgfsyspdfmark {pgfid630}{14288487}{19288368}
\newlabel{arcstuffoncalc}{{166}{402}{Calculators and the Inverse Circular Functions}{examplecounter.166}{}}
\pgfsyspdfmark {pgfid631}{19793511}{19124528}
\newlabel{arccotneg2}{{1c}{402}{Calculators and the Inverse Circular Functions}{Item.2979}{}}
\newlabel{arccotangentcalc}{{2c}{402}{Calculators and the Inverse Circular Functions}{Item.2984}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.12}{\ignorespaces Evaluating $\operatorname  {arccot}(-2)$\relax }}{403}{figure.caption.623}}
\newlabel{fig:arctrig11}{{9.12}{403}{Evaluating $\operatorname {arccot}(-2)$\relax }{figure.caption.623}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.13}{\ignorespaces Evaluating $\operatorname  {arccot}(-2)$\relax }}{403}{figure.caption.624}}
\newlabel{fig:arctrig12}{{9.13}{403}{Evaluating $\operatorname {arccot}(-2)$\relax }{figure.caption.624}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.14}{\ignorespaces Evaluating $\operatorname  {arccsc}\left (-\frac  {3}{2}\right )$\relax }}{403}{figure.caption.625}}
\newlabel{fig:arctrig13}{{9.14}{403}{Evaluating $\operatorname {arccsc}\left (-\frac {3}{2}\right )$\relax }{figure.caption.625}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.15}{\ignorespaces $y=f(x)=\genfrac  {}{}{}0{{"π}}{2}-\qopname  \relax o{arccos}\left (\genfrac  {}{}{}0{x}{5}\right )$\relax }}{403}{figure.caption.626}}
\newlabel{fig:arctrig14}{{9.15}{403}{$y=f(x)=\dfrac {\pi }{2}-\arccos \left (\dfrac {x}{5}\right )$\relax }{figure.caption.626}{}}
\pgfsyspdfmark {pgfid632}{7816480}{23395299}
\@writefile{lof}{\contentsline {figure}{\numberline {9.16}{\ignorespaces $y=f(x)=3\qopname  \relax o{arctan}(4x)$\relax }}{404}{figure.caption.627}}
\newlabel{fig:arctrig15}{{9.16}{404}{$y=f(x)=3\arctan (4x)$\relax }{figure.caption.627}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.17}{\ignorespaces $y=g(x)=\operatorname  {arccot}\left (\genfrac  {}{}{}0{{"π}}{2}\right )+{"π}$\relax }}{404}{figure.caption.628}}
\newlabel{fig:arctrig16}{{9.17}{404}{$y=g(x)=\operatorname {arccot}\left (\dfrac {\pi }{2}\right )+\pi $\relax }{figure.caption.628}{}}
\newlabel{earcstuffoncalc}{{9.1.3}{404}{Calculators and the Inverse Circular Functions}{Item.2993}{}}
\pgfsyspdfmark {pgfid633}{14550631}{8700266}
\@writefile{lof}{\contentsline {figure}{\numberline {9.18}{\ignorespaces Angle of inclination ${"θ}$ for Example \ref  {roofpitchex}\relax }}{405}{figure.caption.629}}
\newlabel{fig:arctrigappex}{{9.18}{405}{Angle of inclination $\theta $ for Example \ref {roofpitchex}\relax }{figure.caption.629}{}}
\pgfsyspdfmark {pgfid634}{4474142}{46871346}
\newlabel{roofpitchex}{{167}{405}{Calculators and the Inverse Circular Functions}{examplecounter.167}{}}
\newlabel{eroofpitchex}{{9.1.3}{405}{Calculators and the Inverse Circular Functions}{examplecounter.167}{}}
\pgfsyspdfmark {pgfid635}{5719326}{28279060}
\@writefile{toc}{\contentsline {subsection}{\numberline {9.1.4}Solving Equations Using the Inverse Trigonometric Functions.}{405}{subsection.9.1.4}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.19}{\ignorespaces Solving $\qopname  \relax o{sin}({"θ})=\frac  {1}{3}$\relax }}{406}{figure.caption.630}}
\newlabel{fig:trigeqn1}{{9.19}{406}{Solving $\sin (\theta )=\frac {1}{3}$\relax }{figure.caption.630}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.20}{\ignorespaces Solving $\qopname  \relax o{tan}(t)=-2$\relax }}{406}{figure.caption.631}}
\newlabel{fig:trigeqn2}{{9.20}{406}{Solving $\tan (t)=-2$\relax }{figure.caption.631}{}}
\pgfsyspdfmark {pgfid636}{14288487}{46871346}
\newlabel{basicinverseeqns}{{168}{406}{Solving Equations Using the Inverse Trigonometric Functions}{examplecounter.168}{}}
\pgfsyspdfmark {pgfid637}{19793511}{46707506}
\newlabel{basicinverseeqnssine}{{1}{406}{Solving Equations Using the Inverse Trigonometric Functions}{Item.2994}{}}
\newlabel{basicinverseeqnstangent}{{2}{406}{Solving Equations Using the Inverse Trigonometric Functions}{Item.2995}{}}
\newlabel{basicinverseeqnssecant}{{3}{406}{Solving Equations Using the Inverse Trigonometric Functions}{Item.2996}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.21}{\ignorespaces Solving $\qopname  \relax o{sec}(x)=-\frac  {5}{3}$\relax }}{407}{figure.caption.632}}
\newlabel{fig:trigeqn3}{{9.21}{407}{Solving $\sec (x)=-\frac {5}{3}$\relax }{figure.caption.632}{}}
\newlabel{ebasicinverseeqns}{{9.1.4}{407}{Solving Equations Using the Inverse Trigonometric Functions}{figure.caption.631}{}}
\pgfsyspdfmark {pgfid638}{4736286}{42644274}
\@writefile{toc}{\contentsline {section}{\numberline {9.2}Trigonometric Equations and Inequalities}{414}{section.9.2}}
\newlabel{TrigEquIneq}{{9.2}{414}{Trigonometric Equations and Inequalities}{section.9.2}{}}
\newlabel{trigeqnstrategy1}{{39}{414}{Trigonometric Equations and Inequalities}{keyideacounter.39}{}}
\pgfsyspdfmark {pgfid640}{14288487}{19215321}
\newlabel{TrigEqnEx1}{{169}{414}{Trigonometric Equations and Inequalities}{examplecounter.169}{}}
\pgfsyspdfmark {pgfid641}{19793511}{19051481}
\newlabel{arctanin02pi}{{5}{414}{Trigonometric Equations and Inequalities}{Item.3247}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.22}{\ignorespaces Solving $\qopname  \relax o{cos}(2x)=-\frac  {\sqrt  {3}}{2}$\relax }}{415}{figure.caption.637}}
\newlabel{fig:trigeg1-1}{{9.22}{415}{Solving $\cos (2x)=-\frac {\sqrt {3}}{2}$\relax }{figure.caption.637}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.23}{\ignorespaces Solving $\qopname  \relax o{csc}\left (\frac  {1}{3}x-{"π}\right )=\sqrt  {2}$\relax }}{415}{figure.caption.638}}
\newlabel{fig:trigeg1-2}{{9.23}{415}{Solving $\csc \left (\frac {1}{3}x-\pi \right )=\sqrt {2}$\relax }{figure.caption.638}{}}
\pgfsyspdfmark {pgfid642}{6374688}{27664326}
\@writefile{lof}{\contentsline {figure}{\numberline {9.24}{\ignorespaces Solving $\qopname  \relax o{cot}(3x)=0$\relax }}{416}{figure.caption.645}}
\newlabel{fig:trigeg1-3}{{9.24}{416}{Solving $\cot (3x)=0$\relax }{figure.caption.645}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.25}{\ignorespaces Solving $\qopname  \relax o{sec}^2(x)=4$\relax }}{416}{figure.caption.646}}
\newlabel{fig:trigeg1-4}{{9.25}{416}{Solving $\sec ^2(x)=4$\relax }{figure.caption.646}{}}
\pgfsyspdfmark {pgfid643}{16189033}{17242852}
\@writefile{lof}{\contentsline {figure}{\numberline {9.26}{\ignorespaces Solving $\qopname  \relax o{tan}(\frac  {x}{2})=-3$\relax }}{417}{figure.caption.657}}
\newlabel{fig:trigeg1-5}{{9.26}{417}{Solving $\tan (\frac {x}{2})=-3$\relax }{figure.caption.657}{}}
\pgfsyspdfmark {pgfid644}{6374688}{13187352}
\@writefile{lof}{\contentsline {figure}{\numberline {9.27}{\ignorespaces Solving $\qopname  \relax o{sin}(2x)=0.87$\relax }}{418}{figure.caption.664}}
\newlabel{fig:trigeg1-6}{{9.27}{418}{Solving $\sin (2x)=0.87$\relax }{figure.caption.664}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.28}{\ignorespaces Solving $3\qopname  \relax o{sin}^3(x)=\qopname  \relax o{sin}^2(x)$\relax }}{418}{figure.caption.665}}
\newlabel{fig:trigeg2-1a}{{9.28}{418}{Solving $3\sin ^3(x)=\sin ^2(x)$\relax }{figure.caption.665}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.29}{\ignorespaces Zooming in on the first two solutions for Example \ref  {TrigEqIdEx1}.1\relax }}{418}{figure.caption.666}}
\newlabel{fig:trigeg2-1b}{{9.29}{418}{Zooming in on the first two solutions for Example \ref {TrigEqIdEx1}.1\relax }{figure.caption.666}{}}
\newlabel{eTrigEqnEx1}{{39}{418}{Trigonometric Equations and Inequalities}{figure.caption.657}{}}
\pgfsyspdfmark {pgfid645}{14550631}{39852026}
\pgfsyspdfmark {pgfid646}{14288487}{34431369}
\newlabel{TrigEqIdEx1}{{170}{418}{Trigonometric Equations and Inequalities}{examplecounter.170}{}}
\pgfsyspdfmark {pgfid647}{19793511}{34267529}
\@writefile{lof}{\contentsline {figure}{\numberline {9.30}{\ignorespaces Solving $\qopname  \relax o{sec}^2(x)=\qopname  \relax o{tan}(x)+3$\relax }}{419}{figure.caption.673}}
\newlabel{fig:trigeg2-2}{{9.30}{419}{Solving $\sec ^2(x)=\tan (x)+3$\relax }{figure.caption.673}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.31}{\ignorespaces Solving $\qopname  \relax o{cos}(2x)=3\qopname  \relax o{cos}(x)-2$\relax }}{419}{figure.caption.674}}
\newlabel{fig:trigeg2-3}{{9.31}{419}{Solving $\cos (2x)=3\cos (x)-2$\relax }{figure.caption.674}{}}
\pgfsyspdfmark {pgfid648}{6374688}{14778453}
\@writefile{lof}{\contentsline {figure}{\numberline {9.32}{\ignorespaces Solving $\qopname  \relax o{cos}(3x)=2-\qopname  \relax o{cos}(x)$\relax }}{420}{figure.caption.677}}
\newlabel{fig:trigeg2-4}{{9.32}{420}{Solving $\cos (3x)=2-\cos (x)$\relax }{figure.caption.677}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.33}{\ignorespaces Solving $\qopname  \relax o{cos}(3x)=\qopname  \relax o{cos}(5x)$\relax }}{420}{figure.caption.678}}
\newlabel{fig:trigeg2-5}{{9.33}{420}{Solving $\cos (3x)=\cos (5x)$\relax }{figure.caption.678}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.34}{\ignorespaces Solving $\qopname  \relax o{sin}(2x)-\sqrt  {3}\qopname  \relax o{cos}(x)$\relax }}{420}{figure.caption.679}}
\newlabel{fig:trigeg2-6}{{9.34}{420}{Solving $\sin (2x)-\sqrt {3}\cos (x)$\relax }{figure.caption.679}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.35}{\ignorespaces Solving $\qopname  \relax o{sin}(x)\qopname  \relax o{cos}(\frac  {x}{2})+\qopname  \relax o{cos}(x)\qopname  \relax o{sin}(\frac  {x}{2})=1$\relax }}{420}{figure.caption.680}}
\newlabel{fig:trigeg2-7}{{9.35}{420}{Solving $\sin (x)\cos (\frac {x}{2})+\cos (x)\sin (\frac {x}{2})=1$\relax }{figure.caption.680}{}}
\pgfsyspdfmark {pgfid649}{16189033}{16036124}
\@writefile{lof}{\contentsline {figure}{\numberline {9.36}{\ignorespaces Solving $\qopname  \relax o{cos}(x)-\sqrt  {3}\qopname  \relax o{sin}(x)=2$\relax }}{421}{figure.caption.681}}
\newlabel{fig:trigeg2-8}{{9.36}{421}{Solving $\cos (x)-\sqrt {3}\sin (x)=2$\relax }{figure.caption.681}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.37}{\ignorespaces Solving $2\qopname  \relax o{sin}(x)\leq 1$\relax }}{421}{figure.caption.682}}
\newlabel{fig:trigeg3-1}{{9.37}{421}{Solving $2\sin (x)\leq 1$\relax }{figure.caption.682}{}}
\newlabel{eTrigEqIdEx1}{{39}{421}{Trigonometric Equations and Inequalities}{Item.3270}{}}
\pgfsyspdfmark {pgfid650}{4736286}{28968873}
\pgfsyspdfmark {pgfid651}{4474142}{23032998}
\newlabel{TrigIneqEx1}{{171}{421}{Trigonometric Equations and Inequalities}{examplecounter.171}{}}
\pgfsyspdfmark {pgfid652}{9979166}{22869158}
\@writefile{lof}{\contentsline {figure}{\numberline {9.38}{\ignorespaces Solving $\qopname  \relax o{sin}(2x)>\qopname  \relax o{cos}(x)$\relax }}{422}{figure.caption.683}}
\newlabel{fig:trigeg3-2}{{9.38}{422}{Solving $\sin (2x)>\cos (x)$\relax }{figure.caption.683}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.39}{\ignorespaces Solving $\qopname  \relax o{tan}(x)\geq 3$\relax }}{422}{figure.caption.684}}
\newlabel{fig:trigeg3-3}{{9.39}{422}{Solving $\tan (x)\geq 3$\relax }{figure.caption.684}{}}
\newlabel{eTrigIneqEx1}{{39}{422}{Trigonometric Equations and Inequalities}{Item.3276}{}}
\pgfsyspdfmark {pgfid653}{14550631}{14337938}
\pgfsyspdfmark {pgfid654}{14288487}{11395943}
\newlabel{TrigDomainEx1}{{172}{422}{Trigonometric Equations and Inequalities}{examplecounter.172}{}}
\pgfsyspdfmark {pgfid655}{19793511}{11232103}
\pgfsyspdfmark {pgfid656}{6374688}{13336464}
\newlabel{eTrigDomainEx1}{{39}{424}{Trigonometric Equations and Inequalities}{Item.3282}{}}
\pgfsyspdfmark {pgfid657}{14550631}{3163422}
\@writefile{lof}{\contentsline {figure}{\numberline {9.40}{\ignorespaces Solving $\qopname  \relax o{arcsin}(2x)=\frac  {{"π}}{3}$\relax }}{425}{figure.caption.691}}
\newlabel{fig:trigeg4-1}{{9.40}{425}{Solving $\arcsin (2x)=\frac {\pi }{3}$\relax }{figure.caption.691}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.41}{\ignorespaces Solving $4\qopname  \relax o{arccos}(x)-3{"π}=0$\relax }}{425}{figure.caption.692}}
\newlabel{fig:trigeg4-2}{{9.41}{425}{Solving $4\arccos (x)-3\pi =0$\relax }{figure.caption.692}{}}
\pgfsyspdfmark {pgfid658}{4474142}{44990354}
\newlabel{ex_trigeglast}{{173}{425}{Trigonometric Equations and Inequalities}{examplecounter.173}{}}
\pgfsyspdfmark {pgfid659}{9979166}{44826514}
\@writefile{lof}{\contentsline {figure}{\numberline {9.42}{\ignorespaces Solving $3\operatorname  {arcsec}(2x-1)+{"π}=2{"π}$\relax }}{426}{figure.caption.697}}
\newlabel{fig:trigeg4-3}{{9.42}{426}{Solving $3\operatorname {arcsec}(2x-1)+\pi =2\pi $\relax }{figure.caption.697}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.43}{\ignorespaces Solving $4\qopname  \relax o{arctan}^2(x)-3{"π}\qopname  \relax o{arctan}(x)-{"π}^2=0$\relax }}{426}{figure.caption.698}}
\newlabel{fig:trigeg4-4}{{9.43}{426}{Solving $4\arctan ^2(x)-3\pi \arctan (x)-\pi ^2=0$\relax }{figure.caption.698}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.44}{\ignorespaces Solving ${"π}^2-4\qopname  \relax o{arccos}^2(x)<0$\relax }}{426}{figure.caption.699}}
