\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}[theorem]{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=0.5in,right=0.5in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} \Large Practice} \lfoot{\small \copyright $\;$ copyright Hidegkuti, 2012} \rfoot{\small Last revised: April 22, 2014} \cfoot{} \chead{\LARGE Riemann Sums} \rhead{\large page \thepage} \textwidth 7.6in \textheight 9.7in \setlength{\headheight}{28pt} \setlength{\parindent}{0pt} \begin{document} \begin{enumerate} \item Consider the function $f\left( x\right) =\ln x$ on the interval $\left[ 1,6\right] .$\FRAME{dtbpFX}{4.1252in}{2.4734in}{0pt}{}{}{Plot}{\special% {language "Scientific Word";type "MAPLEPLOT";width 4.1252in;height 2.4734in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-0.005000";xmax "10.005";xviewmin "-0.005000";xviewmax "10.005";yviewmin "-2";yviewmax "3";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 11;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\ln x$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-0.005000,10.005";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'N4GD4704.wmf';tempfile-properties "XPR";}} a) \ Approximate $\dint\limits_{1}^{6}\ln xdx$ by a left Riemann sum using a uniform partition with $n=5$. \ Is this an underestimation or overestimation of the area under the graph? b) \ Approximate $\dint\limits_{1}^{6}\ln xdx$ by a right Riemann sum using a uniform partition with $n=5$. \ Is this an underestimation or overestimation of the area under the graph? c) \ Approximate $\dint\limits_{1}^{6}\ln xdx$ by a left Riemann sum using a uniform partition with $n=10$. \ Is this an underestimation or overestimation of the area under the graph? d) \ Approximate $\dint\limits_{1}^{6}\ln xdx$ by a right Riemann sum using a uniform partition with $n=10$. \ Is this an underestimation or overestimation of the area under the graph?\bigskip \pagebreak \item Consider the function $f\left( x\right) =\sin x$ on the interval $% \left[ 0,\pi \right] .$\FRAME{dtbpFX}{5.4371in}{2.6455in}{0pt}{}{}{Plot}{% \special{language "Scientific Word";type "MAPLEPLOT";width 5.4371in;height 2.6455in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-0.3";xmax "3.15";xviewmin "-0.3";xviewmax "3.15";yviewmin "-0.1987";yviewmax "1.001987";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-y-ticks 2;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\sin x$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-0.3,3.15";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'N4GD6705.wmf';tempfile-properties "XPR";}}\bigskip \bigskip \bigskip a) \ Compute the left Riemann sum for $\dint\limits_{0}^{\pi }\sin xdx$ on this interval with $n=4$ subintervals. \ Make sure to use radians for angles. b) \ Compute the left Riemann sum for $\dint\limits_{0}^{\pi }\sin xdx$ on this interval with $n=4$ subintervals. \ Make sure to use radians for angles. c) \ Compute the left Riemann sum for $\dint\limits_{0}^{\pi }\sin xdx$ on this interval with $n=6$ subintervals. \ Make sure to use radians for angles. \end{enumerate} \pagebreak \begin{center} {\LARGE Answers\bigskip \bigskip } \end{center} \begin{enumerate} \item a) \ $\ln 2+\ln 3+\ln 4+\ln 5=\ln 120\approx 4.\,\allowbreak 787\,492$ \ \ \ \ underestimate\qquad b) \ $\ln 2+\ln 3+\ln 4+\ln 5+\ln 6=\ln 720\approx \allowbreak 6.\,\allowbreak 579\,25$ \ \ \ \ overestimate c) $\ \allowbreak \dfrac{1}{2}\left( \ln \dfrac{2}{2}+\ln \dfrac{3}{2}+\ln \dfrac{4}{2}+\ln \dfrac{5}{2}+\ln \dfrac{6}{2}+\ln \dfrac{7}{2}+\ln \dfrac{8% }{2}+\ln \dfrac{9}{2}+\ln \dfrac{10}{2}+\ln \dfrac{11}{2}\right) =\dfrac{1}{2% }\ln \left( \dfrac{11!}{2^{10}}\right) \approx 5.\,\allowbreak 285\,418$ \ \ \ \ \qquad underestimate d) \ $\allowbreak \dfrac{1}{2}\left( \ln \dfrac{3}{2}+\ln \dfrac{4}{2}+\ln \dfrac{5}{2}+\ln \dfrac{6}{2}+\ln \dfrac{7}{2}+\ln \dfrac{8}{2}+\ln \dfrac{9% }{2}+\ln \dfrac{10}{2}+\ln \dfrac{11}{2}+\ln \dfrac{12}{2}\right) =\dfrac{1}{% 2}\ln \left( \dfrac{12!}{2^{11}}\right) =\allowbreak 6.\,\allowbreak 181\,297\,8$ \qquad overestimate \item a) \ $\dfrac{1}{4}\pi +\dfrac{1}{4}\pi \sqrt{2}\approx \allowbreak 1.\,\allowbreak 896\,118\,897\,\allowbreak 937\,04$ \ \ \ \ \ \ \ \ \ b) \ \ $\dfrac{1}{4}\pi +\dfrac{1}{4}\pi \sqrt{2}\approx \allowbreak 1.\,\allowbreak 896\,118\,897\,\allowbreak 937\,04$ c) \ $\dfrac{1}{3}\pi +\dfrac{1}{6}\pi \sqrt{3}\approx \allowbreak 1.\,\allowbreak 954\,097\,233\,\allowbreak 313\,71$ \end{enumerate} \vspace{6.2in} {\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}