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      <tr>
        <td id="L5" class="blob-num js-line-number" data-line-number="5"></td>
        <td id="LC5" class="blob-code js-file-line"><span class="k">\usepackage</span><span class="nb">{</span>amsmath<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L6" class="blob-num js-line-number" data-line-number="6"></td>
        <td id="LC6" class="blob-code js-file-line"><span class="k">\usepackage</span><span class="nb">{</span>amsthm<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L7" class="blob-num js-line-number" data-line-number="7"></td>
        <td id="LC7" class="blob-code js-file-line"><span class="k">\usepackage</span><span class="nb">{</span>amsfonts<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L8" class="blob-num js-line-number" data-line-number="8"></td>
        <td id="LC8" class="blob-code js-file-line"><span class="k">\usepackage</span><span class="nb">{</span>tikz<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L9" class="blob-num js-line-number" data-line-number="9"></td>
        <td id="LC9" class="blob-code js-file-line"><span class="k">\usepackage</span><span class="na">[plain]</span><span class="nb">{</span>algorithm<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L10" class="blob-num js-line-number" data-line-number="10"></td>
        <td id="LC10" class="blob-code js-file-line"><span class="k">\usepackage</span><span class="nb">{</span>algpseudocode<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L11" class="blob-num js-line-number" data-line-number="11"></td>
        <td id="LC11" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L12" class="blob-num js-line-number" data-line-number="12"></td>
        <td id="LC12" class="blob-code js-file-line"><span class="k">\usetikzlibrary</span><span class="nb">{</span>automata,positioning<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L13" class="blob-num js-line-number" data-line-number="13"></td>
        <td id="LC13" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L14" class="blob-num js-line-number" data-line-number="14"></td>
        <td id="LC14" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L15" class="blob-num js-line-number" data-line-number="15"></td>
        <td id="LC15" class="blob-code js-file-line"><span class="c">% Basic Document Settings</span></td>
      </tr>
      <tr>
        <td id="L16" class="blob-num js-line-number" data-line-number="16"></td>
        <td id="LC16" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L17" class="blob-num js-line-number" data-line-number="17"></td>
        <td id="LC17" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L18" class="blob-num js-line-number" data-line-number="18"></td>
        <td id="LC18" class="blob-code js-file-line"><span class="k">\topmargin</span>=-0.45in</td>
      </tr>
      <tr>
        <td id="L19" class="blob-num js-line-number" data-line-number="19"></td>
        <td id="LC19" class="blob-code js-file-line"><span class="k">\evensidemargin</span>=0in</td>
      </tr>
      <tr>
        <td id="L20" class="blob-num js-line-number" data-line-number="20"></td>
        <td id="LC20" class="blob-code js-file-line"><span class="k">\oddsidemargin</span>=0in</td>
      </tr>
      <tr>
        <td id="L21" class="blob-num js-line-number" data-line-number="21"></td>
        <td id="LC21" class="blob-code js-file-line"><span class="k">\textwidth</span>=6.5in</td>
      </tr>
      <tr>
        <td id="L22" class="blob-num js-line-number" data-line-number="22"></td>
        <td id="LC22" class="blob-code js-file-line"><span class="k">\textheight</span>=9.0in</td>
      </tr>
      <tr>
        <td id="L23" class="blob-num js-line-number" data-line-number="23"></td>
        <td id="LC23" class="blob-code js-file-line"><span class="k">\headsep</span>=0.25in</td>
      </tr>
      <tr>
        <td id="L24" class="blob-num js-line-number" data-line-number="24"></td>
        <td id="LC24" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L25" class="blob-num js-line-number" data-line-number="25"></td>
        <td id="LC25" class="blob-code js-file-line"><span class="k">\linespread</span><span class="nb">{</span>1.1<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L26" class="blob-num js-line-number" data-line-number="26"></td>
        <td id="LC26" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L27" class="blob-num js-line-number" data-line-number="27"></td>
        <td id="LC27" class="blob-code js-file-line"><span class="k">\pagestyle</span><span class="nb">{</span>fancy<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L28" class="blob-num js-line-number" data-line-number="28"></td>
        <td id="LC28" class="blob-code js-file-line"><span class="k">\lhead</span><span class="nb">{</span><span class="k">\hmwkAuthorName</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L29" class="blob-num js-line-number" data-line-number="29"></td>
        <td id="LC29" class="blob-code js-file-line"><span class="k">\chead</span><span class="nb">{</span><span class="k">\hmwkClass\ </span>(<span class="k">\hmwkClassInstructor\ \hmwkClassTime</span>): <span class="k">\hmwkTitle</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L30" class="blob-num js-line-number" data-line-number="30"></td>
        <td id="LC30" class="blob-code js-file-line"><span class="k">\rhead</span><span class="nb">{</span><span class="k">\firstxmark</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L31" class="blob-num js-line-number" data-line-number="31"></td>
        <td id="LC31" class="blob-code js-file-line"><span class="k">\lfoot</span><span class="nb">{</span><span class="k">\lastxmark</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L32" class="blob-num js-line-number" data-line-number="32"></td>
        <td id="LC32" class="blob-code js-file-line"><span class="k">\cfoot</span><span class="nb">{</span><span class="k">\thepage</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L33" class="blob-num js-line-number" data-line-number="33"></td>
        <td id="LC33" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L34" class="blob-num js-line-number" data-line-number="34"></td>
        <td id="LC34" class="blob-code js-file-line"><span class="k">\renewcommand\headrulewidth</span><span class="nb">{</span>0.4pt<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L35" class="blob-num js-line-number" data-line-number="35"></td>
        <td id="LC35" class="blob-code js-file-line"><span class="k">\renewcommand\footrulewidth</span><span class="nb">{</span>0.4pt<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L36" class="blob-num js-line-number" data-line-number="36"></td>
        <td id="LC36" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L37" class="blob-num js-line-number" data-line-number="37"></td>
        <td id="LC37" class="blob-code js-file-line"><span class="k">\setlength\parindent</span><span class="nb">{</span>0pt<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L38" class="blob-num js-line-number" data-line-number="38"></td>
        <td id="LC38" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L39" class="blob-num js-line-number" data-line-number="39"></td>
        <td id="LC39" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L40" class="blob-num js-line-number" data-line-number="40"></td>
        <td id="LC40" class="blob-code js-file-line"><span class="c">% Create Problem Sections</span></td>
      </tr>
      <tr>
        <td id="L41" class="blob-num js-line-number" data-line-number="41"></td>
        <td id="LC41" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L42" class="blob-num js-line-number" data-line-number="42"></td>
        <td id="LC42" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L43" class="blob-num js-line-number" data-line-number="43"></td>
        <td id="LC43" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\enterProblemHeader</span><span class="nb">}</span>[1]<span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L44" class="blob-num js-line-number" data-line-number="44"></td>
        <td id="LC44" class="blob-code js-file-line">    <span class="k">\nobreak\extramarks</span><span class="nb">{}{</span>Problem <span class="k">\arabic</span><span class="nb">{</span>#1<span class="nb">}</span> continued on next page<span class="k">\ldots</span><span class="nb">}</span><span class="k">\nobreak</span><span class="nb">{}</span></td>
      </tr>
      <tr>
        <td id="L45" class="blob-num js-line-number" data-line-number="45"></td>