\newlabel{fig:trigeg4-5}{{9.44}{426}{Solving $\pi ^2-4\arccos ^2(x)<0$\relax }{figure.caption.699}{}}
\pgfsyspdfmark {pgfid660}{16189033}{29588473}
\@writefile{lof}{\contentsline {figure}{\numberline {9.45}{\ignorespaces Solving $4\operatorname  {arccot}(3x)>{"π}$\relax }}{427}{figure.caption.700}}
\newlabel{fig:trigeg4-6}{{9.45}{427}{Solving $4\operatorname {arccot}(3x)>\pi $\relax }{figure.caption.700}{}}
\newlabel{eex_trigeglast}{{39}{427}{Trigonometric Equations and Inequalities}{figure.caption.699}{}}
\pgfsyspdfmark {pgfid661}{4736286}{39432124}
\@writefile{lof}{\contentsline {figure}{\numberline {9.46}{\ignorespaces Properties of a generic sinusoid \relax }}{431}{figure.caption.701}}
\newlabel{genericsinsuoidfigure}{{9.46}{431}{Properties of a generic sinusoid \index {sinusoid ! graph of}\relax }{figure.caption.701}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.47}{\ignorespaces The Giant Wheel\relax }}{431}{figure.caption.702}}
\newlabel{fig:bigwheel}{{9.47}{431}{The Giant Wheel\relax }{figure.caption.702}{}}
\@writefile{toc}{\contentsline {section}{\numberline {9.3}Applications of Sinusoids}{431}{section.9.3}}
\newlabel{Sinusoid}{{9.3}{431}{Applications of Sinusoids}{section.9.3}{}}
\newlabel{sinesinusoidprops}{{40}{431}{Applications of Sinusoids}{keyideacounter.40}{}}
\pgfsyspdfmark {pgfid663}{4474142}{12802144}
\newlabel{ycoordonwheel}{{174}{431}{Applications of Sinusoids}{examplecounter.174}{}}
\pgfsyspdfmark {pgfid664}{9979166}{12638304}
\@writefile{lof}{\contentsline {figure}{\numberline {9.48}{\ignorespaces The graph $y=h(t)=64\qopname  \relax o{sin}\left (\frac  {4{"π}}{127}t-\frac  {{"π}}{2}\right )+72$\relax }}{432}{figure.caption.703}}
\newlabel{fig:sinusoid1}{{9.48}{432}{The graph $y=h(t)=64\sin \left (\frac {4\pi }{127}t-\frac {\pi }{2}\right )+72$\relax }{figure.caption.703}{}}
\newlabel{eycoordonwheel}{{40}{432}{Applications of Sinusoids}{figure.caption.702}{}}
\pgfsyspdfmark {pgfid665}{15533671}{27756750}
\@writefile{toc}{\contentsline {subsection}{\numberline {9.3.1}Harmonic Motion}{432}{subsection.9.3.1}}
\newlabel{harmomicmotion}{{9.3.1}{432}{Harmonic Motion}{subsection.9.3.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.49}{\ignorespaces A mass on a spring undergoing (approximate) simple harmonic motion\relax }}{433}{figure.caption.704}}
\newlabel{freeundampedmotion}{{78}{434}{Harmonic Motion}{theoremcounter.78}{}}
\pgfsyspdfmark {pgfid667}{14288487}{33602375}
\newlabel{freeudampedex}{{175}{434}{Harmonic Motion}{examplecounter.175}{}}
\pgfsyspdfmark {pgfid668}{19793511}{33438535}
\@writefile{lof}{\contentsline {figure}{\numberline {9.50}{\ignorespaces $y=x(t)=3\qopname  \relax o{sin}(2t+\frac  {{"π}}{2})$ in Example \ref  {freeudampedex}.1\relax }}{435}{figure.caption.705}}
\newlabel{fig:sinusoid2}{{9.50}{435}{$y=x(t)=3\sin (2t+\frac {\pi }{2})$ in Example \ref {freeudampedex}.1\relax }{figure.caption.705}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.51}{\ignorespaces The graph $y=5\qopname  \relax o{sin}(2x+[{"π}-\qopname  \relax o{arcsin}(\frac  {3}{5})])$ in Example \ref  {freeudampedex}.2\relax }}{435}{figure.caption.706}}
\newlabel{fig:sinusoid3}{{9.51}{435}{The graph $y=5\sin (2x+[\pi -\arcsin (\frac {3}{5})])$ in Example \ref {freeudampedex}.2\relax }{figure.caption.706}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.52}{\ignorespaces Graphing $x(t)$ in Example \ref  {underdampedresonance}.1\relax }}{435}{figure.caption.707}}
\newlabel{fig:sinusoid4}{{9.52}{435}{Graphing $x(t)$ in Example \ref {underdampedresonance}.1\relax }{figure.caption.707}{}}
\newlabel{efreeudampedex}{{78}{435}{Harmonic Motion}{Item.3406}{}}
\pgfsyspdfmark {pgfid669}{4736286}{17135924}
\pgfsyspdfmark {pgfid670}{4474142}{11855832}
\newlabel{underdampedresonance}{{176}{435}{Harmonic Motion}{examplecounter.176}{}}
\pgfsyspdfmark {pgfid671}{9979166}{11691992}
\@writefile{lof}{\contentsline {figure}{\numberline {9.53}{\ignorespaces Graphing $x(t)$ in Example \ref  {underdampedresonance}.2\relax }}{436}{figure.caption.708}}
\newlabel{fig:sinusoid5}{{9.53}{436}{Graphing $x(t)$ in Example \ref {underdampedresonance}.2\relax }{figure.caption.708}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.54}{\ignorespaces Graphing $x(t)$ in Example \ref  {underdampedresonance}.2\relax }}{436}{figure.caption.709}}
\newlabel{fig:sinusoid6}{{9.54}{436}{Graphing $x(t)$ in Example \ref {underdampedresonance}.2\relax }{figure.caption.709}{}}
\newlabel{eunderdampedresonance}{{78}{436}{Harmonic Motion}{Item.3412}{}}
\pgfsyspdfmark {pgfid672}{14550631}{4950350}
\newlabel{heightlondoneye}{{3}{437}{Harmonic Motion}{Item.3415}{}}
\newlabel{leftrightlondoneye}{{4}{437}{Harmonic Motion}{Item.3416}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.55}{\ignorespaces The triangle in Example \ref  {righttrianglereviewex}\relax }}{438}{figure.caption.710}}
\newlabel{fig:sines1}{{9.55}{438}{The triangle in Example \ref {righttrianglereviewex}\relax }{figure.caption.710}{}}
\@writefile{toc}{\contentsline {section}{\numberline {9.4}Law of Sines}{438}{section.9.4}}
\newlabel{LawofSines}{{9.4}{438}{Law of Sines}{section.9.4}{}}
\pgfsyspdfmark {pgfid673}{14288487}{35791800}
\newlabel{righttrianglereviewex}{{177}{438}{Law of Sines}{examplecounter.177}{}}
\newlabel{erighttrianglereviewex}{{9.4}{438}{Law of Sines}{examplecounter.177}{}}
\pgfsyspdfmark {pgfid674}{21498695}{19653703}
\newlabel{lawofsines}{{79}{439}{Law of Sines}{theoremcounter.79}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.56}{\ignorespaces $\triangle ABC$ for the first case of the proof of Theorem \ref  {lawofsines}\relax }}{439}{figure.caption.711}}
\newlabel{fig:sines2}{{9.56}{439}{$\triangle ABC$ for the first case of the proof of Theorem \ref {lawofsines}\relax }{figure.caption.711}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.57}{\ignorespaces $\triangle ABC$ for the second case of the proof of Theorem \ref  {lawofsines}\relax }}{439}{figure.caption.712}}
\newlabel{fig:sines3}{{9.57}{439}{$\triangle ABC$ for the second case of the proof of Theorem \ref {lawofsines}\relax }{figure.caption.712}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.59}{\ignorespaces Triangle for Example \ref  {losex} number \ref  {losaas}\relax }}{440}{figure.caption.714}}
\newlabel{fig:sineseg1}{{9.59}{440}{Triangle for Example \ref {losex} number \ref {losaas}\relax }{figure.caption.714}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.60}{\ignorespaces Triangle for Example \ref  {losex} number \ref  {losasa}\relax }}{440}{figure.caption.715}}
\newlabel{fig:sineseg2}{{9.60}{440}{Triangle for Example \ref {losex} number \ref {losasa}\relax }{figure.caption.715}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.58}{\ignorespaces $\triangle ABC$ for the third case of the proof of Theorem \ref  {lawofsines}\relax }}{440}{figure.caption.713}}
\newlabel{fig:sines4}{{9.58}{440}{$\triangle ABC$ for the third case of the proof of Theorem \ref {lawofsines}\relax }{figure.caption.713}{}}
\pgfsyspdfmark {pgfid676}{14288487}{36545960}
\newlabel{losex}{{178}{440}{Law of Sines}{examplecounter.178}{}}
\pgfsyspdfmark {pgfid677}{19793511}{36382120}
\newlabel{losaas}{{1}{440}{Law of Sines}{Item.3428}{}}
\newlabel{losasa}{{2}{440}{Law of Sines}{Item.3429}{}}
\newlabel{losnotriangleex}{{3}{440}{Law of Sines}{Item.3430}{}}
\newlabel{losrighttriangleex}{{4}{440}{Law of Sines}{Item.3431}{}}
\newlabel{lostwotriangleex}{{5}{440}{Law of Sines}{Item.3432}{}}
\newlabel{losonetriangleex}{{6}{440}{Law of Sines}{Item.3433}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.62}{\ignorespaces Triangle for Example \ref  {losex} number \ref  {losnotriangleex}\relax }}{441}{figure.caption.717}}
\newlabel{fig:sineseg3}{{9.62}{441}{Triangle for Example \ref {losex} number \ref {losnotriangleex}\relax }{figure.caption.717}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.63}{\ignorespaces Triangle for Example \ref  {losex} number \ref  {losrighttriangleex}\relax }}{441}{figure.caption.718}}
\newlabel{fig:sineseg4}{{9.63}{441}{Triangle for Example \ref {losex} number \ref {losrighttriangleex}\relax }{figure.caption.718}{}}
\pgfsyspdfmark {pgfid678}{6374688}{35335847}
\@writefile{lof}{\contentsline {figure}{\numberline {9.61}{\ignorespaces Triangle for Example \ref  {losex} number \ref  {lostwotriangleex}\relax }}{441}{figure.caption.716}}
\newlabel{fig:sineseg5}{{9.61}{441}{Triangle for Example \ref {losex} number \ref {lostwotriangleex}\relax }{figure.caption.716}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.65}{\ignorespaces Triangle for Example \ref  {losex} number \ref  {losonetriangleex}\relax }}{442}{figure.caption.720}}
\newlabel{fig:sineseg6}{{9.65}{442}{Triangle for Example \ref {losex} number \ref {losonetriangleex}\relax }{figure.caption.720}{}}
\newlabel{elosex}{{79}{442}{Law of Sines}{figure.caption.718}{}}
\pgfsyspdfmark {pgfid679}{14550631}{41482009}
\newlabel{ASScase}{{80}{442}{Law of Sines}{theoremcounter.80}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.67}{\ignorespaces Diagrams for Example \ref  {losapplication}\relax }}{443}{figure.caption.722}}
\newlabel{fig:sinesappeg}{{9.67}{443}{Diagrams for Example \ref {losapplication}\relax }{figure.caption.722}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.64}{\ignorespaces Illustrating the first two cases in Theorem \ref  {ASScase}\relax }}{443}{figure.caption.719}}
\newlabel{fig:asasines}{{9.64}{443}{Illustrating the first two cases in Theorem \ref {ASScase}\relax }{figure.caption.719}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.66}{\ignorespaces Illustrating the last two cases in Theorem \ref  {ASScase}\relax }}{443}{figure.caption.721}}
\newlabel{fig:asasines2}{{9.66}{443}{Illustrating the last two cases in Theorem \ref {ASScase}\relax }{figure.caption.721}{}}
\pgfsyspdfmark {pgfid681}{4474142}{13577399}
\newlabel{losapplication}{{179}{443}{Law of Sines}{examplecounter.179}{}}
\pgfsyspdfmark {pgfid682}{9979166}{13413559}
\newlabel{elosapplication}{{80}{444}{Law of Sines}{examplecounter.179}{}}
\pgfsyspdfmark {pgfid683}{35191022}{36688584}
\newlabel{areaformulasine}{{81}{444}{Law of Sines}{theoremcounter.81}{}}
\pgfsyspdfmark {pgfid685}{14288487}{25113907}
\newlabel{areaformulasineex}{{180}{444}{Law of Sines}{examplecounter.180}{}}
\newlabel{eareaformulasineex}{{81}{444}{Law of Sines}{examplecounter.180}{}}
\pgfsyspdfmark {pgfid686}{27801191}{17561412}
\newlabel{firstlawofsines}{{1}{445}{Law of Sines}{Item.3440}{}}
\newlabel{secondarea}{{12}{445}{Law of Sines}{Item.3451}{}}
\newlabel{lastlawofsines}{{20}{445}{Law of Sines}{Item.3459}{}}
\newlabel{gradeofroad}{{81}{445}{Law of Sines}{section*.723}{}}
\newlabel{firstroadgrade}{{22}{445}{Law of Sines}{Item.3461}{}}
\newlabel{lastroadgrade}{{24}{445}{Law of Sines}{Item.3463}{}}
\newlabel{bearings}{{81}{446}{Law of Sines}{section*.724}{}}
\@writefile{toc}{\contentsline {section}{\numberline {9.5}Law of Cosines}{447}{section.9.5}}
\newlabel{LawofCosines}{{9.5}{447}{Law of Cosines}{section.9.5}{}}
\newlabel{lawofcosines}{{82}{447}{Law of Cosines}{theoremcounter.82}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.68}{\ignorespaces Generic triangle for the proof of Theorem \ref  {lawofcosines}\relax }}{447}{figure.caption.725}}
\newlabel{fig:cosines1}{{9.68}{447}{Generic triangle for the proof of Theorem \ref {lawofcosines}\relax }{figure.caption.725}{}}
\pgfsyspdfmark {pgfid688}{14288487}{18918479}
\newlabel{locex}{{181}{448}{Law of Cosines}{examplecounter.181}{}}
\pgfsyspdfmark {pgfid689}{19793511}{18754639}
\newlabel{locsas}{{1}{448}{Law of Cosines}{Item.3488}{}}
\newlabel{locsss}{{2}{448}{Law of Cosines}{Item.3489}{}}
\pgfsyspdfmark {pgfid690}{6374688}{23483101}
\@writefile{lof}{\contentsline {figure}{\numberline {9.69}{\ignorespaces Triangle for Example \ref  {locex}.\ref  {locsas}\relax }}{449}{figure.caption.728}}
\newlabel{fig:cosines2}{{9.69}{449}{Triangle for Example \ref {locex}.\ref {locsas}\relax }{figure.caption.728}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.71}{\ignorespaces The pond in Example \ref  {locapplication}\relax }}{450}{figure.caption.732}}
\newlabel{fig:cosines4}{{9.71}{450}{The pond in Example \ref {locapplication}\relax }{figure.caption.732}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.70}{\ignorespaces Triangle for Example \ref  {locex}.\ref  {locsss}\relax }}{450}{figure.caption.729}}
\newlabel{fig:cosines3}{{9.70}{450}{Triangle for Example \ref {locex}.\ref {locsss}\relax }{figure.caption.729}{}}
\newlabel{elocex}{{9.5}{450}{Law of Cosines}{figure.caption.729}{}}
\pgfsyspdfmark {pgfid691}{14550631}{33631550}
\pgfsyspdfmark {pgfid692}{14288487}{26874258}
\newlabel{locapplication}{{182}{450}{Law of Cosines}{examplecounter.182}{}}
