\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.6in,right=0.7in,top=0.7in,bottom=0.7in} \pagestyle{fancy} \lhead{\color{blue} \large Lecture Notes} \chead{\color{black} \LARGE Graphing the Antiderivative} \rhead{\large page \ \thepage} \cfoot{} \lfoot{\small \copyright $\;$ copyright Hidegkuti, Powell, 2009} \rfoot{\small Last revised: October 26, 2009} \textwidth 7.3in \textheight 9.3in \setlength{\headheight}{30pt} \setlength{\parindent}{0in} \begin{document} \begin{enumerate} \item The graph below shows $f^{\prime }$, the first derivative of a function $f.$\FRAME{dtbpFX}{5.0194in}{1.4053in}{0pt}{}{}{Plot}{\special% {language "Scientific Word";type "MAPLEPLOT";width 5.0194in;height 1.4053in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-4.01";xmax "4";xviewmin "-4.01";xviewmax "4";yviewmin "-20";yviewmax "20";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 9;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\dfrac{1}{3}\left( x+3\right) \left( x+2\right) x^{2}\left( x-2\right) \left( x-3\right) $};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-4.01,4";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'MXHXZA03.wmf';tempfile-properties "XPR";}% } \begin{enumerate} \item Find all values of $x$ for which the function $f$ has a local maximum at $x$. \item Find all values of $x$ for which the function $f$ has a local minimum at $x$. \item How many points of inflection does $f$ have? \item Sketch the graph of $f$. \end{enumerate} \item The graph below shows $f^{\prime }$, the first derivative of a function $f.$\FRAME{dtbpFX}{5.0194in}{1.5056in}{0pt}{}{}{Plot}{\special% {language "Scientific Word";type "MAPLEPLOT";width 5.0194in;height 1.5056in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-4.01";xmax "4";xviewmin "-4.01";xviewmax "4";yviewmin "-20";yviewmax "20";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 9;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$-\left( x-1\right) \left( 2-x\right) \left( x+1\right) ^{2}\left( x+3\right) $};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-4.01,4";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'MXHXZJ04.wmf';tempfile-properties "XPR";}% } \begin{enumerate} \item Find all values of $x$ for which the function $f$ has a local maximum at $x$. \item Find all values of $x$ for which the function $f$ has a local minimum at $x$. \item How many points of inflection does $f$ have? \item Sketch the graph of $f$.\pagebreak \end{enumerate} \end{enumerate} \begin{center} {\Large Answers} \end{center} \bigskip 1.) \ a) \ $-3,2$ \ \ \ \ b) \ $-2,3$ \ \ \ \ \ c) \ $5$ \ \ \ \ \ \ \ \ \ d)% \FRAME{dtbpFX}{5.0194in}{2.0081in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 5.0194in;height 2.0081in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-4.01";xmax "4";xviewmin "-4.01";xviewmax "4";yviewmin "-40";yviewmax "20";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 9;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\dfrac{1}{3}\left( x+3\right) \left( x+2\right) x^{2}\left( x-2\right) \left( x-3\right) $};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-4.01,4";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";function \TEXUX{$\dfrac{1}{2}\left( 12x^{3}-\dfrac{13}{5}x^{5}+\dfrac{1}{7}x^{7}\right) -10$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-4.01,4";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'MXHXZJ05.wmf';tempfile-properties "XPR";}}\bigskip \bigskip 2.) \ a) \ $1$ \ \ \ \ b) \ $-3,2$ \ \ \ \ \ c) \ $4$ \ \ \ \ \ \ \ \ \ d)% \FRAME{dtbpFX}{5.0194in}{2.0081in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 5.0194in;height 2.0081in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-4.01";xmax "4";xviewmin "-4.01";xviewmax "4";yviewmin "-50";yviewmax "40";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 9;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$-\left( x-1\right) \left( 2-x\right) \left( x+1\right) ^{2}\left( x+3\right) $};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-4.01,4";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";function \TEXUX{$16+3\left( 6x+\dfrac{5}{2}x^{2}-\dfrac{8}{3}x^{3}-\dfrac{3}{2}x^{4}+\dfrac{2}{5}x^{5}+% \dfrac{1}{6}x^{6}\right) $};linecolor "blue";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-4.01,4";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'MXHXZK06.wmf';tempfile-properties "XPR";}% } \vspace{2.2in} \bigskip \bigskip \bigskip \bigskip {\small %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}