        <td id="LC45" class="blob-code js-file-line">    <span class="k">\nobreak\extramarks</span><span class="nb">{</span>Problem <span class="k">\arabic</span><span class="nb">{</span>#1<span class="nb">}</span> (continued)<span class="nb">}{</span>Problem <span class="k">\arabic</span><span class="nb">{</span>#1<span class="nb">}</span> continued on next page<span class="k">\ldots</span><span class="nb">}</span><span class="k">\nobreak</span><span class="nb">{}</span></td>
      </tr>
      <tr>
        <td id="L46" class="blob-num js-line-number" data-line-number="46"></td>
        <td id="LC46" class="blob-code js-file-line"><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L47" class="blob-num js-line-number" data-line-number="47"></td>
        <td id="LC47" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L48" class="blob-num js-line-number" data-line-number="48"></td>
        <td id="LC48" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\exitProblemHeader</span><span class="nb">}</span>[1]<span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L49" class="blob-num js-line-number" data-line-number="49"></td>
        <td id="LC49" class="blob-code js-file-line">    <span class="k">\nobreak\extramarks</span><span class="nb">{</span>Problem <span class="k">\arabic</span><span class="nb">{</span>#1<span class="nb">}</span> (continued)<span class="nb">}{</span>Problem <span class="k">\arabic</span><span class="nb">{</span>#1<span class="nb">}</span> continued on next page<span class="k">\ldots</span><span class="nb">}</span><span class="k">\nobreak</span><span class="nb">{}</span></td>
      </tr>
      <tr>
        <td id="L50" class="blob-num js-line-number" data-line-number="50"></td>
        <td id="LC50" class="blob-code js-file-line">    <span class="k">\stepcounter</span><span class="nb">{</span>#1<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L51" class="blob-num js-line-number" data-line-number="51"></td>
        <td id="LC51" class="blob-code js-file-line">    <span class="k">\nobreak\extramarks</span><span class="nb">{</span>Problem <span class="k">\arabic</span><span class="nb">{</span>#1<span class="nb">}}{}</span><span class="k">\nobreak</span><span class="nb">{}</span></td>
      </tr>
      <tr>
        <td id="L52" class="blob-num js-line-number" data-line-number="52"></td>
        <td id="LC52" class="blob-code js-file-line"><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L53" class="blob-num js-line-number" data-line-number="53"></td>
        <td id="LC53" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L54" class="blob-num js-line-number" data-line-number="54"></td>
        <td id="LC54" class="blob-code js-file-line"><span class="k">\setcounter</span><span class="nb">{</span>secnumdepth<span class="nb">}{</span>0<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L55" class="blob-num js-line-number" data-line-number="55"></td>
        <td id="LC55" class="blob-code js-file-line"><span class="k">\newcounter</span><span class="nb">{</span>partCounter<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L56" class="blob-num js-line-number" data-line-number="56"></td>
        <td id="LC56" class="blob-code js-file-line"><span class="k">\newcounter</span><span class="nb">{</span>homeworkProblemCounter<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L57" class="blob-num js-line-number" data-line-number="57"></td>
        <td id="LC57" class="blob-code js-file-line"><span class="k">\setcounter</span><span class="nb">{</span>homeworkProblemCounter<span class="nb">}{</span>1<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L58" class="blob-num js-line-number" data-line-number="58"></td>
        <td id="LC58" class="blob-code js-file-line"><span class="k">\nobreak\extramarks</span><span class="nb">{</span>Problem <span class="k">\arabic</span><span class="nb">{</span>homeworkProblemCounter<span class="nb">}}{}</span><span class="k">\nobreak</span><span class="nb">{}</span></td>
      </tr>
      <tr>
        <td id="L59" class="blob-num js-line-number" data-line-number="59"></td>
        <td id="LC59" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L60" class="blob-num js-line-number" data-line-number="60"></td>
        <td id="LC60" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L61" class="blob-num js-line-number" data-line-number="61"></td>
        <td id="LC61" class="blob-code js-file-line"><span class="c">% Homework Problem Environment</span></td>
      </tr>
      <tr>
        <td id="L62" class="blob-num js-line-number" data-line-number="62"></td>
        <td id="LC62" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L63" class="blob-num js-line-number" data-line-number="63"></td>
        <td id="LC63" class="blob-code js-file-line"><span class="c">% This environment takes an optional argument. When given, it will adjust the</span></td>
      </tr>
      <tr>
        <td id="L64" class="blob-num js-line-number" data-line-number="64"></td>
        <td id="LC64" class="blob-code js-file-line"><span class="c">% problem counter. This is useful for when the problems given for your</span></td>
      </tr>
      <tr>
        <td id="L65" class="blob-num js-line-number" data-line-number="65"></td>
        <td id="LC65" class="blob-code js-file-line"><span class="c">% assignment aren&#39;t sequential. See the last 3 problems of this template for an</span></td>
      </tr>
      <tr>
        <td id="L66" class="blob-num js-line-number" data-line-number="66"></td>
        <td id="LC66" class="blob-code js-file-line"><span class="c">% example.</span></td>
      </tr>
      <tr>
        <td id="L67" class="blob-num js-line-number" data-line-number="67"></td>
        <td id="LC67" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L68" class="blob-num js-line-number" data-line-number="68"></td>
        <td id="LC68" class="blob-code js-file-line"><span class="k">\newenvironment</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span>[1][-1]<span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L69" class="blob-num js-line-number" data-line-number="69"></td>
        <td id="LC69" class="blob-code js-file-line">    <span class="k">\ifnum</span>#1&gt;0</td>
      </tr>
      <tr>
        <td id="L70" class="blob-num js-line-number" data-line-number="70"></td>
        <td id="LC70" class="blob-code js-file-line">        <span class="k">\setcounter</span><span class="nb">{</span>homeworkProblemCounter<span class="nb">}{</span>#1<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L71" class="blob-num js-line-number" data-line-number="71"></td>
        <td id="LC71" class="blob-code js-file-line">    <span class="k">\fi</span></td>
      </tr>
      <tr>
        <td id="L72" class="blob-num js-line-number" data-line-number="72"></td>
        <td id="LC72" class="blob-code js-file-line">    <span class="k">\section</span><span class="nb">{</span>Problem <span class="k">\arabic</span><span class="nb">{</span>homeworkProblemCounter<span class="nb">}}</span></td>
      </tr>
      <tr>
        <td id="L73" class="blob-num js-line-number" data-line-number="73"></td>
        <td id="LC73" class="blob-code js-file-line">    <span class="k">\setcounter</span><span class="nb">{</span>partCounter<span class="nb">}{</span>1<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L74" class="blob-num js-line-number" data-line-number="74"></td>
        <td id="LC74" class="blob-code js-file-line">    <span class="k">\enterProblemHeader</span><span class="nb">{</span>homeworkProblemCounter<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L75" class="blob-num js-line-number" data-line-number="75"></td>
        <td id="LC75" class="blob-code js-file-line"><span class="nb">}{</span></td>
      </tr>
      <tr>
        <td id="L76" class="blob-num js-line-number" data-line-number="76"></td>
        <td id="LC76" class="blob-code js-file-line">    <span class="k">\exitProblemHeader</span><span class="nb">{</span>homeworkProblemCounter<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L77" class="blob-num js-line-number" data-line-number="77"></td>
        <td id="LC77" class="blob-code js-file-line"><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L78" class="blob-num js-line-number" data-line-number="78"></td>
        <td id="LC78" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L79" class="blob-num js-line-number" data-line-number="79"></td>
        <td id="LC79" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L80" class="blob-num js-line-number" data-line-number="80"></td>
        <td id="LC80" class="blob-code js-file-line"><span class="c">% Homework Details</span></td>
      </tr>
      <tr>
        <td id="L81" class="blob-num js-line-number" data-line-number="81"></td>
        <td id="LC81" class="blob-code js-file-line"><span class="c">%   - Title</span></td>
      </tr>
      <tr>
        <td id="L82" class="blob-num js-line-number" data-line-number="82"></td>