\newlabel{elocapplication}{{9.5}{450}{Law of Cosines}{examplecounter.182}{}}
\pgfsyspdfmark {pgfid693}{21865284}{18059666}
\newlabel{HeronsFormula}{{83}{450}{Law of Cosines}{theoremcounter.83}{}}
\pgfsyspdfmark {pgfid695}{14288487}{38751374}
\newlabel{heronex}{{183}{452}{Law of Cosines}{examplecounter.183}{}}
\newlabel{eheronex}{{83}{452}{Law of Cosines}{examplecounter.183}{}}
\pgfsyspdfmark {pgfid696}{28375829}{33659325}
\newlabel{firstlawofcosines}{{1}{453}{Law of Cosines}{Item.3492}{}}
\newlabel{firstherons}{{6}{453}{Law of Cosines}{Item.3497}{}}
\newlabel{secondherons}{{8}{453}{Law of Cosines}{Item.3499}{}}
\newlabel{thirdherons}{{10}{453}{Law of Cosines}{Item.3501}{}}
\newlabel{lastlawofcosines}{{10}{453}{Law of Cosines}{Item.3501}{}}
\newlabel{ambigfirst}{{11}{453}{Law of Cosines}{Item.3502}{}}
\newlabel{anylawfirst}{{11}{453}{Law of Cosines}{Item.3502}{}}
\newlabel{ambigsecond}{{13}{453}{Law of Cosines}{Item.3504}{}}
\newlabel{ambigthird}{{14}{453}{Law of Cosines}{Item.3505}{}}
\newlabel{anylawlast}{{16}{453}{Law of Cosines}{Item.3507}{}}
\newlabel{lofcosinesbearingexercise}{{20}{453}{Law of Cosines}{Item.3511}{}}
\@writefile{toc}{\contentsline {section}{\numberline {9.6}Polar Coordinates}{455}{section.9.6}}
\newlabel{IntroPolar}{{9.6}{455}{Polar Coordinates}{section.9.6}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.72}{\ignorespaces Rectangular vs. Polar Coordinates\relax }}{455}{figure.caption.739}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.73}{\ignorespaces Locating a point using polar coordinates\relax }}{455}{figure.caption.740}}
\newlabel{fig:polar1}{{9.73}{455}{Locating a point using polar coordinates\relax }{figure.caption.740}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.74}{\ignorespaces Performing the rotation first\relax }}{456}{figure.caption.741}}
\newlabel{fig:polar2}{{9.74}{456}{Performing the rotation first\relax }{figure.caption.741}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.75}{\ignorespaces Using polar coordinates when $r<0$\relax }}{456}{figure.caption.742}}
\newlabel{fig:polar3}{{9.75}{456}{Using polar coordinates when $r<0$\relax }{figure.caption.742}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.76}{\ignorespaces Performing the rotation first to plot the point in Figure \ref  {fig:polar3}\relax }}{456}{figure.caption.743}}
\newlabel{fig:polar4}{{9.76}{456}{Performing the rotation first to plot the point in Figure \ref {fig:polar3}\relax }{figure.caption.743}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.77}{\ignorespaces ${"θ}=-\frac  {3{"π}}{4}<0$ produces a clockwise rotation\relax }}{456}{figure.caption.744}}
\newlabel{fig:polar5}{{9.77}{456}{$\theta =-\frac {3\pi }{4}<0$ produces a clockwise rotation\relax }{figure.caption.744}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.78}{\ignorespaces Rotating first with ${"θ}<0$\relax }}{457}{figure.caption.745}}
\newlabel{fig:polar6}{{9.78}{457}{Rotating first with $\theta <0$\relax }{figure.caption.745}{}}
\pgfsyspdfmark {pgfid697}{4474142}{36747250}
\newlabel{plotpolarex}{{184}{457}{Polar Coordinates}{examplecounter.184}{}}
\pgfsyspdfmark {pgfid698}{9979166}{36583410}
\@writefile{lof}{\contentsline {figure}{\numberline {9.79}{\ignorespaces Plotting $P(2,240^\circ )$\relax }}{457}{figure.caption.746}}
\newlabel{fig:polar7}{{9.79}{457}{Plotting $P(2,240^\circ )$\relax }{figure.caption.746}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.80}{\ignorespaces Alternate polar reprentations of $P(2,240^\circ )$\relax }}{457}{figure.caption.747}}
\newlabel{fig:polar8}{{9.80}{457}{Alternate polar reprentations of $P(2,240^\circ )$\relax }{figure.caption.747}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.81}{\ignorespaces Plotting $P(-4,\frac  {7{"π}}{6})$\relax }}{458}{figure.caption.748}}
\newlabel{fig:polar9}{{9.81}{458}{Plotting $P(-4,\frac {7\pi }{6})$\relax }{figure.caption.748}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.82}{\ignorespaces Alternate polar representations of $P(-4,\frac  {7{"π}}{6})$\relax }}{458}{figure.caption.749}}
\newlabel{fig:polar10}{{9.82}{458}{Alternate polar representations of $P(-4,\frac {7\pi }{6})$\relax }{figure.caption.749}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.83}{\ignorespaces Plotting $P(117,-\frac  {5{"π}}{2})$\relax }}{458}{figure.caption.750}}
\newlabel{fig:polar11}{{9.83}{458}{Plotting $P(117,-\frac {5\pi }{2})$\relax }{figure.caption.750}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.84}{\ignorespaces Alternate polar representations of $P(117,-\frac  {5{"π}}{2})$\relax }}{459}{figure.caption.751}}
\newlabel{fig:polar12}{{9.84}{459}{Alternate polar representations of $P(117,-\frac {5\pi }{2})$\relax }{figure.caption.751}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.85}{\ignorespaces Plotting $P(-3,-\frac  {{"π}}{4})$\relax }}{459}{figure.caption.752}}
\newlabel{fig:polar13}{{9.85}{459}{Plotting $P(-3,-\frac {\pi }{4})$\relax }{figure.caption.752}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.86}{\ignorespaces Alternate polar representations of $P(-3,-\frac  {{"π}}{4})$\relax }}{459}{figure.caption.753}}
\newlabel{fig:polar14}{{9.86}{459}{Alternate polar representations of $P(-3,-\frac {\pi }{4})$\relax }{figure.caption.753}{}}
\newlabel{eplotpolarex}{{9.6}{459}{Polar Coordinates}{figure.caption.753}{}}
\pgfsyspdfmark {pgfid699}{4736286}{10856848}
\newlabel{idea:polarrep}{{41}{460}{Polar Coordinates}{keyideacounter.41}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.87}{\ignorespaces $P$ has rectangular coordinates $(2,-2\sqrt  {3})$ and polar coordinates $(4,\frac  {5{"π}}{3})$\relax }}{461}{figure.caption.754}}
\newlabel{fig:polar15}{{9.87}{461}{$P$ has rectangular coordinates $(2,-2\sqrt {3})$ and polar coordinates $(4,\frac {5\pi }{3})$\relax }{figure.caption.754}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.88}{\ignorespaces $Q$ has rectangular coordinates $(-3,-3)$ and polar coordinates $(3\sqrt  {2},\frac  {5{"π}}{4})$\relax }}{461}{figure.caption.755}}
\newlabel{fig:polar16}{{9.88}{461}{$Q$ has rectangular coordinates $(-3,-3)$ and polar coordinates $(3\sqrt {2},\frac {5\pi }{4})$\relax }{figure.caption.755}{}}
\newlabel{polarrectangularconversion}{{84}{461}{Polar Coordinates}{theoremcounter.84}{}}
\pgfsyspdfmark {pgfid702}{4474142}{29222225}
\newlabel{pointconversionex}{{185}{461}{Polar Coordinates}{examplecounter.185}{}}
\pgfsyspdfmark {pgfid703}{9979166}{29058385}
\@writefile{lof}{\contentsline {figure}{\numberline {9.89}{\ignorespaces $R$ has rectangular coordinates $(0,-3)$ and polar coordinates $(-3,\frac  {3{"π}}{2})$\relax }}{462}{figure.caption.760}}
\newlabel{fig:polar17}{{9.89}{462}{$R$ has rectangular coordinates $(0,-3)$ and polar coordinates $(-3,\frac {3\pi }{2})$\relax }{figure.caption.760}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.90}{\ignorespaces $S$ has rectangular coordinates $(-3,4)$ and polar coordinates $(5,{"π}-\qopname  \relax o{arctan}(\frac  {4}{3}))$\relax }}{462}{figure.caption.761}}
\newlabel{fig:polar18}{{9.90}{462}{$S$ has rectangular coordinates $(-3,4)$ and polar coordinates $(5,\pi -\arctan (\frac {4}{3}))$\relax }{figure.caption.761}{}}
\newlabel{epointconversionex}{{84}{462}{Polar Coordinates}{Item.3534}{}}
\pgfsyspdfmark {pgfid704}{15533671}{25162293}
\pgfsyspdfmark {pgfid705}{14288487}{18751642}
\newlabel{recteqntopolar}{{186}{462}{Polar Coordinates}{examplecounter.186}{}}
\pgfsyspdfmark {pgfid706}{19793511}{18587802}
\newlabel{ris6costheta}{{1}{462}{Polar Coordinates}{Item.3535}{}}
\newlabel{yisnegx}{{2}{462}{Polar Coordinates}{Item.3536}{}}
\newlabel{erecteqntopolar}{{84}{463}{Polar Coordinates}{Item.3540}{}}
\pgfsyspdfmark {pgfid707}{4736286}{16217199}
\pgfsyspdfmark {pgfid708}{4474142}{15181979}
\newlabel{polareqntorect}{{187}{463}{Polar Coordinates}{examplecounter.187}{}}
\pgfsyspdfmark {pgfid709}{9979166}{15018139}
\newlabel{risneg3}{{1}{463}{Polar Coordinates}{Item.3541}{}}
\newlabel{cardioidtorect}{{3}{463}{Polar Coordinates}{Item.3543}{}}
\newlabel{epolareqntorect}{{84}{464}{Polar Coordinates}{AMS.765}{}}
\pgfsyspdfmark {pgfid710}{14550631}{3163422}
\newlabel{polarpointgraphfirst}{{1}{466}{Polar Coordinates}{Item.3547}{}}
\newlabel{polarpointgraphlast}{{16}{466}{Polar Coordinates}{Item.3562}{}}
\newlabel{polartorectfirst}{{17}{466}{Polar Coordinates}{Item.3563}{}}
\newlabel{polartorectlast}{{36}{466}{Polar Coordinates}{Item.3582}{}}
\newlabel{recttopolarfirst}{{37}{466}{Polar Coordinates}{Item.3583}{}}
\newlabel{recttopolarlast}{{56}{467}{Polar Coordinates}{Item.3602}{}}
\newlabel{equrecttopolarfirst}{{57}{467}{Polar Coordinates}{Item.3603}{}}
\newlabel{solvethetaone}{{60}{467}{Polar Coordinates}{Item.3606}{}}
\newlabel{solvethetafour}{{63}{467}{Polar Coordinates}{Item.3609}{}}
\newlabel{equrecttopolarlast}{{76}{467}{Polar Coordinates}{Item.3622}{}}
\newlabel{equpolartorectfirst}{{77}{467}{Polar Coordinates}{Item.3623}{}}
\newlabel{equpolartorectlast}{{96}{468}{Polar Coordinates}{Item.3642}{}}
\@writefile{toc}{\contentsline {section}{\numberline {9.7}The Polar Form of Complex Numbers}{469}{section.9.7}}
\newlabel{PolarComplex}{{9.7}{469}{The Polar Form of Complex Numbers}{section.9.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.91}{\ignorespaces The complex plane\relax }}{469}{figure.caption.766}}
\newlabel{fig:complexplane}{{9.91}{469}{The complex plane\relax }{figure.caption.766}{}}
\newlabel{modulusargumentdefn}{{56}{469}{The Polar Form of Complex Numbers}{definitioncounter.56}{}}
\pgfsyspdfmark {pgfid712}{14288487}{29977248}
\newlabel{plotmodargex}{{188}{470}{The Polar Form of Complex Numbers}{examplecounter.188}{}}
\pgfsyspdfmark {pgfid713}{19793511}{29813408}
\@writefile{lof}{\contentsline {figure}{\numberline {9.92}{\ignorespaces Plots of the four complex numbers in Example \ref  {plotmodargex}\relax }}{471}{figure.caption.767}}
\newlabel{fig:polarcom1}{{9.92}{471}{Plots of the four complex numbers in Example \ref {plotmodargex}\relax }{figure.caption.767}{}}
\newlabel{eplotmodargex}{{56}{471}{The Polar Form of Complex Numbers}{figure.caption.767}{}}
\pgfsyspdfmark {pgfid714}{4736286}{26353109}
\newlabel{modprops}{{85}{471}{The Polar Form of Complex Numbers}{theoremcounter.85}{}}
\newlabel{argprops}{{86}{473}{The Polar Form of Complex Numbers}{theoremcounter.86}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.93}{\ignorespaces Polar coordinates, $(r, {"θ})$ associated with $z = a+bi$ with $r \geq 0$.\relax }}{473}{figure.caption.790}}
\newlabel{fig:polarcom2}{{9.93}{473}{Polar coordinates, $(r, \theta )$ associated with $z = a+bi$ with $r \geq 0$.\relax }{figure.caption.790}{}}
\newlabel{polarformcomplex}{{57}{474}{The Polar Form of Complex Numbers}{definitioncounter.57}{}}
\pgfsyspdfmark {pgfid718}{14288487}{36779058}
\newlabel{polarcomplexex}{{189}{474}{The Polar Form of Complex Numbers}{examplecounter.189}{}}
\pgfsyspdfmark {pgfid719}{19793511}{36615218}
\newlabel{epolarcomplexex}{{57}{475}{The Polar Form of Complex Numbers}{Item.3672}{}}
\pgfsyspdfmark {pgfid720}{4736286}{37512698}
\newlabel{prodquotpolarcomplex}{{87}{475}{The Polar Form of Complex Numbers}{theoremcounter.87}{}}
\pgfsyspdfmark {pgfid722}{4474142}{37533332}
\newlabel{polararithmeticex}{{190}{477}{The Polar Form of Complex Numbers}{examplecounter.190}{}}
\newlabel{epolararithmeticex}{{87}{477}{The Polar Form of Complex Numbers}{Item.3678}{}}
\pgfsyspdfmark {pgfid723}{4736286}{9040099}
\@writefile{lof}{\contentsline {figure}{\numberline {9.94}{\ignorespaces Visualizing $zw$ for $z = 4\operatorname  {cis}\left (\frac  {{"π}}{6}\right )$ and $w = 2 \operatorname  {cis}\left (\frac  {2{"π}}{3}\right )$.\relax }}{478}{figure.caption.823}}
\newlabel{fig:polarcomp}{{9.94}{478}{Visualizing $zw$ for $z = 4\operatorname {cis}\left (\frac {\pi }{6}\right )$ and $w = 2 \operatorname {cis}\left (\frac {2\pi }{3}\right )$.\relax }{figure.caption.823}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.95}{\ignorespaces Visualizing $\genfrac  {}{}{}0{z}{w}$ for $z = 4\operatorname  {cis}\left (\frac  {{"π}}{6}\right )$ and $w = 2 \operatorname  {cis}\left (\frac  {2{"π}}{3}\right )$.\relax }}{478}{figure.caption.824}}
\newlabel{nthrootcomplex}{{58}{479}{The Polar Form of Complex Numbers}{definitioncounter.58}{}}
\newlabel{nthrootscomplexthm}{{88}{480}{The Polar Form of Complex Numbers}{theoremcounter.88}{}}
\pgfsyspdfmark {pgfid726}{14288487}{16116688}
\newlabel{nthrootscomplexex}{{191}{480}{The Polar Form of Complex Numbers}{examplecounter.191}{}}
\pgfsyspdfmark {pgfid727}{19793511}{15952848}
\newlabel{fourthrootsneg16}{{2}{480}{The Polar Form of Complex Numbers}{Item.3680}{}}
\newlabel{halfanglecuberoot}{{3}{480}{The Polar Form of Complex Numbers}{Item.3681}{}}
\newlabel{calculatorfifthroot}{{4}{480}{The Polar Form of Complex Numbers}{Item.3682}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.96}{\ignorespaces The two square roots of $z=-2+2\sqrt  {3}i$\relax }}{481}{figure.caption.829}}