        <td id="LC82" class="blob-code js-file-line"><span class="c">%   - Due date</span></td>
      </tr>
      <tr>
        <td id="L83" class="blob-num js-line-number" data-line-number="83"></td>
        <td id="LC83" class="blob-code js-file-line"><span class="c">%   - Class</span></td>
      </tr>
      <tr>
        <td id="L84" class="blob-num js-line-number" data-line-number="84"></td>
        <td id="LC84" class="blob-code js-file-line"><span class="c">%   - Section/Time</span></td>
      </tr>
      <tr>
        <td id="L85" class="blob-num js-line-number" data-line-number="85"></td>
        <td id="LC85" class="blob-code js-file-line"><span class="c">%   - Instructor</span></td>
      </tr>
      <tr>
        <td id="L86" class="blob-num js-line-number" data-line-number="86"></td>
        <td id="LC86" class="blob-code js-file-line"><span class="c">%   - Author</span></td>
      </tr>
      <tr>
        <td id="L87" class="blob-num js-line-number" data-line-number="87"></td>
        <td id="LC87" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L88" class="blob-num js-line-number" data-line-number="88"></td>
        <td id="LC88" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L89" class="blob-num js-line-number" data-line-number="89"></td>
        <td id="LC89" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\hmwkTitle</span><span class="nb">}{</span>Homework<span class="k">\ \#</span>2<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L90" class="blob-num js-line-number" data-line-number="90"></td>
        <td id="LC90" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\hmwkDueDate</span><span class="nb">}{</span>February 12, 2014<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L91" class="blob-num js-line-number" data-line-number="91"></td>
        <td id="LC91" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\hmwkClass</span><span class="nb">}{</span>Calculus<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L92" class="blob-num js-line-number" data-line-number="92"></td>
        <td id="LC92" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\hmwkClassTime</span><span class="nb">}{</span>Section A<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L93" class="blob-num js-line-number" data-line-number="93"></td>
        <td id="LC93" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\hmwkClassInstructor</span><span class="nb">}{</span>Professor Isaac Newton<span class="nb">}</span></td>
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      <tr>
        <td id="L94" class="blob-num js-line-number" data-line-number="94"></td>
        <td id="LC94" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\hmwkAuthorName</span><span class="nb">}{</span>Josh Davis<span class="nb">}</span></td>
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        <td id="L95" class="blob-num js-line-number" data-line-number="95"></td>
        <td id="LC95" class="blob-code js-file-line">
</td>
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        <td id="L96" class="blob-num js-line-number" data-line-number="96"></td>
        <td id="LC96" class="blob-code js-file-line"><span class="c">%</span></td>
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      <tr>
        <td id="L97" class="blob-num js-line-number" data-line-number="97"></td>
        <td id="LC97" class="blob-code js-file-line"><span class="c">% Title Page</span></td>
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      <tr>
        <td id="L98" class="blob-num js-line-number" data-line-number="98"></td>
        <td id="LC98" class="blob-code js-file-line"><span class="c">%</span></td>
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      <tr>
        <td id="L99" class="blob-num js-line-number" data-line-number="99"></td>
        <td id="LC99" class="blob-code js-file-line">
</td>
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        <td id="L100" class="blob-num js-line-number" data-line-number="100"></td>
        <td id="LC100" class="blob-code js-file-line"><span class="k">\title</span><span class="nb">{</span></td>
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      <tr>
        <td id="L101" class="blob-num js-line-number" data-line-number="101"></td>
        <td id="LC101" class="blob-code js-file-line">    <span class="k">\vspace</span><span class="nb">{</span>2in<span class="nb">}</span></td>
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      <tr>
        <td id="L102" class="blob-num js-line-number" data-line-number="102"></td>
        <td id="LC102" class="blob-code js-file-line">    <span class="k">\textmd</span><span class="nb">{</span><span class="k">\textbf</span><span class="nb">{</span><span class="k">\hmwkClass</span>:<span class="k">\ \hmwkTitle</span><span class="nb">}}</span><span class="k">\\</span></td>
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        <td id="L103" class="blob-num js-line-number" data-line-number="103"></td>
        <td id="LC103" class="blob-code js-file-line">    <span class="k">\normalsize\vspace</span><span class="nb">{</span>0.1in<span class="nb">}</span><span class="k">\small</span><span class="nb">{</span>Due<span class="k">\ </span>on<span class="k">\ \hmwkDueDate\ </span>at 3:10pm<span class="nb">}</span><span class="k">\\</span></td>
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      <tr>
        <td id="L104" class="blob-num js-line-number" data-line-number="104"></td>
        <td id="LC104" class="blob-code js-file-line">    <span class="k">\vspace</span><span class="nb">{</span>0.1in<span class="nb">}</span><span class="k">\large</span><span class="nb">{</span><span class="k">\textit</span><span class="nb">{</span><span class="k">\hmwkClassInstructor\ \hmwkClassTime</span><span class="nb">}}</span></td>
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        <td id="L105" class="blob-num js-line-number" data-line-number="105"></td>
        <td id="LC105" class="blob-code js-file-line">    <span class="k">\vspace</span><span class="nb">{</span>3in<span class="nb">}</span></td>
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      <tr>
        <td id="L106" class="blob-num js-line-number" data-line-number="106"></td>
        <td id="LC106" class="blob-code js-file-line"><span class="nb">}</span></td>
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        <td id="L107" class="blob-num js-line-number" data-line-number="107"></td>
        <td id="LC107" class="blob-code js-file-line">
</td>
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        <td id="L108" class="blob-num js-line-number" data-line-number="108"></td>
        <td id="LC108" class="blob-code js-file-line"><span class="k">\author</span><span class="nb">{</span><span class="k">\textbf</span><span class="nb">{</span><span class="k">\hmwkAuthorName</span><span class="nb">}}</span></td>
      </tr>
      <tr>
        <td id="L109" class="blob-num js-line-number" data-line-number="109"></td>
        <td id="LC109" class="blob-code js-file-line"><span class="k">\date</span><span class="nb">{}</span></td>
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      <tr>
        <td id="L110" class="blob-num js-line-number" data-line-number="110"></td>
        <td id="LC110" class="blob-code js-file-line">
</td>
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        <td id="L111" class="blob-num js-line-number" data-line-number="111"></td>
        <td id="LC111" class="blob-code js-file-line"><span class="k">\renewcommand</span><span class="nb">{</span><span class="k">\part</span><span class="nb">}</span>[1]<span class="nb">{</span><span class="k">\textbf</span><span class="nb">{</span><span class="k">\large</span> Part <span class="k">\Alph</span><span class="nb">{</span>partCounter<span class="nb">}}</span><span class="k">\stepcounter</span><span class="nb">{</span>partCounter<span class="nb">}</span><span class="k">\\</span><span class="nb">}</span></td>
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      <tr>
        <td id="L112" class="blob-num js-line-number" data-line-number="112"></td>
        <td id="LC112" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L113" class="blob-num js-line-number" data-line-number="113"></td>
        <td id="LC113" class="blob-code js-file-line"><span class="c">%</span></td>
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      <tr>
        <td id="L114" class="blob-num js-line-number" data-line-number="114"></td>
        <td id="LC114" class="blob-code js-file-line"><span class="c">% Various Helper Commands</span></td>
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      <tr>
        <td id="L115" class="blob-num js-line-number" data-line-number="115"></td>
        <td id="LC115" class="blob-code js-file-line"><span class="c">%</span></td>
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      <tr>
        <td id="L116" class="blob-num js-line-number" data-line-number="116"></td>
        <td id="LC116" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L117" class="blob-num js-line-number" data-line-number="117"></td>
        <td id="LC117" class="blob-code js-file-line"><span class="c">% Useful for algorithms</span></td>
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      <tr>
        <td id="L118" class="blob-num js-line-number" data-line-number="118"></td>
        <td id="LC118" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\alg</span><span class="nb">}</span>[1]<span class="nb">{</span><span class="k">\textsc</span><span class="nb">{</span><span class="k">\bfseries</span> <span class="k">\footnotesize</span> #1<span class="nb">}}</span></td>
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      <tr>
        <td id="L119" class="blob-num js-line-number" data-line-number="119"></td>
        <td id="LC119" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L120" class="blob-num js-line-number" data-line-number="120"></td>
        <td id="LC120" class="blob-code js-file-line"><span class="c">% For derivatives</span></td>
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      <tr>
        <td id="L121" class="blob-num js-line-number" data-line-number="121"></td>
        <td id="LC121" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\deriv</span><span class="nb">}</span>[1]<span class="nb">{</span><span class="k">\frac</span><span class="nb">{</span><span class="k">\mathrm</span><span class="nb">{</span>d<span class="nb">}}{</span><span class="k">\mathrm</span><span class="nb">{</span>d<span class="nb">}</span>x<span class="nb">}</span> (#1)<span class="nb">}</span></td>
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        <td id="L122" class="blob-num js-line-number" data-line-number="122"></td>
        <td id="LC122" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L123" class="blob-num js-line-number" data-line-number="123"></td>
        <td id="LC123" class="blob-code js-file-line"><span class="c">% For partial derivatives</span></td>
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      <tr>
        <td id="L124" class="blob-num js-line-number" data-line-number="124"></td>
        <td id="LC124" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\pderiv</span><span class="nb">}</span>[2]<span class="nb">{</span><span class="k">\frac</span><span class="nb">{</span><span class="k">\partial</span><span class="nb">}{</span><span class="k">\partial</span> #1<span class="nb">}</span> (#2)<span class="nb">}</span></td>
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        <td id="L125" class="blob-num js-line-number" data-line-number="125"></td>
        <td id="LC125" class="blob-code js-file-line">
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      <tr>
        <td id="L126" class="blob-num js-line-number" data-line-number="126"></td>
        <td id="LC126" class="blob-code js-file-line"><span class="c">% Integral dx</span></td>
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      <tr>
        <td id="L127" class="blob-num js-line-number" data-line-number="127"></td>
        <td id="LC127" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\dx</span><span class="nb">}{</span><span class="k">\mathrm</span><span class="nb">{</span>d<span class="nb">}</span>x<span class="nb">}</span></td>
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      <tr>
        <td id="L128" class="blob-num js-line-number" data-line-number="128"></td>
        <td id="LC128" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L129" class="blob-num js-line-number" data-line-number="129"></td>
        <td id="LC129" class="blob-code js-file-line"><span class="c">% Alias for the Solution section header</span></td>
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      <tr>
        <td id="L130" class="blob-num js-line-number" data-line-number="130"></td>
        <td id="LC130" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\solution</span><span class="nb">}{</span><span class="k">\textbf</span><span class="nb">{</span><span class="k">\large</span> Solution<span class="nb">}}</span></td>
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      <tr>
        <td id="L131" class="blob-num js-line-number" data-line-number="131"></td>
        <td id="LC131" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L132" class="blob-num js-line-number" data-line-number="132"></td>
        <td id="LC132" class="blob-code js-file-line"><span class="c">% Probability commands: Expectation, Variance, Covariance, Bias</span></td>
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      <tr>
        <td id="L133" class="blob-num js-line-number" data-line-number="133"></td>
        <td id="LC133" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\E</span><span class="nb">}{</span><span class="k">\mathrm</span><span class="nb">{</span>E<span class="nb">}}</span></td>
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      <tr>
        <td id="L134" class="blob-num js-line-number" data-line-number="134"></td>
        <td id="LC134" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\Var</span><span class="nb">}{</span><span class="k">\mathrm</span><span class="nb">{</span>Var<span class="nb">}}</span></td>
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      <tr>
        <td id="L135" class="blob-num js-line-number" data-line-number="135"></td>
        <td id="LC135" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\Cov</span><span class="nb">}{</span><span class="k">\mathrm</span><span class="nb">{</span>Cov<span class="nb">}}</span></td>
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      <tr>
        <td id="L136" class="blob-num js-line-number" data-line-number="136"></td>
        <td id="LC136" class="blob-code js-file-line"><span class="k">\newcommand</span><span class="nb">{</span><span class="k">\Bias</span><span class="nb">}{</span><span class="k">\mathrm</span><span class="nb">{</span>Bias<span class="nb">}}</span></td>
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      <tr>
        <td id="L137" class="blob-num js-line-number" data-line-number="137"></td>
        <td id="LC137" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L138" class="blob-num js-line-number" data-line-number="138"></td>
        <td id="LC138" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>document<span class="nb">}</span></td>
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      <tr>
        <td id="L139" class="blob-num js-line-number" data-line-number="139"></td>
        <td id="LC139" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L140" class="blob-num js-line-number" data-line-number="140"></td>
        <td id="LC140" class="blob-code js-file-line"><span class="k">\maketitle</span></td>
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      <tr>
        <td id="L141" class="blob-num js-line-number" data-line-number="141"></td>
        <td id="LC141" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L142" class="blob-num js-line-number" data-line-number="142"></td>
        <td id="LC142" class="blob-code js-file-line"><span class="k">\pagebreak</span></td>
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      <tr>
        <td id="L143" class="blob-num js-line-number" data-line-number="143"></td>
        <td id="LC143" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L144" class="blob-num js-line-number" data-line-number="144"></td>
        <td id="LC144" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
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      <tr>
        <td id="L145" class="blob-num js-line-number" data-line-number="145"></td>
        <td id="LC145" class="blob-code js-file-line">    Give an appropriate positive constant <span class="s">\(</span><span class="nb">c</span><span class="s">\)</span> such that <span class="s">\(</span><span class="nb">f</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="nv">\leq</span><span class="nb"> c </span><span class="nv">\cdot</span><span class="nb"></span></td>
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      <tr>
        <td id="L146" class="blob-num js-line-number" data-line-number="146"></td>
        <td id="LC146" class="blob-code js-file-line"><span class="nb">    g</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="s">\)</span> for all <span class="s">\(</span><span class="nb">n &gt; </span><span class="m">1</span><span class="s">\)</span>.</td>
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      <tr>