\newlabel{fig:polarroot1}{{9.96}{481}{The two square roots of $z=-2+2\sqrt {3}i$\relax }{figure.caption.829}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.97}{\ignorespaces The four fourth roots of $z=-16$\relax }}{481}{figure.caption.830}}
\newlabel{fig:polarroot2}{{9.97}{481}{The four fourth roots of $z=-16$\relax }{figure.caption.830}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.98}{\ignorespaces The three third roots of $z=\sqrt  {2}+i\sqrt  {2}$\relax }}{481}{figure.caption.831}}
\newlabel{fig:polarroot3}{{9.98}{481}{The three third roots of $z=\sqrt {2}+i\sqrt {2}$\relax }{figure.caption.831}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {9.99}{\ignorespaces The five fifth roots of $1$\relax }}{481}{figure.caption.832}}
\newlabel{fig:polarroot4}{{9.99}{481}{The five fifth roots of $1$\relax }{figure.caption.832}{}}
\newlabel{enthrootscomplexex}{{88}{481}{The Polar Form of Complex Numbers}{Item.3686}{}}
\pgfsyspdfmark {pgfid728}{4736286}{21167926}
\newlabel{polarcompbasicfirst}{{1}{482}{The Polar Form of Complex Numbers}{Item.3687}{}}
\newlabel{polarcompbasiclast}{{20}{482}{The Polar Form of Complex Numbers}{Item.3706}{}}
\newlabel{rectcomplast}{{40}{482}{The Polar Form of Complex Numbers}{Item.3726}{}}
\newlabel{demoivrefirst}{{53}{483}{The Polar Form of Complex Numbers}{Item.3739}{}}
\newlabel{polarrootsfirst}{{65}{483}{The Polar Form of Complex Numbers}{Item.3751}{}}
\newlabel{polarrootslast}{{76}{483}{The Polar Form of Complex Numbers}{Item.3762}{}}
\newlabel{eulerformulaexercise}{{82}{484}{The Polar Form of Complex Numbers}{Item.3776}{}}
\newlabel{expformcosandsin}{{82f}{484}{The Polar Form of Complex Numbers}{Item.3782}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.1}{\ignorespaces $\qopname  \relax o{sin}(x)/x$ near $x=1$.\relax }}{485}{figure.caption.833}}
\newlabel{fig:zoom_sinx_over_x}{{10.1}{485}{$\sin (x)/x$ near $x=1$.\relax }{figure.caption.833}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.2}{\ignorespaces $\qopname  \relax o{sin}(x)/x$ near $x=0$.\relax }}{485}{figure.caption.834}}
\newlabel{fig:sinx_over_x}{{10.2}{485}{$\sin (x)/x$ near $x=0$.\relax }{figure.caption.834}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {10}Limits}{485}{chapter.10}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{chapter:limits}{{10}{485}{Limits}{chapter.10}{}}
\@writefile{toc}{\contentsline {section}{\numberline {10.1}An Introduction To Limits}{485}{section.10.1}}
\newlabel{sec:limit_intro}{{10.1}{485}{An Introduction To Limits}{section.10.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.3}{\ignorespaces Values of $\qopname  \relax o{sin}(x)/x$ with $x$ near 1.\relax }}{486}{figure.caption.835}}
\newlabel{table:sinx_1}{{10.3}{486}{Values of $\sin (x)/x$ with $x$ near 1.\relax }{figure.caption.835}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.4}{\ignorespaces Values of $\qopname  \relax o{sin}(x)/x$ with $x$ near 0.\relax }}{486}{figure.caption.836}}
\newlabel{table:sinx_2}{{10.4}{486}{Values of $\sin (x)/x$ with $x$ near 0.\relax }{figure.caption.836}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.5}{\ignorespaces Graphically approximating a limit in Example \ref  {ex_limit1}.\relax }}{486}{figure.caption.837}}
\newlabel{fig:limit1}{{10.5}{486}{Graphically approximating a limit in Example \ref {ex_limit1}.\relax }{figure.caption.837}{}}
\pgfsyspdfmark {pgfid729}{14288487}{10856824}
\newlabel{ex_limit1}{{192}{486}{An Introduction To Limits}{examplecounter.192}{}}
\pgfsyspdfmark {pgfid730}{19793511}{10692984}
\@writefile{lof}{\contentsline {figure}{\numberline {10.6}{\ignorespaces Numerically approximating a limit in Example \ref  {ex_limit1}.\relax }}{487}{figure.caption.838}}
\newlabel{table:limit1}{{10.6}{487}{Numerically approximating a limit in Example \ref {ex_limit1}.\relax }{figure.caption.838}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.7}{\ignorespaces Graphically approximating a limit in Example \ref  {ex_limit2}.\relax }}{487}{figure.caption.839}}
\newlabel{fig:limit2}{{10.7}{487}{Graphically approximating a limit in Example \ref {ex_limit2}.\relax }{figure.caption.839}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.8}{\ignorespaces Numerically approximating a limit in Example \ref  {ex_limit2}.\relax }}{487}{figure.caption.840}}
\newlabel{table:limit2}{{10.8}{487}{Numerically approximating a limit in Example \ref {ex_limit2}.\relax }{figure.caption.840}{}}
\newlabel{eex_limit1}{{10.1}{487}{An Introduction To Limits}{figure.caption.837}{}}
\pgfsyspdfmark {pgfid731}{5719326}{40521252}
\pgfsyspdfmark {pgfid732}{4474142}{23597842}
\newlabel{ex_limit2}{{193}{487}{An Introduction To Limits}{examplecounter.193}{}}
\newlabel{eex_limit2}{{10.1}{487}{An Introduction To Limits}{examplecounter.193}{}}
\pgfsyspdfmark {pgfid733}{12919432}{10679896}
\@writefile{lof}{\contentsline {figure}{\numberline {10.10}{\ignorespaces Observing no limit as $x\to 1$ in Example \ref  {ex_no_limit1}.\relax }}{488}{figure.caption.842}}
\newlabel{fig:nolimit1}{{10.10}{488}{Observing no limit as $x\to 1$ in Example \ref {ex_no_limit1}.\relax }{figure.caption.842}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.11}{\ignorespaces Values of $f(x)$ near $x=1$ in Example \ref  {ex_no_limit1}.\relax }}{488}{figure.caption.843}}
\newlabel{table:nolimit1}{{10.11}{488}{Values of $f(x)$ near $x=1$ in Example \ref {ex_no_limit1}.\relax }{figure.caption.843}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.12}{\ignorespaces Observing no limit as $x\to 1$ in Example \ref  {ex_no_limit2}.\relax }}{488}{figure.caption.844}}
\newlabel{fig:nolimit2}{{10.12}{488}{Observing no limit as $x\to 1$ in Example \ref {ex_no_limit2}.\relax }{figure.caption.844}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.13}{\ignorespaces Values of $f(x)$ near $x=1$ in Example \ref  {ex_no_limit2}.\relax }}{488}{figure.caption.845}}
\newlabel{table:nolimit2}{{10.13}{488}{Values of $f(x)$ near $x=1$ in Example \ref {ex_no_limit2}.\relax }{figure.caption.845}{}}
\pgfsyspdfmark {pgfid734}{14288487}{41809231}
\newlabel{ex_no_limit1}{{194}{488}{An Introduction To Limits}{examplecounter.194}{}}
\newlabel{eex_no_limit1}{{10.1}{488}{An Introduction To Limits}{examplecounter.194}{}}
\pgfsyspdfmark {pgfid735}{26209509}{29754121}
\pgfsyspdfmark {pgfid736}{14288487}{28334924}
\newlabel{ex_no_limit2}{{195}{488}{An Introduction To Limits}{examplecounter.195}{}}
\newlabel{eex_no_limit2}{{10.1}{488}{An Introduction To Limits}{examplecounter.195}{}}
\pgfsyspdfmark {pgfid737}{18610577}{15236093}
\pgfsyspdfmark {pgfid738}{14288487}{13816897}
\newlabel{ex_no_limit3}{{196}{488}{An Introduction To Limits}{examplecounter.196}{}}
\pgfsyspdfmark {pgfid739}{19793511}{13653057}
\@writefile{lof}{\contentsline {figure}{\numberline {10.14}{\ignorespaces Interpreting a difference quotient as the slope of a secant line.\relax }}{489}{figure.caption.846}}
\newlabel{fig:diffquot1}{{10.14}{489}{Interpreting a difference quotient as the slope of a secant line.\relax }{figure.caption.846}{}}
\newlabel{eex_no_limit3}{{10.1}{489}{An Introduction To Limits}{examplecounter.196}{}}
\pgfsyspdfmark {pgfid740}{12997226}{46707506}
\@writefile{lof}{\contentsline {figure}{\numberline {10.9}{\ignorespaces Observing that $f(x) = \qopname  \relax o{sin}(1/x)$ has no limit as $x\to 0$ in Example \ref  {ex_no_limit3}.\relax }}{489}{figure.caption.841}}
\newlabel{fig:nolimit3}{{10.9}{489}{Observing that $f(x) = \sin (1/x)$ has no limit as $x\to 0$ in Example \ref {ex_no_limit3}.\relax }{figure.caption.841}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.15}{\ignorespaces Secant lines of $f(x)$ at $x=1$ and $x=1+h$, for shrinking values of $h$ (i.e., $h\rightarrow 0$).\relax }}{490}{figure.caption.847}}
\newlabel{fig:diff_quot_small_h}{{10.15}{490}{Secant lines of $f(x)$ at $x=1$ and $x=1+h$, for shrinking values of $h$ (i.e., $h\rightarrow 0$).\relax }{figure.caption.847}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.16}{\ignorespaces The difference quotient evaluated at values of $h$ near 0.\relax }}{490}{figure.caption.848}}
\newlabel{table:diff_quot_smallh}{{10.16}{490}{The difference quotient evaluated at values of $h$ near 0.\relax }{figure.caption.848}{}}
\newlabel{def:limit}{{59}{490}{An Introduction To Limits}{definitioncounter.59}{}}
\@writefile{toc}{\contentsline {section}{\numberline {10.2}Finding Limits Analytically}{492}{section.10.2}}
\newlabel{sec:limit_analytically}{{10.2}{492}{Finding Limits Analytically}{section.10.2}{}}
\newlabel{thm:limit_algebra}{{89}{492}{Finding Limits Analytically}{theoremcounter.89}{}}
\pgfsyspdfmark {pgfid743}{14288487}{11723246}
\newlabel{ex_basic_limit_1}{{197}{492}{Finding Limits Analytically}{examplecounter.197}{}}
\pgfsyspdfmark {pgfid744}{19793511}{11559406}
\newlabel{eex_basic_limit_1}{{89}{493}{Finding Limits Analytically}{Item.3825}{}}
\pgfsyspdfmark {pgfid745}{4736286}{33990045}
\newlabel{thm:poly_rat}{{90}{493}{Finding Limits Analytically}{theoremcounter.90}{}}
\pgfsyspdfmark {pgfid747}{4474142}{17795255}
\newlabel{ex_limit_rat}{{198}{493}{Finding Limits Analytically}{examplecounter.198}{}}
\newlabel{eex_limit_rat}{{90}{493}{Finding Limits Analytically}{examplecounter.198}{}}
\pgfsyspdfmark {pgfid748}{5719326}{10049158}
\newlabel{thm:lim_continuous}{{91}{494}{Finding Limits Analytically}{theoremcounter.91}{}}
\pgfsyspdfmark {pgfid750}{14288487}{33321199}
\newlabel{ex_limit_1}{{199}{494}{Finding Limits Analytically}{examplecounter.199}{}}
\pgfsyspdfmark {pgfid751}{19793511}{33157359}
\@writefile{lof}{\contentsline {figure}{\numberline {10.17}{\ignorespaces The unit circle and related triangles.\relax }}{495}{figure.caption.849}}
\newlabel{fig:squeeze_sinx}{{10.17}{495}{The unit circle and related triangles.\relax }{figure.caption.849}{}}
\newlabel{eex_limit_1}{{90}{495}{Finding Limits Analytically}{Item.3846}{}}
\pgfsyspdfmark {pgfid752}{4736286}{41123855}
\newlabel{thm:sqz}{{92}{495}{Finding Limits Analytically}{theoremcounter.92}{}}
\pgfsyspdfmark {pgfid754}{4474142}{12708419}
\newlabel{ex_limit_sinx_prove}{{200}{495}{Finding Limits Analytically}{examplecounter.200}{}}
\pgfsyspdfmark {pgfid755}{9979166}{12544579}
\newlabel{eex_limit_sinx_prove}{{92}{496}{Finding Limits Analytically}{figure.caption.849}{}}
\pgfsyspdfmark {pgfid756}{14550631}{15798939}
\@writefile{lof}{\contentsline {figure}{\numberline {10.18}{\ignorespaces Graphing $f$ in Example \ref  {ex_limit_onept} to understand a limit.\relax }}{497}{figure.caption.850}}
\newlabel{fig:limitxplus1}{{10.18}{497}{Graphing $f$ in Example \ref {ex_limit_onept} to understand a limit.\relax }{figure.caption.850}{}}
\newlabel{thm:special_limits}{{93}{497}{Finding Limits Analytically}{theoremcounter.93}{}}
\pgfsyspdfmark {pgfid758}{4474142}{27612532}
\newlabel{ex_limit_onept}{{201}{497}{Finding Limits Analytically}{examplecounter.201}{}}
\newlabel{eex_limit_onept}{{93}{497}{Finding Limits Analytically}{examplecounter.201}{}}
\pgfsyspdfmark {pgfid759}{5719326}{5522718}
\newlabel{thm:limit_allbut1}{{94}{498}{Finding Limits Analytically}{theoremcounter.94}{}}
\pgfsyspdfmark {pgfid761}{14288487}{30544331}
\newlabel{ex_limit_allbut1}{{202}{498}{Finding Limits Analytically}{examplecounter.202}{}}
\newlabel{eex_limit_allbut1}{{94}{498}{Finding Limits Analytically}{examplecounter.202}{}}
\pgfsyspdfmark {pgfid762}{15533671}{14581339}
\pgfsyspdfmark {pgfid763}{14288487}{8612958}
\newlabel{ex_limit_diffquot}{{203}{498}{Finding Limits Analytically}{examplecounter.203}{}}
\pgfsyspdfmark {pgfid764}{19793511}{8449118}
\newlabel{eex_limit_diffquot}{{94}{499}{Finding Limits Analytically}{examplecounter.203}{}}
\pgfsyspdfmark {pgfid765}{17064072}{34929280}
\@writefile{toc}{\contentsline {section}{\numberline {10.3}One Sided Limits}{502}{section.10.3}}
\newlabel{sec:limit_continuity}{{10.3}{502}{One Sided Limits}{section.10.3}{}}
\newlabel{def:onesidedlimit}{{60}{502}{One Sided Limits}{definitioncounter.60}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.19}{\ignorespaces A graph of $f$ in Example \ref  {ex_onesidea}.\relax }}{503}{figure.caption.851}}
\newlabel{fig:onesided1}{{10.19}{503}{A graph of $f$ in Example \ref {ex_onesidea}.\relax }{figure.caption.851}{}}
\pgfsyspdfmark {pgfid767}{4474142}{46871346}
\newlabel{ex_onesidea}{{204}{503}{One Sided Limits}{examplecounter.204}{}}
\newlabel{eex_onesidea}{{60}{503}{One Sided Limits}{Item.3912}{}}
\pgfsyspdfmark {pgfid768}{4736286}{17699535}
\newlabel{thm:leftrightlimits}{{95}{503}{One Sided Limits}{theoremcounter.95}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.20}{\ignorespaces A graph of $f$ from Example \ref  {ex_onesideb}\relax }}{504}{figure.caption.852}}