        <td id="L147" class="blob-num js-line-number" data-line-number="147"></td>
        <td id="LC147" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L148" class="blob-num js-line-number" data-line-number="148"></td>
        <td id="LC148" class="blob-code js-file-line">    <span class="k">\begin</span><span class="nb">{</span>enumerate<span class="nb">}</span></td>
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      <tr>
        <td id="L149" class="blob-num js-line-number" data-line-number="149"></td>
        <td id="LC149" class="blob-code js-file-line">        <span class="k">\item</span> <span class="s">\(</span><span class="nb">f</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"> </span><span class="o">+</span><span class="nb"> n </span><span class="o">+</span><span class="nb"> </span><span class="m">1</span><span class="s">\)</span>, <span class="s">\(</span><span class="nb">g</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">2</span><span class="nb">n^</span><span class="m">3</span><span class="s">\)</span></td>
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      <tr>
        <td id="L150" class="blob-num js-line-number" data-line-number="150"></td>
        <td id="LC150" class="blob-code js-file-line">        <span class="k">\item</span> <span class="s">\(</span><span class="nb">f</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb"> n</span><span class="nv">\sqrt</span><span class="nb">{n} </span><span class="o">+</span><span class="nb"> n^</span><span class="m">2</span><span class="s">\)</span>, <span class="s">\(</span><span class="nb">g</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb"> n^</span><span class="m">2</span><span class="s">\)</span></td>
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      <tr>
        <td id="L151" class="blob-num js-line-number" data-line-number="151"></td>
        <td id="LC151" class="blob-code js-file-line">        <span class="k">\item</span> <span class="s">\(</span><span class="nb">f</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"> </span><span class="o">-</span><span class="nb"> n </span><span class="o">+</span><span class="nb"> </span><span class="m">1</span><span class="s">\)</span>, <span class="s">\(</span><span class="nb">g</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"> </span><span class="o">/</span><span class="nb"> </span><span class="m">2</span><span class="s">\)</span></td>
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      <tr>
        <td id="L152" class="blob-num js-line-number" data-line-number="152"></td>
        <td id="LC152" class="blob-code js-file-line">    <span class="k">\end</span><span class="nb">{</span>enumerate<span class="nb">}</span></td>
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      <tr>
        <td id="L153" class="blob-num js-line-number" data-line-number="153"></td>
        <td id="LC153" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L154" class="blob-num js-line-number" data-line-number="154"></td>
        <td id="LC154" class="blob-code js-file-line">    <span class="k">\textbf</span><span class="nb">{</span>Solution<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L155" class="blob-num js-line-number" data-line-number="155"></td>
        <td id="LC155" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L156" class="blob-num js-line-number" data-line-number="156"></td>
        <td id="LC156" class="blob-code js-file-line">    We solve each solution algebraically to determine a possible constant</td>
      </tr>
      <tr>
        <td id="L157" class="blob-num js-line-number" data-line-number="157"></td>
        <td id="LC157" class="blob-code js-file-line">    <span class="s">\(</span><span class="nb">c</span><span class="s">\)</span>.</td>
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      <tr>
        <td id="L158" class="blob-num js-line-number" data-line-number="158"></td>
        <td id="LC158" class="blob-code js-file-line">    <span class="k">\\</span></td>
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      <tr>
        <td id="L159" class="blob-num js-line-number" data-line-number="159"></td>
        <td id="LC159" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L160" class="blob-num js-line-number" data-line-number="160"></td>
        <td id="LC160" class="blob-code js-file-line">    <span class="k">\textbf</span><span class="nb">{</span>Part One<span class="nb">}</span></td>
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      <tr>
        <td id="L161" class="blob-num js-line-number" data-line-number="161"></td>
        <td id="LC161" class="blob-code js-file-line">
</td>
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      <tr>
        <td id="L162" class="blob-num js-line-number" data-line-number="162"></td>
        <td id="LC162" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
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      <tr>
        <td id="L163" class="blob-num js-line-number" data-line-number="163"></td>
        <td id="LC163" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L164" class="blob-num js-line-number" data-line-number="164"></td>
        <td id="LC164" class="blob-code js-file-line"><span class="nb">            n^</span><span class="m">2</span><span class="nb"> </span><span class="o">+</span><span class="nb"> n </span><span class="o">+</span><span class="nb"> </span><span class="m">1</span><span class="nb"> &amp;</span><span class="o">=</span><span class="nb"></span></td>
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      <tr>
        <td id="L165" class="blob-num js-line-number" data-line-number="165"></td>
        <td id="LC165" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
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      <tr>
        <td id="L166" class="blob-num js-line-number" data-line-number="166"></td>
        <td id="LC166" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="nv">\leq</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"> </span><span class="o">+</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"> </span><span class="o">+</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L167" class="blob-num js-line-number" data-line-number="167"></td>
        <td id="LC167" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L168" class="blob-num js-line-number" data-line-number="168"></td>
        <td id="LC168" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="m">3</span><span class="nb">n^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L169" class="blob-num js-line-number" data-line-number="169"></td>
        <td id="LC169" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L170" class="blob-num js-line-number" data-line-number="170"></td>
        <td id="LC170" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="nv">\leq</span><span class="nb"> c </span><span class="nv">\cdot</span><span class="nb"> </span><span class="m">2</span><span class="nb">n^</span><span class="m">3</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L171" class="blob-num js-line-number" data-line-number="171"></td>
        <td id="LC171" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L172" class="blob-num js-line-number" data-line-number="172"></td>
        <td id="LC172" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L173" class="blob-num js-line-number" data-line-number="173"></td>
        <td id="LC173" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L174" class="blob-num js-line-number" data-line-number="174"></td>
        <td id="LC174" class="blob-code js-file-line">    Thus a valid <span class="s">\(</span><span class="nb">c</span><span class="s">\)</span> could be when <span class="s">\(</span><span class="nb">c </span><span class="o">=</span><span class="nb"> </span><span class="m">2</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L175" class="blob-num js-line-number" data-line-number="175"></td>
        <td id="LC175" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L176" class="blob-num js-line-number" data-line-number="176"></td>
        <td id="LC176" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L177" class="blob-num js-line-number" data-line-number="177"></td>
        <td id="LC177" class="blob-code js-file-line">    <span class="k">\textbf</span><span class="nb">{</span>Part Two<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L178" class="blob-num js-line-number" data-line-number="178"></td>
        <td id="LC178" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L179" class="blob-num js-line-number" data-line-number="179"></td>
        <td id="LC179" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L180" class="blob-num js-line-number" data-line-number="180"></td>
        <td id="LC180" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L181" class="blob-num js-line-number" data-line-number="181"></td>
        <td id="LC181" class="blob-code js-file-line"><span class="nb">            n^</span><span class="m">2</span><span class="nb"> </span><span class="o">+</span><span class="nb"> n</span><span class="nv">\sqrt</span><span class="nb">{n} &amp;</span><span class="o">=</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L182" class="blob-num js-line-number" data-line-number="182"></td>
        <td id="LC182" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L183" class="blob-num js-line-number" data-line-number="183"></td>
        <td id="LC183" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"> </span><span class="o">+</span><span class="nb"> n^{</span><span class="m">3</span><span class="o">/</span><span class="m">2</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L184" class="blob-num js-line-number" data-line-number="184"></td>
        <td id="LC184" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L185" class="blob-num js-line-number" data-line-number="185"></td>
        <td id="LC185" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="nv">\leq</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"> </span><span class="o">+</span><span class="nb"> n^{</span><span class="m">4</span><span class="o">/</span><span class="m">2</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L186" class="blob-num js-line-number" data-line-number="186"></td>
        <td id="LC186" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L187" class="blob-num js-line-number" data-line-number="187"></td>
        <td id="LC187" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"> </span><span class="o">+</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L188" class="blob-num js-line-number" data-line-number="188"></td>
        <td id="LC188" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L189" class="blob-num js-line-number" data-line-number="189"></td>
        <td id="LC189" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="m">2</span><span class="nb">n^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L190" class="blob-num js-line-number" data-line-number="190"></td>
        <td id="LC190" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L191" class="blob-num js-line-number" data-line-number="191"></td>
        <td id="LC191" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="nv">\leq</span><span class="nb"> c </span><span class="nv">\cdot</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L192" class="blob-num js-line-number" data-line-number="192"></td>
        <td id="LC192" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L193" class="blob-num js-line-number" data-line-number="193"></td>
        <td id="LC193" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L194" class="blob-num js-line-number" data-line-number="194"></td>
        <td id="LC194" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L195" class="blob-num js-line-number" data-line-number="195"></td>
        <td id="LC195" class="blob-code js-file-line">    Thus a valid <span class="s">\(</span><span class="nb">c</span><span class="s">\)</span> is <span class="s">\(</span><span class="nb">c </span><span class="o">=</span><span class="nb"> </span><span class="m">2</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L196" class="blob-num js-line-number" data-line-number="196"></td>
        <td id="LC196" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L197" class="blob-num js-line-number" data-line-number="197"></td>
        <td id="LC197" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L198" class="blob-num js-line-number" data-line-number="198"></td>
        <td id="LC198" class="blob-code js-file-line">    <span class="k">\textbf</span><span class="nb">{</span>Part Three<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L199" class="blob-num js-line-number" data-line-number="199"></td>
        <td id="LC199" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L200" class="blob-num js-line-number" data-line-number="200"></td>
        <td id="LC200" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L201" class="blob-num js-line-number" data-line-number="201"></td>
        <td id="LC201" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L202" class="blob-num js-line-number" data-line-number="202"></td>
        <td id="LC202" class="blob-code js-file-line"><span class="nb">            n^</span><span class="m">2</span><span class="nb"> </span><span class="o">-</span><span class="nb"> n </span><span class="o">+</span><span class="nb"> </span><span class="m">1</span><span class="nb"> &amp;</span><span class="o">=</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L203" class="blob-num js-line-number" data-line-number="203"></td>
        <td id="LC203" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L204" class="blob-num js-line-number" data-line-number="204"></td>
        <td id="LC204" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="nv">\leq</span><span class="nb"> n^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L205" class="blob-num js-line-number" data-line-number="205"></td>
        <td id="LC205" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L206" class="blob-num js-line-number" data-line-number="206"></td>
        <td id="LC206" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="nv">\leq</span><span class="nb"> c </span><span class="nv">\cdot</span><span class="nb"> n^</span><span class="m">2</span><span class="o">/</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L207" class="blob-num js-line-number" data-line-number="207"></td>
        <td id="LC207" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L208" class="blob-num js-line-number" data-line-number="208"></td>
        <td id="LC208" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L209" class="blob-num js-line-number" data-line-number="209"></td>
        <td id="LC209" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L210" class="blob-num js-line-number" data-line-number="210"></td>
        <td id="LC210" class="blob-code js-file-line">    Thus a valid <span class="s">\(</span><span class="nb">c</span><span class="s">\)</span> is <span class="s">\(</span><span class="nb">c </span><span class="o">=</span><span class="nb"> </span><span class="m">2</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L211" class="blob-num js-line-number" data-line-number="211"></td>
        <td id="LC211" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L212" class="blob-num js-line-number" data-line-number="212"></td>
        <td id="LC212" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L213" class="blob-num js-line-number" data-line-number="213"></td>
        <td id="LC213" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L214" class="blob-num js-line-number" data-line-number="214"></td>
        <td id="LC214" class="blob-code js-file-line"><span class="k">\pagebreak</span></td>
      </tr>
      <tr>
        <td id="L215" class="blob-num js-line-number" data-line-number="215"></td>
        <td id="LC215" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L216" class="blob-num js-line-number" data-line-number="216"></td>
        <td id="LC216" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L217" class="blob-num js-line-number" data-line-number="217"></td>
        <td id="LC217" class="blob-code js-file-line">    Let <span class="s">\(</span><span class="nv">\Sigma</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="nv">\{</span><span class="m">0</span><span class="nb">, </span><span class="m">1</span><span class="nv">\}</span><span class="s">\)</span>. Construct a DFA <span class="s">\(</span><span class="nb">A</span><span class="s">\)</span> that recognizes the</td>
      </tr>
      <tr>
        <td id="L218" class="blob-num js-line-number" data-line-number="218"></td>
        <td id="LC218" class="blob-code js-file-line">    language that consists of all binary numbers that can be divided by 5.</td>
      </tr>
      <tr>
        <td id="L219" class="blob-num js-line-number" data-line-number="219"></td>
        <td id="LC219" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L220" class="blob-num js-line-number" data-line-number="220"></td>
        <td id="LC220" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L221" class="blob-num js-line-number" data-line-number="221"></td>
        <td id="LC221" class="blob-code js-file-line">    Let the state <span class="s">\(</span><span class="nb">q_k</span><span class="s">\)</span> indicate the remainder of <span class="s">\(</span><span class="nb">k</span><span class="s">\)</span> divided by 5. For</td>