\newlabel{fig:onesidedb}{{10.20}{504}{A graph of $f$ from Example \ref {ex_onesideb}\relax }{figure.caption.852}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.21}{\ignorespaces Graphing $f$ in Example \ref  {ex_onesidec}\relax }}{504}{figure.caption.853}}
\newlabel{fig:onesidedc}{{10.21}{504}{Graphing $f$ in Example \ref {ex_onesidec}\relax }{figure.caption.853}{}}
\pgfsyspdfmark {pgfid770}{14288487}{40474064}
\newlabel{ex_onesideb}{{205}{504}{One Sided Limits}{examplecounter.205}{}}
\newlabel{eex_onesideb}{{95}{504}{One Sided Limits}{Item.3928}{}}
\pgfsyspdfmark {pgfid771}{14550631}{12032750}
\pgfsyspdfmark {pgfid772}{14288487}{10570813}
\newlabel{ex_onesidec}{{206}{504}{One Sided Limits}{examplecounter.206}{}}
\pgfsyspdfmark {pgfid773}{19793511}{10406973}
\@writefile{lof}{\contentsline {figure}{\numberline {10.22}{\ignorespaces Graphing $f$ in Example \ref  {ex_onesided}\relax }}{505}{figure.caption.854}}
\newlabel{fig:onesidedd}{{10.22}{505}{Graphing $f$ in Example \ref {ex_onesided}\relax }{figure.caption.854}{}}
\newlabel{eex_onesidec}{{95}{505}{One Sided Limits}{Item.3932}{}}
\pgfsyspdfmark {pgfid774}{5719326}{44033642}
\pgfsyspdfmark {pgfid775}{4474142}{42624618}
\newlabel{ex_onesided}{{207}{505}{One Sided Limits}{examplecounter.207}{}}
\newlabel{eex_onesided}{{95}{505}{One Sided Limits}{Item.3936}{}}
\pgfsyspdfmark {pgfid776}{4736286}{31547260}
\@writefile{lof}{\contentsline {figure}{\numberline {10.23}{\ignorespaces A graph of $f$ in Example \ref  {ex_contint1}.\relax }}{508}{figure.caption.855}}
\newlabel{fig_continuous1}{{10.23}{508}{A graph of $f$ in Example \ref {ex_contint1}.\relax }{figure.caption.855}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.24}{\ignorespaces A graph of the step function in Example \ref  {ex_contint2}.\relax }}{508}{figure.caption.856}}
\newlabel{fig:continuous2}{{10.24}{508}{A graph of the step function in Example \ref {ex_contint2}.\relax }{figure.caption.856}{}}
\@writefile{toc}{\contentsline {section}{\numberline {10.4}Continuity}{508}{section.10.4}}
\newlabel{sec:continuity}{{10.4}{508}{Continuity}{section.10.4}{}}
\newlabel{def:continuous}{{61}{508}{Continuity}{definitioncounter.61}{}}
\pgfsyspdfmark {pgfid778}{14288487}{25741503}
\newlabel{ex_contint1}{{208}{508}{Continuity}{examplecounter.208}{}}
\newlabel{eex_contint1}{{61}{508}{Continuity}{Item.4052}{}}
\pgfsyspdfmark {pgfid779}{25975513}{14631622}
\pgfsyspdfmark {pgfid780}{14288487}{13222598}
\newlabel{ex_contint2}{{209}{508}{Continuity}{examplecounter.209}{}}
\pgfsyspdfmark {pgfid781}{19793511}{13058758}
\newlabel{eex_contint2}{{61}{509}{Continuity}{figure.caption.856}{}}
\pgfsyspdfmark {pgfid782}{5719326}{40417049}
\newlabel{def:closed_continuity}{{62}{509}{Continuity}{definitioncounter.62}{}}
\pgfsyspdfmark {pgfid784}{4474142}{23966440}
\newlabel{ex_cont_funct1}{{210}{509}{Continuity}{examplecounter.210}{}}
\pgfsyspdfmark {pgfid785}{9979166}{23802600}
\newlabel{eex_cont_funct1}{{62}{510}{Continuity}{Item.4068}{}}
\pgfsyspdfmark {pgfid786}{14550631}{42835016}
\newlabel{thm:continuity_algebra}{{96}{510}{Continuity}{theoremcounter.96}{}}
\newlabel{thm:continuous_functions}{{97}{510}{Continuity}{theoremcounter.97}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.25}{\ignorespaces A graph of $f$ in Example \ref  {ex_cont_funct}(a).\relax }}{511}{figure.caption.857}}
\newlabel{fig_continuous3}{{10.25}{511}{A graph of $f$ in Example \ref {ex_cont_funct}(a).\relax }{figure.caption.857}{}}
\pgfsyspdfmark {pgfid789}{4474142}{46871346}
\newlabel{ex_cont_funct}{{211}{511}{Continuity}{examplecounter.211}{}}
\newlabel{eex_cont_funct}{{97}{511}{Continuity}{Item.4092}{}}
\pgfsyspdfmark {pgfid790}{4736286}{24613391}
\newlabel{thm:IVT}{{98}{511}{Continuity}{theoremcounter.98}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.26}{\ignorespaces Graphing a root of $f(x) = x-\qopname  \relax o{cos}x$.\relax }}{512}{figure.caption.858}}
\newlabel{fig:xminuscosx}{{10.26}{512}{Graphing a root of $f(x) = x-\cos x$.\relax }{figure.caption.858}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.27}{\ignorespaces Iterations of the Bisection Method of Root Finding\relax }}{512}{figure.caption.859}}
\newlabel{table:rootfinding}{{10.27}{512}{Iterations of the Bisection Method of Root Finding\relax }{figure.caption.859}{}}
\pgfsyspdfmark {pgfid792}{14288487}{27212705}
\newlabel{ex_bisect_method}{{212}{512}{Continuity}{examplecounter.212}{}}
\pgfsyspdfmark {pgfid793}{19793511}{27048865}
\newlabel{eex_bisect_method}{{98}{513}{Continuity}{examplecounter.212}{}}
\pgfsyspdfmark {pgfid794}{10336072}{45134642}
\@writefile{lof}{\contentsline {figure}{\numberline {10.28}{\ignorespaces Graphing $f(x) = 1/x^2$ for values of $x$ near 0.\relax }}{516}{figure.caption.860}}
\newlabel{fig:oneoverxsquared}{{10.28}{516}{Graphing $f(x) = 1/x^2$ for values of $x$ near 0.\relax }{figure.caption.860}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.29}{\ignorespaces Observing infinite limit as $x\to 1$ in Example \ref  {ex_inflim1}.\relax }}{516}{figure.caption.861}}
\newlabel{fig:inflim1}{{10.29}{516}{Observing infinite limit as $x\to 1$ in Example \ref {ex_inflim1}.\relax }{figure.caption.861}{}}
\@writefile{toc}{\contentsline {section}{\numberline {10.5}Limits Involving Infinity}{516}{section.10.5}}
\newlabel{sec:limits_infty}{{10.5}{516}{Limits Involving Infinity}{section.10.5}{}}
\newlabel{def:limit_of_infinity}{{63}{516}{Limits Involving Infinity}{definitioncounter.63}{}}
\pgfsyspdfmark {pgfid796}{14288487}{13693715}
\newlabel{ex_inflim1}{{213}{516}{Limits Involving Infinity}{examplecounter.213}{}}
\newlabel{eex_inflim1}{{63}{516}{Limits Involving Infinity}{examplecounter.213}{}}
\pgfsyspdfmark {pgfid797}{27265034}{5522718}
\@writefile{lof}{\contentsline {figure}{\numberline {10.30}{\ignorespaces Evaluating $\displaystyle \qopname  \relax m{lim}_{x\rightarrow 0}\frac  1x$.\relax }}{517}{figure.caption.862}}
\newlabel{fig:oneoverx}{{10.30}{517}{Evaluating $\ds \lim _{x\rightarrow 0}\frac 1x$.\relax }{figure.caption.862}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.31}{\ignorespaces Graphing $f(x) = \genfrac  {}{}{}0{3x}{x^2-4}$.\relax }}{517}{figure.caption.863}}
\newlabel{fig:multipleasymptotes}{{10.31}{517}{Graphing $f(x) = \dfrac {3x}{x^2-4}$.\relax }{figure.caption.863}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.32}{\ignorespaces Graphically showing that $f(x) = \genfrac  {}{}{}0{x^2-1}{x-1}$ does not have an asymptote at $x=1$.\relax }}{517}{figure.caption.864}}
\newlabel{fig:noasy}{{10.32}{517}{Graphically showing that $f(x) = \dfrac {x^2-1}{x-1}$ does not have an asymptote at $x=1$.\relax }{figure.caption.864}{}}
\pgfsyspdfmark {pgfid798}{4474142}{46871346}
\newlabel{ex_inflim2}{{214}{517}{Limits Involving Infinity}{examplecounter.214}{}}
\newlabel{eex_inflim2}{{63}{517}{Limits Involving Infinity}{examplecounter.214}{}}
\pgfsyspdfmark {pgfid799}{5719326}{39764717}
\pgfsyspdfmark {pgfid800}{4474142}{34260240}
\newlabel{ex_vertasy1}{{215}{517}{Limits Involving Infinity}{examplecounter.215}{}}
\newlabel{eex_vertasy1}{{63}{517}{Limits Involving Infinity}{examplecounter.215}{}}
\pgfsyspdfmark {pgfid801}{5719326}{25632376}
\newlabel{def:limit_at_infinity}{{64}{518}{Limits Involving Infinity}{definitioncounter.64}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.34}{\ignorespaces Using a graph and a table to approximate a horizontal asymptote in Example \ref  {ex_hzasy1}.\relax }}{519}{figure.caption.866}}
\newlabel{fig:hzasy1}{{10.34}{519}{Using a graph and a table to approximate a horizontal asymptote in Example \ref {ex_hzasy1}.\relax }{figure.caption.866}{}}
\pgfsyspdfmark {pgfid803}{4474142}{44285788}
\newlabel{ex_hzasy1}{{216}{519}{Limits Involving Infinity}{examplecounter.216}{}}
\newlabel{eex_hzasy1}{{64}{519}{Limits Involving Infinity}{examplecounter.216}{}}
\pgfsyspdfmark {pgfid804}{20039006}{34254925}
\@writefile{lof}{\contentsline {figure}{\numberline {10.33}{\ignorespaces Considering different types of horizontal asymptotes.\relax }}{519}{figure.caption.865}}
\newlabel{fig:hzasy}{{10.33}{519}{Considering different types of horizontal asymptotes.\relax }{figure.caption.865}{}}
\newlabel{thm:lim_rational_fn_at_infty}{{99}{520}{Limits Involving Infinity}{theoremcounter.99}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10.36}{\ignorespaces Visualizing the functions in Example \ref  {ex_hzasy3}.\relax }}{521}{figure.caption.867}}
\newlabel{fig:hzasy3}{{10.36}{521}{Visualizing the functions in Example \ref {ex_hzasy3}.\relax }{figure.caption.867}{}}
\pgfsyspdfmark {pgfid806}{4474142}{40993031}
\newlabel{ex_hzasy2}{{217}{521}{Limits Involving Infinity}{examplecounter.217}{}}
\newlabel{eex_hzasy2}{{99}{521}{Limits Involving Infinity}{examplecounter.217}{}}
\pgfsyspdfmark {pgfid807}{25707806}{26004805}
\pgfsyspdfmark {pgfid808}{4474142}{24595781}
\newlabel{ex_hzasy3}{{218}{521}{Limits Involving Infinity}{examplecounter.218}{}}
\newlabel{eex_hzasy3}{{99}{521}{Limits Involving Infinity}{Item.4167}{}}
\pgfsyspdfmark {pgfid809}{4736286}{7437207}
\@writefile{toc}{\contentsline {chapter}{\numberline {11}Derivatives}{525}{chapter.11}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{chapter:derivatives}{{11}{525}{Derivatives}{chapter.11}{}}
\@writefile{toc}{\contentsline {section}{\numberline {11.1}Instantaneous Rates of Change: The Derivative}{525}{section.11.1}}
\newlabel{sec:derivative}{{11.1}{525}{Instantaneous Rates of Change: The Derivative}{section.11.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.2}{\ignorespaces Approximating the instantaneous velocity with average velocities over a small time period $h$.\relax }}{526}{figure.caption.869}}
\newlabel{table:falling}{{11.2}{526}{Approximating the instantaneous velocity with average velocities over a small time period $h$.\relax }{figure.caption.869}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.1}{\ignorespaces Parts (a), (b) and (c) show the secant line to $f(x)$ with $h=1$, zoomed in different amounts. Part (d) shows the tangent line to $f$ at $x=2$.\relax }}{527}{figure.caption.868}}
\newlabel{fig:derivfalling}{{11.1}{527}{Parts (a), (b) and (c) show the secant line to $f(x)$ with $h=1$, zoomed in different amounts. Part (d) shows the tangent line to $f$ at $x=2$.\relax }{figure.caption.868}{}}
\newlabel{def:derivative_at_a_point}{{65}{527}{Instantaneous Rates of Change: The Derivative}{definitioncounter.65}{}}
\newlabel{def:tangent_line}{{66}{527}{Instantaneous Rates of Change: The Derivative}{definitioncounter.66}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.3}{\ignorespaces A graph of $f(x) = 3x^2+5x-7$ and its tangent lines at $x=1$ and $x=3$.\relax }}{528}{figure.caption.870}}
\newlabel{fig:tangent1}{{11.3}{528}{A graph of $f(x) = 3x^2+5x-7$ and its tangent lines at $x=1$ and $x=3$.\relax }{figure.caption.870}{}}
\pgfsyspdfmark {pgfid812}{14288487}{45826265}
\newlabel{ex_derv_point1}{{219}{528}{Instantaneous Rates of Change: The Derivative}{examplecounter.219}{}}
\newlabel{eex_derv_point1}{{66}{528}{Instantaneous Rates of Change: The Derivative}{Item.4228}{}}
\pgfsyspdfmark {pgfid813}{16403423}{8140663}
\@writefile{lof}{\contentsline {figure}{\numberline {11.4}{\ignorespaces A graph of $f(x)=3x^2+5x-7$, along with its normal line at $x=1$.\relax }}{529}{figure.caption.871}}
\newlabel{fig:normal1}{{11.4}{529}{A graph of $f(x)=3x^2+5x-7$, along with its normal line at $x=1$.\relax }{figure.caption.871}{}}
\newlabel{def:normal_line}{{67}{529}{Instantaneous Rates of Change: The Derivative}{definitioncounter.67}{}}
\pgfsyspdfmark {pgfid815}{4474142}{37353037}
\newlabel{ex_normal1}{{220}{529}{Instantaneous Rates of Change: The Derivative}{examplecounter.220}{}}
\newlabel{eex_normal1}{{67}{529}{Instantaneous Rates of Change: The Derivative}{examplecounter.220}{}}
\pgfsyspdfmark {pgfid816}{5719326}{19129352}
\pgfsyspdfmark {pgfid817}{4474142}{8301784}
\newlabel{ex_der_line}{{221}{529}{Instantaneous Rates of Change: The Derivative}{examplecounter.221}{}}
\pgfsyspdfmark {pgfid818}{9979166}{8137944}
\@writefile{lof}{\contentsline {figure}{\numberline {11.5}{\ignorespaces $f(x) = \qopname  \relax o{sin}x$ graphed with an approximation to its tangent line at $x=0$.\relax }}{530}{figure.caption.872}}
\newlabel{fig:tangentsinx}{{11.5}{530}{$f(x) = \sin x$ graphed with an approximation to its tangent line at $x=0$.\relax }{figure.caption.872}{}}
\newlabel{eex_der_line}{{67}{530}{Instantaneous Rates of Change: The Derivative}{figure.caption.871}{}}
\pgfsyspdfmark {pgfid819}{24137183}{31691580}
\pgfsyspdfmark {pgfid820}{14288487}{26350396}
\newlabel{ex_der_num_approx}{{222}{530}{Instantaneous Rates of Change: The Derivative}{examplecounter.222}{}}
\newlabel{eex_der_num_approx}{{67}{530}{Instantaneous Rates of Change: The Derivative}{examplecounter.222}{}}
\pgfsyspdfmark {pgfid821}{22619537}{13711692}
\newlabel{def:the_derivative}{{68}{531}{Instantaneous Rates of Change: The Derivative}{definitioncounter.68}{}}