      </tr>
      <tr>
        <td id="L222" class="blob-num js-line-number" data-line-number="222"></td>
        <td id="LC222" class="blob-code js-file-line">    example, the remainder of 2 would correlate to state <span class="s">\(</span><span class="nb">q_</span><span class="m">2</span><span class="s">\)</span> because <span class="s">\(</span><span class="m">7</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L223" class="blob-num js-line-number" data-line-number="223"></td>
        <td id="LC223" class="blob-code js-file-line"><span class="nb">    </span><span class="nv">\mod</span><span class="nb"> </span><span class="m">5</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">2</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L224" class="blob-num js-line-number" data-line-number="224"></td>
        <td id="LC224" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L225" class="blob-num js-line-number" data-line-number="225"></td>
        <td id="LC225" class="blob-code js-file-line">    <span class="k">\begin</span><span class="nb">{</span>figure<span class="nb">}</span>[here]</td>
      </tr>
      <tr>
        <td id="L226" class="blob-num js-line-number" data-line-number="226"></td>
        <td id="LC226" class="blob-code js-file-line">        <span class="k">\centering</span></td>
      </tr>
      <tr>
        <td id="L227" class="blob-num js-line-number" data-line-number="227"></td>
        <td id="LC227" class="blob-code js-file-line">        <span class="k">\begin</span><span class="nb">{</span>tikzpicture<span class="nb">}</span>[shorten &gt;=1pt,node distance=2cm,on grid,auto]</td>
      </tr>
      <tr>
        <td id="L228" class="blob-num js-line-number" data-line-number="228"></td>
        <td id="LC228" class="blob-code js-file-line">            <span class="k">\node</span><span class="na">[state, accepting, initial]</span> (q<span class="nb">_</span>0)   <span class="nb">{</span><span class="s">$</span><span class="nb">q_</span><span class="m">0</span><span class="s">$</span><span class="nb">}</span>;</td>
      </tr>
      <tr>
        <td id="L229" class="blob-num js-line-number" data-line-number="229"></td>
        <td id="LC229" class="blob-code js-file-line">            <span class="k">\node</span><span class="na">[state]</span> (q<span class="nb">_</span>1) [right=of q<span class="nb">_</span>0] <span class="nb">{</span><span class="s">$</span><span class="nb">q_</span><span class="m">1</span><span class="s">$</span><span class="nb">}</span>;</td>
      </tr>
      <tr>
        <td id="L230" class="blob-num js-line-number" data-line-number="230"></td>
        <td id="LC230" class="blob-code js-file-line">            <span class="k">\node</span><span class="na">[state]</span> (q<span class="nb">_</span>2) [right=of q<span class="nb">_</span>1] <span class="nb">{</span><span class="s">$</span><span class="nb">q_</span><span class="m">2</span><span class="s">$</span><span class="nb">}</span>;</td>
      </tr>
      <tr>
        <td id="L231" class="blob-num js-line-number" data-line-number="231"></td>
        <td id="LC231" class="blob-code js-file-line">            <span class="k">\node</span><span class="na">[state]</span> (q<span class="nb">_</span>3) [right=of q<span class="nb">_</span>2] <span class="nb">{</span><span class="s">$</span><span class="nb">q_</span><span class="m">3</span><span class="s">$</span><span class="nb">}</span>;</td>
      </tr>
      <tr>
        <td id="L232" class="blob-num js-line-number" data-line-number="232"></td>
        <td id="LC232" class="blob-code js-file-line">            <span class="k">\node</span><span class="na">[state]</span> (q<span class="nb">_</span>4) [right=of q<span class="nb">_</span>3] <span class="nb">{</span><span class="s">$</span><span class="nb">q_</span><span class="m">4</span><span class="s">$</span><span class="nb">}</span>;</td>
      </tr>
      <tr>
        <td id="L233" class="blob-num js-line-number" data-line-number="233"></td>
        <td id="LC233" class="blob-code js-file-line">            <span class="k">\path</span><span class="na">[-&gt;]</span></td>
      </tr>
      <tr>
        <td id="L234" class="blob-num js-line-number" data-line-number="234"></td>
        <td id="LC234" class="blob-code js-file-line">                (q<span class="nb">_</span>0)</td>
      </tr>
      <tr>
        <td id="L235" class="blob-num js-line-number" data-line-number="235"></td>
        <td id="LC235" class="blob-code js-file-line">                    edge [loop above] node <span class="nb">{</span>0<span class="nb">}</span> (q<span class="nb">_</span>0)</td>
      </tr>
      <tr>
        <td id="L236" class="blob-num js-line-number" data-line-number="236"></td>
        <td id="LC236" class="blob-code js-file-line">                    edge node <span class="nb">{</span>1<span class="nb">}</span> (q<span class="nb">_</span>1)</td>
      </tr>
      <tr>
        <td id="L237" class="blob-num js-line-number" data-line-number="237"></td>
        <td id="LC237" class="blob-code js-file-line">                (q<span class="nb">_</span>1)</td>
      </tr>
      <tr>
        <td id="L238" class="blob-num js-line-number" data-line-number="238"></td>
        <td id="LC238" class="blob-code js-file-line">                    edge node <span class="nb">{</span>0<span class="nb">}</span> (q<span class="nb">_</span>2)</td>
      </tr>
      <tr>
        <td id="L239" class="blob-num js-line-number" data-line-number="239"></td>
        <td id="LC239" class="blob-code js-file-line">                    edge [bend right=-30] node <span class="nb">{</span>1<span class="nb">}</span> (q<span class="nb">_</span>3)</td>
      </tr>
      <tr>
        <td id="L240" class="blob-num js-line-number" data-line-number="240"></td>
        <td id="LC240" class="blob-code js-file-line">                (q<span class="nb">_</span>2)</td>
      </tr>
      <tr>
        <td id="L241" class="blob-num js-line-number" data-line-number="241"></td>
        <td id="LC241" class="blob-code js-file-line">                    edge [bend left] node <span class="nb">{</span>1<span class="nb">}</span> (q<span class="nb">_</span>0)</td>
      </tr>
      <tr>
        <td id="L242" class="blob-num js-line-number" data-line-number="242"></td>
        <td id="LC242" class="blob-code js-file-line">                    edge [bend right=-30] node <span class="nb">{</span>0<span class="nb">}</span> (q<span class="nb">_</span>4)</td>
      </tr>
      <tr>
        <td id="L243" class="blob-num js-line-number" data-line-number="243"></td>
        <td id="LC243" class="blob-code js-file-line">                (q<span class="nb">_</span>3)</td>
      </tr>
      <tr>
        <td id="L244" class="blob-num js-line-number" data-line-number="244"></td>
        <td id="LC244" class="blob-code js-file-line">                    edge node <span class="nb">{</span>1<span class="nb">}</span> (q<span class="nb">_</span>2)</td>
      </tr>
      <tr>
        <td id="L245" class="blob-num js-line-number" data-line-number="245"></td>
        <td id="LC245" class="blob-code js-file-line">                    edge [bend left] node <span class="nb">{</span>0<span class="nb">}</span> (q<span class="nb">_</span>1)</td>
      </tr>
      <tr>
        <td id="L246" class="blob-num js-line-number" data-line-number="246"></td>
        <td id="LC246" class="blob-code js-file-line">                (q<span class="nb">_</span>4)</td>
      </tr>
      <tr>
        <td id="L247" class="blob-num js-line-number" data-line-number="247"></td>
        <td id="LC247" class="blob-code js-file-line">                    edge node <span class="nb">{</span>0<span class="nb">}</span> (q<span class="nb">_</span>3)</td>
      </tr>
      <tr>
        <td id="L248" class="blob-num js-line-number" data-line-number="248"></td>
        <td id="LC248" class="blob-code js-file-line">                    edge [loop below] node <span class="nb">{</span>1<span class="nb">}</span> (q<span class="nb">_</span>4);</td>
      </tr>
      <tr>
        <td id="L249" class="blob-num js-line-number" data-line-number="249"></td>
        <td id="LC249" class="blob-code js-file-line">        <span class="k">\end</span><span class="nb">{</span>tikzpicture<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L250" class="blob-num js-line-number" data-line-number="250"></td>
        <td id="LC250" class="blob-code js-file-line">        <span class="k">\caption</span><span class="nb">{</span>DFA, <span class="s">\(</span><span class="nb">A</span><span class="s">\)</span>, this is really beautiful, ya know?<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L251" class="blob-num js-line-number" data-line-number="251"></td>