\pgfsyspdfmark {pgfid825}{4474142}{17738556}
\newlabel{ex_deriv1}{{223}{531}{Instantaneous Rates of Change: The Derivative}{examplecounter.223}{}}
\newlabel{eex_deriv1}{{68}{531}{Instantaneous Rates of Change: The Derivative}{examplecounter.223}{}}
\pgfsyspdfmark {pgfid826}{17744327}{5522718}
\pgfsyspdfmark {pgfid827}{14288487}{46871346}
\newlabel{ex_deriv2}{{224}{532}{Instantaneous Rates of Change: The Derivative}{examplecounter.224}{}}
\newlabel{eex_deriv2}{{68}{532}{Instantaneous Rates of Change: The Derivative}{examplecounter.224}{}}
\pgfsyspdfmark {pgfid828}{15533671}{21899427}
\pgfsyspdfmark {pgfid829}{14288487}{20490403}
\newlabel{ex_deriv_sinx}{{225}{532}{Instantaneous Rates of Change: The Derivative}{examplecounter.225}{}}
\pgfsyspdfmark {pgfid830}{19793511}{20326563}
\@writefile{lof}{\contentsline {figure}{\numberline {11.6}{\ignorespaces The absolute value function, $f(x) = |x|$. Notice how the slope of the lines (and hence the tangent lines) abruptly changes at $x=0$.\relax }}{533}{figure.caption.873}}
\newlabel{fig:absolutevalue}{{11.6}{533}{The absolute value function, $f(x) = |x|$. Notice how the slope of the lines (and hence the tangent lines) abruptly changes at $x=0$.\relax }{figure.caption.873}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.7}{\ignorespaces A graph of the derivative of $f(x) = |x|$.\relax }}{533}{figure.caption.874}}
\newlabel{fig:absolutevalueprime}{{11.7}{533}{A graph of the derivative of $f(x) = |x|$.\relax }{figure.caption.874}{}}
\newlabel{eex_deriv_sinx}{{68}{533}{Instantaneous Rates of Change: The Derivative}{examplecounter.225}{}}
\pgfsyspdfmark {pgfid831}{16626302}{43561778}
\pgfsyspdfmark {pgfid832}{4474142}{42152754}
\newlabel{ex_not_diff}{{226}{533}{Instantaneous Rates of Change: The Derivative}{examplecounter.226}{}}
\newlabel{eex_not_diff}{{68}{533}{Instantaneous Rates of Change: The Derivative}{examplecounter.226}{}}
\pgfsyspdfmark {pgfid833}{19039214}{7095582}
\@writefile{lof}{\contentsline {figure}{\numberline {11.8}{\ignorespaces A graph of $f(x)$ as defined in Example \ref  {ex_diff_piecewise}.\relax }}{534}{figure.caption.875}}
\newlabel{fig:piecewisesinx1}{{11.8}{534}{A graph of $f(x)$ as defined in Example \ref {ex_diff_piecewise}.\relax }{figure.caption.875}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.9}{\ignorespaces A graph of $\ensuremath  {f\tmspace  +\thinmuskip {.1667em}'}(x)$ in Example \ref  {ex_diff_piecewise}.\relax }}{534}{figure.caption.876}}
\newlabel{fig:piecewisecosx1}{{11.9}{534}{A graph of $\fp (x)$ in Example \ref {ex_diff_piecewise}.\relax }{figure.caption.876}{}}
\pgfsyspdfmark {pgfid834}{14288487}{45298482}
\newlabel{ex_diff_piecewise}{{227}{534}{Instantaneous Rates of Change: The Derivative}{examplecounter.227}{}}
\newlabel{eex_diff_piecewise}{{68}{534}{Instantaneous Rates of Change: The Derivative}{figure.caption.874}{}}
\pgfsyspdfmark {pgfid835}{26201617}{18615832}
\newlabel{exer:02_01_10}{{6}{535}{Instantaneous Rates of Change: The Derivative}{Item.4234}{}}
\newlabel{exer:02_01_16}{{12}{535}{Instantaneous Rates of Change: The Derivative}{Item.4240}{}}
\@writefile{toc}{\contentsline {section}{\numberline {11.2}Interpretations of the Derivative}{537}{section.11.2}}
\newlabel{sec:interp_deriv}{{11.2}{537}{Interpretations of the Derivative}{section.11.2}{}}
\pgfsyspdfmark {pgfid836}{14288487}{44060133}
\newlabel{ex_der_meaning1}{{228}{538}{Interpretations of the Derivative}{examplecounter.228}{}}
\newlabel{eex_der_meaning1}{{11.2}{538}{Interpretations of the Derivative}{examplecounter.228}{}}
\pgfsyspdfmark {pgfid837}{35522151}{39177701}
\pgfsyspdfmark {pgfid838}{14288487}{37768677}
\newlabel{ex_der_meaning2}{{229}{538}{Interpretations of the Derivative}{examplecounter.229}{}}
\newlabel{eex_der_meaning2}{{11.2}{538}{Interpretations of the Derivative}{examplecounter.229}{}}
\pgfsyspdfmark {pgfid839}{32569937}{23923073}
\@writefile{lof}{\contentsline {figure}{\numberline {11.10}{\ignorespaces A graph of $f(x)=x^2$.\relax }}{539}{figure.caption.877}}
\newlabel{fig:xsquared}{{11.10}{539}{A graph of $f(x)=x^2$.\relax }{figure.caption.877}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.11}{\ignorespaces A graph of $f(x)=x^2$ and tangent lines.\relax }}{539}{figure.caption.878}}
\newlabel{fig:xsquaredwithgrid}{{11.11}{539}{A graph of $f(x)=x^2$ and tangent lines.\relax }{figure.caption.878}{}}
\newlabel{idea:motion}{{42}{539}{Interpretations of the Derivative}{keyideacounter.42}{}}
\pgfsyspdfmark {pgfid841}{4474142}{7259422}
\newlabel{ex_der_meaning3}{{230}{539}{Interpretations of the Derivative}{examplecounter.230}{}}
\pgfsyspdfmark {pgfid842}{9979166}{7095582}
\@writefile{lof}{\contentsline {figure}{\numberline {11.12}{\ignorespaces Graphs of $f$ and $\ensuremath  {f\tmspace  +\thinmuskip {.1667em}'}$ in Example \ref  {ex_der_meaning4}, along with tangent lines in (b).\relax }}{540}{figure.caption.879}}
\newlabel{fig:fwithderiv}{{11.12}{540}{Graphs of $f$ and $\fp $ in Example \ref {ex_der_meaning4}, along with tangent lines in (b).\relax }{figure.caption.879}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.13}{\ignorespaces Zooming in on $f$ at $x=3$ for the function given in Examples \ref  {ex_der_meaning4} and \ref  {ex_der_meaning5}.\relax }}{540}{figure.caption.880}}
\newlabel{fig:fwithderivzoom3}{{11.13}{540}{Zooming in on $f$ at $x=3$ for the function given in Examples \ref {ex_der_meaning4} and \ref {ex_der_meaning5}.\relax }{figure.caption.880}{}}
\newlabel{eex_der_meaning3}{{42}{540}{Interpretations of the Derivative}{figure.caption.878}{}}
\pgfsyspdfmark {pgfid843}{33282349}{41179697}
\pgfsyspdfmark {pgfid844}{14288487}{39747889}
\newlabel{ex_der_meaning4}{{231}{540}{Interpretations of the Derivative}{examplecounter.231}{}}
\newlabel{eex_der_meaning4}{{42}{540}{Interpretations of the Derivative}{examplecounter.231}{}}
\pgfsyspdfmark {pgfid845}{31083712}{30032941}
\pgfsyspdfmark {pgfid846}{14288487}{28601133}
\newlabel{ex_der_meaning5}{{232}{540}{Interpretations of the Derivative}{examplecounter.232}{}}
\newlabel{eex_der_meaning5}{{42}{540}{Interpretations of the Derivative}{examplecounter.232}{}}
\pgfsyspdfmark {pgfid847}{23679018}{13637669}
\@writefile{toc}{\contentsline {section}{\numberline {11.3}Basic Differentiation Rules}{543}{section.11.3}}
\newlabel{sec:basic_diff_rules}{{11.3}{543}{Basic Differentiation Rules}{section.11.3}{}}
\newlabel{thm:deriv_common}{{100}{543}{Basic Differentiation Rules}{theoremcounter.100}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.14}{\ignorespaces A graph of $f(x) = x^3$, along with its derivative $\ensuremath  {f\tmspace  +\thinmuskip {.1667em}'}(x) = 3x^2$ and its tangent line at $x=-1$.\relax }}{544}{figure.caption.881}}
\newlabel{fig:xcubedwithderiv}{{11.14}{544}{A graph of $f(x) = x^3$, along with its derivative $\fp (x) = 3x^2$ and its tangent line at $x=-1$.\relax }{figure.caption.881}{}}
\pgfsyspdfmark {pgfid849}{14288487}{43725618}
\newlabel{ex_deriv_rule1}{{233}{544}{Basic Differentiation Rules}{examplecounter.233}{}}
\newlabel{eex_deriv_rule1}{{11.3}{544}{Basic Differentiation Rules}{Item.4327}{}}
\pgfsyspdfmark {pgfid850}{14550631}{26512773}
\newlabel{thm:deriv_prop}{{101}{544}{Basic Differentiation Rules}{theoremcounter.101}{}}
\pgfsyspdfmark {pgfid852}{4474142}{44167509}
\newlabel{ex_der2}{{234}{545}{Basic Differentiation Rules}{examplecounter.234}{}}
\newlabel{eex_der2}{{101}{545}{Basic Differentiation Rules}{examplecounter.234}{}}
\pgfsyspdfmark {pgfid853}{20188658}{23978054}
\newlabel{def:Higher_Deriv}{{69}{545}{Basic Differentiation Rules}{definitioncounter.69}{}}
\pgfsyspdfmark {pgfid855}{14288487}{42939186}
\newlabel{ex_high_order}{{235}{546}{Basic Differentiation Rules}{examplecounter.235}{}}
\newlabel{eex_high_order}{{69}{546}{Basic Differentiation Rules}{Item.4338}{}}
\pgfsyspdfmark {pgfid856}{14550631}{23825163}
\@writefile{lof}{\contentsline {figure}{\numberline {11.15}{\ignorespaces A graph of $y = 5x^2\qopname  \relax o{sin}x$ and its tangent line at $x={"π}/2$.\relax }}{549}{figure.caption.890}}
\newlabel{fig:5xsquaredsinx}{{11.15}{549}{A graph of $y = 5x^2\sin x$ and its tangent line at $x=\pi /2$.\relax }{figure.caption.890}{}}
\@writefile{toc}{\contentsline {section}{\numberline {11.4}The Product and Quotient Rules}{549}{section.11.4}}
\newlabel{sec:prod_quot_rules}{{11.4}{549}{The Product and Quotient Rules}{section.11.4}{}}
\newlabel{thm:ProductRule}{{102}{549}{The Product and Quotient Rules}{theoremcounter.102}{}}
\pgfsyspdfmark {pgfid858}{4474142}{26991862}
\newlabel{ex_prod1}{{236}{549}{The Product and Quotient Rules}{examplecounter.236}{}}
\newlabel{eex_prod1}{{102}{549}{The Product and Quotient Rules}{examplecounter.236}{}}
\pgfsyspdfmark {pgfid859}{20107058}{14171780}
\pgfsyspdfmark {pgfid860}{4474142}{11189892}
\newlabel{ex_prove_product}{{237}{549}{The Product and Quotient Rules}{examplecounter.237}{}}
\pgfsyspdfmark {pgfid861}{9979166}{11026052}
\newlabel{eex_prove_product}{{102}{550}{The Product and Quotient Rules}{figure.caption.890}{}}
\pgfsyspdfmark {pgfid862}{15533671}{33981510}
\pgfsyspdfmark {pgfid863}{14288487}{28472274}
\newlabel{ex_prod2}{{238}{550}{The Product and Quotient Rules}{examplecounter.238}{}}
\newlabel{eex_prod2}{{102}{550}{The Product and Quotient Rules}{examplecounter.238}{}}
\pgfsyspdfmark {pgfid864}{27551252}{13199959}
\pgfsyspdfmark {pgfid865}{14288487}{11706909}
\newlabel{ex_prod10}{{239}{550}{The Product and Quotient Rules}{examplecounter.239}{}}
\pgfsyspdfmark {pgfid866}{19793511}{11543069}
\newlabel{eex_prod10}{{102}{551}{The Product and Quotient Rules}{examplecounter.239}{}}
\pgfsyspdfmark {pgfid867}{7576392}{38050009}
\pgfsyspdfmark {pgfid868}{4474142}{35068121}
\newlabel{ex_deriv_ln}{{240}{551}{The Product and Quotient Rules}{examplecounter.240}{}}
\newlabel{eex_deriv_ln}{{102}{551}{The Product and Quotient Rules}{Item.4385}{}}
\pgfsyspdfmark {pgfid869}{11340552}{20770258}
\newlabel{thm:QuotientRule}{{103}{551}{The Product and Quotient Rules}{theoremcounter.103}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.16}{\ignorespaces A graph of $y=\qopname  \relax o{tan}x$ along with its tangent line at $x={"π}/4$.\relax }}{552}{figure.caption.891}}
\newlabel{fig:tanx}{{11.16}{552}{A graph of $y=\tan x$ along with its tangent line at $x=\pi /4$.\relax }{figure.caption.891}{}}
\pgfsyspdfmark {pgfid871}{14288487}{41330792}
\newlabel{ex_quot1}{{241}{552}{The Product and Quotient Rules}{examplecounter.241}{}}
\newlabel{eex_quot1}{{103}{552}{The Product and Quotient Rules}{examplecounter.241}{}}
\pgfsyspdfmark {pgfid872}{15533671}{31854755}
\pgfsyspdfmark {pgfid873}{14288487}{26478041}
\newlabel{ex_der_tan}{{242}{552}{The Product and Quotient Rules}{examplecounter.242}{}}
\newlabel{eex_der_tan}{{103}{552}{The Product and Quotient Rules}{examplecounter.242}{}}
\pgfsyspdfmark {pgfid874}{27539003}{9865859}
\newlabel{thm:deriv_trig}{{104}{553}{The Product and Quotient Rules}{theoremcounter.104}{}}
\pgfsyspdfmark {pgfid876}{4474142}{35083097}
\newlabel{ex_prod_quot}{{243}{553}{The Product and Quotient Rules}{examplecounter.243}{}}
\newlabel{eex_prod_quot}{{104}{553}{The Product and Quotient Rules}{examplecounter.243}{}}
\pgfsyspdfmark {pgfid877}{8548872}{16271060}
\pgfsyspdfmark {pgfid878}{4474142}{13266908}
\newlabel{ex_deriv_power}{{244}{553}{The Product and Quotient Rules}{examplecounter.244}{}}
\pgfsyspdfmark {pgfid879}{9979166}{13103068}
\newlabel{eex_deriv_power}{{104}{554}{The Product and Quotient Rules}{Item.4395}{}}
\pgfsyspdfmark {pgfid880}{14550631}{45390346}
\newlabel{thm:PowerRule}{{105}{554}{The Product and Quotient Rules}{theoremcounter.105}{}}
\pgfsyspdfmark {pgfid882}{14288487}{20318501}
\newlabel{ex_multiple_deriv}{{245}{554}{The Product and Quotient Rules}{examplecounter.245}{}}
\pgfsyspdfmark {pgfid883}{19793511}{20154661}
\newlabel{eex_multiple_deriv}{{105}{555}{The Product and Quotient Rules}{Item.4401}{}}
\pgfsyspdfmark {pgfid884}{4736286}{33452604}
\@writefile{toc}{\contentsline {section}{\numberline {11.5}The Chain Rule}{558}{section.11.5}}
\newlabel{sec:chainrule}{{11.5}{558}{The Chain Rule}{section.11.5}{}}
\pgfsyspdfmark {pgfid885}{14288487}{28732728}
\newlabel{ex_chain1}{{246}{558}{The Chain Rule}{examplecounter.246}{}}
\newlabel{eex_chain1}{{11.5}{558}{The Chain Rule}{examplecounter.246}{}}
\pgfsyspdfmark {pgfid886}{28623911}{5522718}
\newlabel{thm:chain_rule}{{106}{559}{The Chain Rule}{theoremcounter.106}{}}
\pgfsyspdfmark {pgfid888}{4474142}{38830119}
\newlabel{ex_chain2}{{247}{559}{The Chain Rule}{examplecounter.247}{}}
\newlabel{eex_chain2}{{106}{559}{The Chain Rule}{examplecounter.247}{}}
\pgfsyspdfmark {pgfid889}{5719326}{13203469}
\newlabel{thm:gen_power_rule}{{107}{559}{The Chain Rule}{theoremcounter.107}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.17}{\ignorespaces $f(x) = \qopname  \relax o{cos}x^2$ sketched along with its tangent line at $x=1$.\relax }}{560}{figure.caption.892}}
\newlabel{fig:chain7}{{11.17}{560}{$f(x) = \cos x^2$ sketched along with its tangent line at $x=1$.\relax }{figure.caption.892}{}}
\pgfsyspdfmark {pgfid891}{14288487}{38662768}
\newlabel{ex_chain3}{{248}{560}{The Chain Rule}{examplecounter.248}{}}
\newlabel{eex_chain3}{{107}{560}{The Chain Rule}{Item.4452}{}}
\pgfsyspdfmark {pgfid892}{14550631}{18824347}
\pgfsyspdfmark {pgfid893}{14288487}{17415323}
\newlabel{ex_chain7}{{249}{560}{The Chain Rule}{examplecounter.249}{}}
\newlabel{eex_chain7}{{107}{560}{The Chain Rule}{examplecounter.249}{}}
\pgfsyspdfmark {pgfid894}{30843643}{9812500}
\pgfsyspdfmark {pgfid895}{4474142}{26014933}
\newlabel{ex_chain4}{{250}{561}{The Chain Rule}{examplecounter.250}{}}
\newlabel{eex_chain4}{{107}{561}{The Chain Rule}{Item.4459}{}}
\pgfsyspdfmark {pgfid896}{4736286}{11027742}
\pgfsyspdfmark {pgfid897}{14288487}{44512050}
\newlabel{ex_chain6}{{251}{562}{The Chain Rule}{examplecounter.251}{}}
\newlabel{eex_chain6}{{107}{562}{The Chain Rule}{examplecounter.251}{}}
\pgfsyspdfmark {pgfid898}{16101031}{25798793}
\pgfsyspdfmark {pgfid899}{14288487}{21244041}
\newlabel{ex_chain5}{{252}{562}{The Chain Rule}{examplecounter.252}{}}
\newlabel{eex_chain5}{{107}{562}{The Chain Rule}{examplecounter.252}{}}
\pgfsyspdfmark {pgfid900}{32151377}{5522718}
\pgfsyspdfmark {pgfid901}{4474142}{43630238}
\newlabel{ex_chain8}{{253}{563}{The Chain Rule}{examplecounter.253}{}}
\newlabel{eex_chain8}{{107}{563}{The Chain Rule}{examplecounter.253}{}}
\pgfsyspdfmark {pgfid902}{15337992}{20277150}
\newlabel{thm:exponentials}{{108}{563}{The Chain Rule}{theoremcounter.108}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.18}{\ignorespaces A series of gears to demonstrate the Chain Rule. Note how $\genfrac  {}{}{}0{dy}{dx} = \genfrac  {}{}{}0{dy}{du}\cdot \genfrac  {}{}{}0{du}{dx}$\relax }}{564}{figure.caption.893}}
\newlabel{fig:chainrulegears}{{11.18}{564}{A series of gears to demonstrate the Chain Rule. Note how $\dfrac {dy}{dx} = \dfrac {dy}{du}\cdot \dfrac {du}{dx}$\relax }{figure.caption.893}{}}
\newlabel{exer:02_05_06}{{7}{565}{The Chain Rule}{Item.4466}{}}
\newlabel{exer:02_05_09}{{10}{565}{The Chain Rule}{Item.4469}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.1}{\ignorespaces Graphs of functions with and without extreme values.\relax }}{567}{figure.caption.894}}
\newlabel{fig:extreme}{{12.1}{567}{Graphs of functions with and without extreme values.\relax }{figure.caption.894}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {12}The Graphical Behavior of Functions}{567}{chapter.12}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{chapter:graphbehavior}{{12}{567}{The Graphical Behavior of Functions}{chapter.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.1}Extreme Values}{567}{section.12.1}}
\newlabel{sec:extreme_values}{{12.1}{567}{Extreme Values}{section.12.1}{}}
\newlabel{def:extreme_values}{{70}{567}{Extreme Values}{definitioncounter.70}{}}
\newlabel{thm:extreme_val}{{109}{567}{Extreme Values}{theoremcounter.109}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.2}{\ignorespaces A graph of $f(x) = 2x^3-9x^2$ as in Example \ref  {ex_extval1}.\relax }}{568}{figure.caption.895}}
\newlabel{fig:extval1}{{12.2}{568}{A graph of $f(x) = 2x^3-9x^2$ as in Example \ref {ex_extval1}.\relax }{figure.caption.895}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.3}{\ignorespaces A graph of $f(x) = (3x^4-4x^3-12x^2+5)/5$ as in Example \ref  {ex_extval2}.\relax }}{568}{figure.caption.896}}
\newlabel{fig:extval2}{{12.3}{568}{A graph of $f(x) = (3x^4-4x^3-12x^2+5)/5$ as in Example \ref {ex_extval2}.\relax }{figure.caption.896}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.4}{\ignorespaces A graph of $f(x) = (x-1)^{2/3}+2$ as in Example \ref  {ex_extval3}.\relax }}{568}{figure.caption.897}}
\newlabel{fig:extval3}{{12.4}{568}{A graph of $f(x) = (x-1)^{2/3}+2$ as in Example \ref {ex_extval3}.\relax }{figure.caption.897}{}}
\pgfsyspdfmark {pgfid906}{14288487}{46871346}
\newlabel{ex_extval1}{{254}{568}{Extreme Values}{examplecounter.254}{}}
\newlabel{eex_extval1}{{109}{568}{Extreme Values}{figure.caption.894}{}}
\pgfsyspdfmark {pgfid907}{28310703}{39606619}
\newlabel{def:rel_ext}{{71}{568}{Extreme Values}{definitioncounter.71}{}}
\pgfsyspdfmark {pgfid909}{14288487}{19957327}
\newlabel{ex_extval2}{{255}{568}{Extreme Values}{examplecounter.255}{}}
\newlabel{eex_extval2}{{71}{568}{Extreme Values}{examplecounter.255}{}}
\pgfsyspdfmark {pgfid910}{22798984}{11073739}
\pgfsyspdfmark {pgfid911}{14288487}{9641717}
\newlabel{ex_extval3}{{256}{568}{Extreme Values}{examplecounter.256}{}}
\pgfsyspdfmark {pgfid912}{19793511}{9477877}
\@writefile{lof}{\contentsline {figure}{\numberline {12.5}{\ignorespaces A graph of $f(x)=x^3$ which has a critical value of $x=0$, but no relative extrema.\relax }}{569}{figure.caption.898}}
\newlabel{fig:extreme4}{{12.5}{569}{A graph of $f(x)=x^3$ which has a critical value of $x=0$, but no relative extrema.\relax }{figure.caption.898}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.6}{\ignorespaces A graph of $f(x) = 2x^3+3x^2-12x$ on $[0,3]$ as in Example \ref  {ex_extval4}.\relax }}{569}{figure.caption.899}}
\newlabel{fig:extval4}{{12.6}{569}{A graph of $f(x) = 2x^3+3x^2-12x$ on $[0,3]$ as in Example \ref {ex_extval4}.\relax }{figure.caption.899}{}}
\newlabel{eex_extval3}{{71}{569}{Extreme Values}{examplecounter.256}{}}
\pgfsyspdfmark {pgfid913}{22908491}{46707506}
\newlabel{def:criticalnum}{{72}{569}{Extreme Values}{definitioncounter.72}{}}
\newlabel{thm:criticalpts}{{110}{569}{Extreme Values}{theoremcounter.110}{}}
\newlabel{idea:extrema}{{43}{569}{Extreme Values}{keyideacounter.43}{}}
\pgfsyspdfmark {pgfid917}{4474142}{8172430}
\newlabel{ex_extval4}{{257}{569}{Extreme Values}{examplecounter.257}{}}
\pgfsyspdfmark {pgfid918}{9979166}{8008590}
\@writefile{lof}{\contentsline {figure}{\numberline {12.7}{\ignorespaces Finding the extreme values of $f$ in Example \ref  {ex_extval4}.\relax }}{570}{figure.caption.900}}
\newlabel{table:ext4}{{12.7}{570}{Finding the extreme values of $f$ in Example \ref {ex_extval4}.\relax }{figure.caption.900}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.8}{\ignorespaces Finding the extreme values of $f$ in Example \ref  {ex_extval5}.\relax }}{570}{figure.caption.901}}
\newlabel{table:ext5}{{12.8}{570}{Finding the extreme values of $f$ in Example \ref {ex_extval5}.\relax }{figure.caption.901}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.9}{\ignorespaces A graph of $f(x)$ on $[-4,2]$ as in Example \ref  {ex_extval5}.\relax }}{570}{figure.caption.902}}
\newlabel{fig:extval5}{{12.9}{570}{A graph of $f(x)$ on $[-4,2]$ as in Example \ref {ex_extval5}.\relax }{figure.caption.902}{}}
\newlabel{eex_extval4}{{43}{570}{Extreme Values}{figure.caption.899}{}}
\pgfsyspdfmark {pgfid919}{15533671}{37442342}
\pgfsyspdfmark {pgfid920}{14288487}{30602017}
\newlabel{ex_extval5}{{258}{570}{Extreme Values}{examplecounter.258}{}}
\newlabel{eex_extval5}{{43}{570}{Extreme Values}{examplecounter.258}{}}
\pgfsyspdfmark {pgfid921}{22666149}{5522718}
\@writefile{lof}{\contentsline {figure}{\numberline {12.10}{\ignorespaces Finding the extrema of $f(x)= \qopname  \relax o{cos}(x^2)$ in Example \ref  {ex_extval6}.\relax }}{571}{figure.caption.903}}
\newlabel{table:ext6}{{12.10}{571}{Finding the extrema of $f(x)= \cos (x^2)$ in Example \ref {ex_extval6}.\relax }{figure.caption.903}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.11}{\ignorespaces A graph of $f(x)=\qopname  \relax o{cos}(x^2)$ on $[-2,2]$ as in Example \ref  {ex_extval6}.\relax }}{571}{figure.caption.904}}
\newlabel{fig:extval6}{{12.11}{571}{A graph of $f(x)=\cos (x^2)$ on $[-2,2]$ as in Example \ref {ex_extval6}.\relax }{figure.caption.904}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.12}{\ignorespaces Finding the extrema of the half--circle in Example \ref  {ex_extval7}.\relax }}{571}{figure.caption.905}}
\newlabel{table:ext7}{{12.12}{571}{Finding the extrema of the half--circle in Example \ref {ex_extval7}.\relax }{figure.caption.905}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.13}{\ignorespaces A graph of $f(x)=\sqrt  {1-x^2}$ on $[-1,1]$ as in Example \ref  {ex_extval7}.\relax }}{571}{figure.caption.906}}
\newlabel{fig:extval7}{{12.13}{571}{A graph of $f(x)=\sqrt {1-x^2}$ on $[-1,1]$ as in Example \ref {ex_extval7}.\relax }{figure.caption.906}{}}
\pgfsyspdfmark {pgfid922}{4474142}{46871346}
\newlabel{ex_extval6}{{259}{571}{Extreme Values}{examplecounter.259}{}}
\newlabel{eex_extval6}{{43}{571}{Extreme Values}{examplecounter.259}{}}
\pgfsyspdfmark {pgfid923}{23530170}{34124594}
\pgfsyspdfmark {pgfid924}{4474142}{31142706}
\newlabel{ex_extval7}{{260}{571}{Extreme Values}{examplecounter.260}{}}
\newlabel{eex_extval7}{{43}{571}{Extreme Values}{examplecounter.260}{}}
\pgfsyspdfmark {pgfid925}{15220552}{22464683}
\@writefile{lof}{\contentsline {figure}{\numberline {12.14}{\ignorespaces A graph of a function $f$ used to illustrate the concepts of \textit  {increasing} and \textit  {decreasing}.\relax }}{574}{figure.caption.907}}
\newlabel{fig:incr0}{{12.14}{574}{A graph of a function $f$ used to illustrate the concepts of \textit {increasing} and \textit {decreasing}.\relax }{figure.caption.907}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.15}{\ignorespaces Examining the secant line of an increasing function.\relax }}{574}{figure.caption.908}}
\newlabel{fig:incr00}{{12.15}{574}{Examining the secant line of an increasing function.\relax }{figure.caption.908}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.2}Increasing and Decreasing Functions}{574}{section.12.2}}
\newlabel{sec:incr_decr}{{12.2}{574}{Increasing and Decreasing Functions}{section.12.2}{}}
\newlabel{def:incr_decr}{{73}{574}{Increasing and Decreasing Functions}{definitioncounter.73}{}}
\newlabel{thm:incr_decr}{{111}{575}{Increasing and Decreasing Functions}{theoremcounter.111}{}}
\newlabel{idea:incr_decr}{{44}{575}{Increasing and Decreasing Functions}{keyideacounter.44}{}}
\pgfsyspdfmark {pgfid929}{4474142}{21235012}
\newlabel{ex_incr1}{{261}{575}{Increasing and Decreasing Functions}{examplecounter.261}{}}
\pgfsyspdfmark {pgfid930}{9979166}{21071172}
\@writefile{lof}{\contentsline {figure}{\numberline {12.16}{\ignorespaces Number line for $f$ in Example \ref  {ex_incr1}.\relax }}{575}{figure.caption.909}}
\newlabel{fig:incrline1}{{12.16}{575}{Number line for $f$ in Example \ref {ex_incr1}.\relax }{figure.caption.909}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.17}{\ignorespaces A graph of $f(x)$ in Example \ref  {ex_incr1}, showing where $f$ is increasing and decreasing.\relax }}{576}{figure.caption.910}}
\newlabel{fig:incr1}{{12.17}{576}{A graph of $f(x)$ in Example \ref {ex_incr1}, showing where $f$ is increasing and decreasing.\relax }{figure.caption.910}{}}
\newlabel{eex_incr1}{{44}{576}{Increasing and Decreasing Functions}{figure.caption.909}{}}
\pgfsyspdfmark {pgfid931}{20512443}{31226920}
\newlabel{thm:first_der}{{112}{577}{Increasing and Decreasing Functions}{theoremcounter.112}{}}
\pgfsyspdfmark {pgfid933}{4474142}{28584999}
\newlabel{ex_incr2}{{262}{577}{Increasing and Decreasing Functions}{examplecounter.262}{}}
\pgfsyspdfmark {pgfid934}{9979166}{28421159}
\@writefile{lof}{\contentsline {figure}{\numberline {12.19}{\ignorespaces A graph of $f(x)$ in Example \ref  {ex_incr2}, showing where $f$ is increasing and decreasing.\relax }}{578}{figure.caption.912}}
\newlabel{fig:incr2}{{12.19}{578}{A graph of $f(x)$ in Example \ref {ex_incr2}, showing where $f$ is increasing and decreasing.\relax }{figure.caption.912}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.18}{\ignorespaces Number line for $f$ in Example \ref  {ex_incr2}.\relax }}{578}{figure.caption.911}}
\newlabel{fig:incrline2}{{12.18}{578}{Number line for $f$ in Example \ref {ex_incr2}.\relax }{figure.caption.911}{}}
\newlabel{eex_incr2}{{112}{578}{Increasing and Decreasing Functions}{figure.caption.911}{}}
\pgfsyspdfmark {pgfid935}{16612672}{26353083}
\pgfsyspdfmark {pgfid936}{14288487}{19439035}
\newlabel{ex_incr3}{{263}{578}{Increasing and Decreasing Functions}{examplecounter.263}{}}
\pgfsyspdfmark {pgfid937}{19793511}{19275195}
\@writefile{lof}{\contentsline {figure}{\numberline {12.21}{\ignorespaces A graph of $f(x)$ in Example \ref  {ex_incr3}, showing where $f$ is increasing and decreasing.\relax }}{579}{figure.caption.914}}
\newlabel{fig:incr3}{{12.21}{579}{A graph of $f(x)$ in Example \ref {ex_incr3}, showing where $f$ is increasing and decreasing.\relax }{figure.caption.914}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.20}{\ignorespaces Number line for $f$ in Example \ref  {ex_incr3}.\relax }}{579}{figure.caption.913}}
\newlabel{fig:incrline3}{{12.20}{579}{Number line for $f$ in Example \ref {ex_incr3}.\relax }{figure.caption.913}{}}
\newlabel{eex_incr3}{{112}{579}{Increasing and Decreasing Functions}{figure.caption.913}{}}
\pgfsyspdfmark {pgfid938}{15195088}{19544517}
\@writefile{lof}{\contentsline {figure}{\numberline {12.22}{\ignorespaces A function $f$ with a concave up graph. Notice how the slopes of the tangent lines, when looking from left to right, are increasing.\relax }}{581}{figure.caption.915}}
\newlabel{fig:concavity1}{{12.22}{581}{A function $f$ with a concave up graph. Notice how the slopes of the tangent lines, when looking from left to right, are increasing.\relax }{figure.caption.915}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.23}{\ignorespaces A function $f$ with a concave down graph. Notice how the slopes of the tangent lines, when looking from left to right, are decreasing.\relax }}{581}{figure.caption.916}}
\newlabel{fig:concavity2}{{12.23}{581}{A function $f$ with a concave down graph. Notice how the slopes of the tangent lines, when looking from left to right, are decreasing.\relax }{figure.caption.916}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.3}Concavity and the Second Derivative}{581}{section.12.3}}
\newlabel{sec:concavity}{{12.3}{581}{Concavity and the Second Derivative}{section.12.3}{}}
\newlabel{def:concavity}{{74}{581}{Concavity and the Second Derivative}{definitioncounter.74}{}}
\newlabel{thm:concavity}{{113}{581}{Concavity and the Second Derivative}{theoremcounter.113}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.24}{\ignorespaces Demonstrating the 4 ways that concavity interacts with increasing/decreasing, along with the relationships with the first and second derivatives.\relax }}{582}{figure.caption.917}}
\newlabel{fig:concavity3}{{12.24}{582}{Demonstrating the 4 ways that concavity interacts with increasing/decreasing, along with the relationships with the first and second derivatives.\relax }{figure.caption.917}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.25}{\ignorespaces A graph of a function with its inflection points marked. The intervals where concave up/down are also indicated.\relax }}{582}{figure.caption.918}}
\newlabel{fig:concavity4}{{12.25}{582}{A graph of a function with its inflection points marked. The intervals where concave up/down are also indicated.\relax }{figure.caption.918}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.26}{\ignorespaces A number line determining the concavity of $f$ in Example \ref  {ex_conc1}.\relax }}{582}{figure.caption.919}}
\newlabel{fig:concline1}{{12.26}{582}{A number line determining the concavity of $f$ in Example \ref {ex_conc1}.\relax }{figure.caption.919}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.27}{\ignorespaces A graph of $f(x)$ used in Example \ref  {ex_conc1}.\relax }}{582}{figure.caption.920}}
\newlabel{fig:conc1}{{12.27}{582}{A graph of $f(x)$ used in Example \ref {ex_conc1}.\relax }{figure.caption.920}{}}
\newlabel{def:infl}{{75}{582}{Concavity and the Second Derivative}{definitioncounter.75}{}}
\newlabel{thm:inflection}{{114}{582}{Concavity and the Second Derivative}{theoremcounter.114}{}}
\pgfsyspdfmark {pgfid943}{14288487}{25259156}
\newlabel{ex_conc1}{{264}{582}{Concavity and the Second Derivative}{examplecounter.264}{}}
\newlabel{eex_conc1}{{114}{582}{Concavity and the Second Derivative}{examplecounter.264}{}}
\pgfsyspdfmark {pgfid944}{17952427}{12438518}
\pgfsyspdfmark {pgfid945}{14288487}{11004865}
\newlabel{ex_conc2}{{265}{582}{Concavity and the Second Derivative}{examplecounter.265}{}}
\pgfsyspdfmark {pgfid946}{19793511}{10841025}
\@writefile{lof}{\contentsline {figure}{\numberline {12.29}{\ignorespaces A graph of $f(x)$ and $\ensuremath  {f\tmspace  +\thinmuskip {.1667em}''}(x)$ in Example \ref  {ex_conc2}.\relax }}{583}{figure.caption.922}}
\newlabel{fig:conc2}{{12.29}{583}{A graph of $f(x)$ and $\fpp (x)$ in Example \ref {ex_conc2}.\relax }{figure.caption.922}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.30}{\ignorespaces A graph of $S(t)$ in Example \ref  {ex_conc3}, modeling the sale of a product over time.\relax }}{583}{figure.caption.923}}
\newlabel{fig:conc3}{{12.30}{583}{A graph of $S(t)$ in Example \ref {ex_conc3}, modeling the sale of a product over time.\relax }{figure.caption.923}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.28}{\ignorespaces Number line for $f$ in Example \ref  {ex_conc2}.\relax }}{583}{figure.caption.921}}
\newlabel{fig:concline2}{{12.28}{583}{Number line for $f$ in Example \ref {ex_conc2}.\relax }{figure.caption.921}{}}
\newlabel{eex_conc2}{{114}{583}{Concavity and the Second Derivative}{figure.caption.921}{}}
\pgfsyspdfmark {pgfid947}{25707806}{17555845}
\pgfsyspdfmark {pgfid948}{4474142}{7259422}
\newlabel{ex_conc3}{{266}{583}{Concavity and the Second Derivative}{examplecounter.266}{}}
\pgfsyspdfmark {pgfid949}{9979166}{7095582}
\@writefile{lof}{\contentsline {figure}{\numberline {12.31}{\ignorespaces A graph of $S(t)$ in Example \ref  {ex_conc3} along with $S\hskip .75pt'(t)$.\relax }}{584}{figure.caption.924}}
\newlabel{fig:conc3b}{{12.31}{584}{A graph of $S(t)$ in Example \ref {ex_conc3} along with $S\primeskip '(t)$.\relax }{figure.caption.924}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.32}{\ignorespaces A graph of $f(x) = x^4$. Clearly $f$ is always concave up, despite the fact that $\ensuremath  {f\tmspace  +\thinmuskip {.1667em}''}(x) = 0$ when $x=0$. It this example, the \textit  {possible} point of inflection $(0,0)$ is not a point of inflection.\relax }}{584}{figure.caption.925}}
\newlabel{fig:concavity5}{{12.32}{584}{A graph of $f(x) = x^4$. Clearly $f$ is always concave up, despite the fact that $\fpp (x) = 0$ when $x=0$. It this example, the \textit {possible} point of inflection $(0,0)$ is not a point of inflection.\relax }{figure.caption.925}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.33}{\ignorespaces Demonstrating the fact that relative maxima occur when the graph is concave down and relative minima occur when the graph is concave up.\relax }}{584}{figure.caption.926}}
\newlabel{fig:concavity6}{{12.33}{584}{Demonstrating the fact that relative maxima occur when the graph is concave down and relative minima occur when the graph is concave up.\relax }{figure.caption.926}{}}
\newlabel{eex_conc3}{{114}{584}{Concavity and the Second Derivative}{figure.caption.923}{}}
\pgfsyspdfmark {pgfid950}{29150097}{37270322}
\newlabel{thm:second_der}{{115}{584}{Concavity and the Second Derivative}{theoremcounter.115}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.34}{\ignorespaces A graph of $f(x)$ in Example \ref  {ex_conc4}. The second derivative is evaluated at each critical point. When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum.\relax }}{585}{figure.caption.927}}
\newlabel{fig:conc4}{{12.34}{585}{A graph of $f(x)$ in Example \ref {ex_conc4}. The second derivative is evaluated at each critical point. When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum.\relax }{figure.caption.927}{}}
\pgfsyspdfmark {pgfid952}{4474142}{46871346}
\newlabel{ex_conc4}{{267}{585}{Concavity and the Second Derivative}{examplecounter.267}{}}
\newlabel{eex_conc4}{{115}{585}{Concavity and the Second Derivative}{examplecounter.267}{}}
\pgfsyspdfmark {pgfid953}{19929310}{39629618}
\newlabel{exer:03_04_ex_16}{{16}{586}{Concavity and the Second Derivative}{Item.4598}{}}
\newlabel{exer:03_04_ex_28}{{28}{586}{Concavity and the Second Derivative}{Item.4610}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.4}Curve Sketching}{588}{section.12.4}}
\newlabel{sec:sketch}{{12.4}{588}{Curve Sketching}{section.12.4}{}}
\newlabel{idea:sketch}{{45}{588}{Curve Sketching}{keyideacounter.45}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.36}{\ignorespaces Sketching $f$ in Example \ref  {ex_sketch1}.\relax }}{589}{figure.caption.929}}
\newlabel{fig:sketch1}{{12.36}{589}{Sketching $f$ in Example \ref {ex_sketch1}.\relax }{figure.caption.929}{}}
\newlabel{idea:sketchb}{{45}{589}{Curve Sketching}{keyideacounter.45}{}}
\pgfsyspdfmark {pgfid956}{4474142}{36531458}
\newlabel{ex_sketch1}{{268}{589}{Curve Sketching}{examplecounter.268}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.35}{\ignorespaces Number line for $f$ in Example \ref  {ex_sketch1}.\relax }}{589}{figure.caption.928}}
\newlabel{fig:sketchline1}{{12.35}{589}{Number line for $f$ in Example \ref {ex_sketch1}.\relax }{figure.caption.928}{}}
\newlabel{eex_sketch1}{{45}{589}{Curve Sketching}{Item.4650}{}}
\pgfsyspdfmark {pgfid957}{5719326}{3163422}
\@writefile{lof}{\contentsline {figure}{\numberline {12.38}{\ignorespaces Sketching $f$ in Example \ref  {ex_sketch2}.\relax }}{590}{figure.caption.931}}
\newlabel{fig:sketch2}{{12.38}{590}{Sketching $f$ in Example \ref {ex_sketch2}.\relax }{figure.caption.931}{}}
\pgfsyspdfmark {pgfid958}{14288487}{46871346}
\newlabel{ex_sketch2}{{269}{590}{Curve Sketching}{examplecounter.269}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.37}{\ignorespaces Number line for $f$ in Example \ref  {ex_sketch2}.\relax }}{590}{figure.caption.930}}
\newlabel{fig:sketchline2}{{12.37}{590}{Number line for $f$ in Example \ref {ex_sketch2}.\relax }{figure.caption.930}{}}
\newlabel{eex_sketch2}{{45}{590}{Curve Sketching}{Item.4657}{}}
\pgfsyspdfmark {pgfid959}{15533671}{3163422}
\@writefile{lof}{\contentsline {figure}{\numberline {12.40}{\ignorespaces Sketching $f$ in Example \ref  {ex_sketch3}.\relax }}{591}{figure.caption.933}}
\newlabel{fig:sketch3}{{12.40}{591}{Sketching $f$ in Example \ref {ex_sketch3}.\relax }{figure.caption.933}{}}
\pgfsyspdfmark {pgfid960}{4474142}{46871346}
\newlabel{ex_sketch3}{{270}{591}{Curve Sketching}{examplecounter.270}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.39}{\ignorespaces Number line for $f$ in Example \ref  {ex_sketch3}.\relax }}{591}{figure.caption.932}}
\newlabel{fig:sketchline3}{{12.39}{591}{Number line for $f$ in Example \ref {ex_sketch3}.\relax }{figure.caption.932}{}}
\newlabel{eex_sketch3}{{45}{591}{Curve Sketching}{Item.4664}{}}
\pgfsyspdfmark {pgfid961}{5719326}{14969178}
\@writefile{lof}{\contentsline {figure}{\numberline {12.41}{\ignorespaces A graph of $y=\qopname  \relax o{sin}x$ generated by \textit  {Mathematica}.\relax }}{592}{figure.caption.934}}
\newlabel{fig:mathematica_sinx}{{12.41}{592}{A graph of $y=\sin x$ generated by \textit {Mathematica}.\relax }{figure.caption.934}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.5}Antiderivatives and Indefinite Integration}{594}{section.12.5}}
\newlabel{sec:antider}{{12.5}{594}{Antiderivatives and Indefinite Integration}{section.12.5}{}}
\newlabel{def:antider}{{76}{594}{Antiderivatives and Indefinite Integration}{definitioncounter.76}{}}
\newlabel{thm:antideriv_const}{{116}{594}{Antiderivatives and Indefinite Integration}{theoremcounter.116}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {12.42}{\ignorespaces Understanding the indefinite integral notation.\relax }}{595}{figure.caption.935}}
\newlabel{fig:anti1}{{12.42}{595}{Understanding the indefinite integral notation.\relax }{figure.caption.935}{}}
\pgfsyspdfmark {pgfid964}{4474142}{26038040}
\newlabel{ex_anti2}{{271}{595}{Antiderivatives and Indefinite Integration}{examplecounter.271}{}}
\newlabel{eex_anti2}{{116}{595}{Antiderivatives and Indefinite Integration}{examplecounter.271}{}}
\pgfsyspdfmark {pgfid965}{5719326}{18429638}
\pgfsyspdfmark {pgfid966}{14288487}{41533149}
\newlabel{ex_anti3}{{272}{596}{Antiderivatives and Indefinite Integration}{examplecounter.272}{}}
\newlabel{eex_anti3}{{116}{596}{Antiderivatives and Indefinite Integration}{examplecounter.272}{}}
\pgfsyspdfmark {pgfid967}{23315597}{17562580}
\newlabel{thm:indef_alg}{{117}{597}{Antiderivatives and Indefinite Integration}{theoremcounter.117}{}}
\pgfsyspdfmark {pgfid969}{14288487}{20436809}
\newlabel{ex_anti4}{{273}{598}{Antiderivatives and Indefinite Integration}{examplecounter.273}{}}
\pgfsyspdfmark {pgfid970}{19793511}{20272969}
\newlabel{eex_anti4}{{117}{599}{Antiderivatives and Indefinite Integration}{examplecounter.273}{}}
\pgfsyspdfmark {pgfid971}{23147592}{40189005}
\pgfsyspdfmark {pgfid972}{4474142}{38779981}
\newlabel{ex_anti5}{{274}{599}{Antiderivatives and Indefinite Integration}{examplecounter.274}{}}
\newlabel{eex_anti5}{{117}{599}{Antiderivatives and Indefinite Integration}{examplecounter.274}{}}
\pgfsyspdfmark {pgfid973}{12627358}{21054935}
\HyPL@Entry{610<</P(A.)/S/D>>}
\@writefile{toc}{\contentsline {chapter}{\numberline {A}Answers To Selected Problems}{A.1}{appendix.A}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{sinusoidexercise1}{{35}{A.38}{Answers To Selected Problems}{Item.5743}{}}
\@writefile{toc}{\contentsline {chapter}{Index}{A.51}{section*.936}}
\zref@newlabel{LastPage}{\default{A}\page{A.64}\abspage{674}}