        <td id="LC251" class="blob-code js-file-line">        <span class="k">\label</span><span class="nb">{</span>fig:multiple5<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L252" class="blob-num js-line-number" data-line-number="252"></td>
        <td id="LC252" class="blob-code js-file-line">    <span class="k">\end</span><span class="nb">{</span>figure<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L253" class="blob-num js-line-number" data-line-number="253"></td>
        <td id="LC253" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L254" class="blob-num js-line-number" data-line-number="254"></td>
        <td id="LC254" class="blob-code js-file-line">    <span class="k">\textbf</span><span class="nb">{</span>Justification<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L255" class="blob-num js-line-number" data-line-number="255"></td>
        <td id="LC255" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L256" class="blob-num js-line-number" data-line-number="256"></td>
        <td id="LC256" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L257" class="blob-num js-line-number" data-line-number="257"></td>
        <td id="LC257" class="blob-code js-file-line">    Take a given binary number, <span class="s">\(</span><span class="nb">x</span><span class="s">\)</span>. Since there are only two inputs to our</td>
      </tr>
      <tr>
        <td id="L258" class="blob-num js-line-number" data-line-number="258"></td>
        <td id="LC258" class="blob-code js-file-line">    state machine, <span class="s">\(</span><span class="nb">x</span><span class="s">\)</span> can either become <span class="s">\(</span><span class="nb">x</span><span class="m">0</span><span class="s">\)</span> or <span class="s">\(</span><span class="nb">x</span><span class="m">1</span><span class="s">\)</span>. When a 0 comes</td>
      </tr>
      <tr>
        <td id="L259" class="blob-num js-line-number" data-line-number="259"></td>
        <td id="LC259" class="blob-code js-file-line">    into the state machine, it is the same as taking the binary number and</td>
      </tr>
      <tr>
        <td id="L260" class="blob-num js-line-number" data-line-number="260"></td>
        <td id="LC260" class="blob-code js-file-line">    multiplying it by two. When a 1 comes into the machine, it is the same as</td>
      </tr>
      <tr>
        <td id="L261" class="blob-num js-line-number" data-line-number="261"></td>
        <td id="LC261" class="blob-code js-file-line">    multipying by two and adding one.</td>
      </tr>
      <tr>
        <td id="L262" class="blob-num js-line-number" data-line-number="262"></td>
        <td id="LC262" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L263" class="blob-num js-line-number" data-line-number="263"></td>
        <td id="LC263" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L264" class="blob-num js-line-number" data-line-number="264"></td>
        <td id="LC264" class="blob-code js-file-line">    Using this knowledge, we can construct a transition table that tell us</td>
      </tr>
      <tr>
        <td id="L265" class="blob-num js-line-number" data-line-number="265"></td>
        <td id="LC265" class="blob-code js-file-line">    where to go:</td>
      </tr>
      <tr>
        <td id="L266" class="blob-num js-line-number" data-line-number="266"></td>
        <td id="LC266" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L267" class="blob-num js-line-number" data-line-number="267"></td>
        <td id="LC267" class="blob-code js-file-line">    <span class="k">\begin</span><span class="nb">{</span>table<span class="nb">}</span>[ht]</td>
      </tr>
      <tr>
        <td id="L268" class="blob-num js-line-number" data-line-number="268"></td>
        <td id="LC268" class="blob-code js-file-line">        <span class="k">\centering</span></td>
      </tr>
      <tr>
        <td id="L269" class="blob-num js-line-number" data-line-number="269"></td>
        <td id="LC269" class="blob-code js-file-line">        <span class="k">\begin</span><span class="nb">{</span>tabular<span class="nb">}{</span>c || c | c | c | c | c<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L270" class="blob-num js-line-number" data-line-number="270"></td>
        <td id="LC270" class="blob-code js-file-line">            <span class="nb">&amp;</span> <span class="s">\(</span><span class="nb">x </span><span class="nv">\mod</span><span class="nb"> </span><span class="m">5</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">0</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L271" class="blob-num js-line-number" data-line-number="271"></td>
        <td id="LC271" class="blob-code js-file-line">            <span class="nb">&amp;</span> <span class="s">\(</span><span class="nb">x </span><span class="nv">\mod</span><span class="nb"> </span><span class="m">5</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L272" class="blob-num js-line-number" data-line-number="272"></td>
        <td id="LC272" class="blob-code js-file-line">            <span class="nb">&amp;</span> <span class="s">\(</span><span class="nb">x </span><span class="nv">\mod</span><span class="nb"> </span><span class="m">5</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">2</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L273" class="blob-num js-line-number" data-line-number="273"></td>
        <td id="LC273" class="blob-code js-file-line">            <span class="nb">&amp;</span> <span class="s">\(</span><span class="nb">x </span><span class="nv">\mod</span><span class="nb"> </span><span class="m">5</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">3</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L274" class="blob-num js-line-number" data-line-number="274"></td>
        <td id="LC274" class="blob-code js-file-line">            <span class="nb">&amp;</span> <span class="s">\(</span><span class="nb">x </span><span class="nv">\mod</span><span class="nb"> </span><span class="m">5</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">4</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L275" class="blob-num js-line-number" data-line-number="275"></td>
        <td id="LC275" class="blob-code js-file-line">            <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L276" class="blob-num js-line-number" data-line-number="276"></td>
        <td id="LC276" class="blob-code js-file-line">            <span class="k">\hline</span></td>
      </tr>
      <tr>
        <td id="L277" class="blob-num js-line-number" data-line-number="277"></td>
        <td id="LC277" class="blob-code js-file-line">            <span class="s">\(</span><span class="nb">x</span><span class="m">0</span><span class="s">\)</span> <span class="nb">&amp;</span> 0 <span class="nb">&amp;</span> 2 <span class="nb">&amp;</span> 4 <span class="nb">&amp;</span> 1 <span class="nb">&amp;</span> 3 <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L278" class="blob-num js-line-number" data-line-number="278"></td>
        <td id="LC278" class="blob-code js-file-line">            <span class="s">\(</span><span class="nb">x</span><span class="m">1</span><span class="s">\)</span> <span class="nb">&amp;</span> 1 <span class="nb">&amp;</span> 3 <span class="nb">&amp;</span> 0 <span class="nb">&amp;</span> 2 <span class="nb">&amp;</span> 4 <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L279" class="blob-num js-line-number" data-line-number="279"></td>
        <td id="LC279" class="blob-code js-file-line">        <span class="k">\end</span><span class="nb">{</span>tabular<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L280" class="blob-num js-line-number" data-line-number="280"></td>
        <td id="LC280" class="blob-code js-file-line">    <span class="k">\end</span><span class="nb">{</span>table<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L281" class="blob-num js-line-number" data-line-number="281"></td>
        <td id="LC281" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L282" class="blob-num js-line-number" data-line-number="282"></td>
        <td id="LC282" class="blob-code js-file-line">    Therefore on state <span class="s">\(</span><span class="nb">q_</span><span class="m">0</span><span class="s">\)</span> or (<span class="s">\(</span><span class="nb">x </span><span class="nv">\mod</span><span class="nb"> </span><span class="m">5</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">0</span><span class="s">\)</span>), a transition line should</td>
      </tr>
      <tr>
        <td id="L283" class="blob-num js-line-number" data-line-number="283"></td>
        <td id="LC283" class="blob-code js-file-line">    go to state <span class="s">\(</span><span class="nb">q_</span><span class="m">0</span><span class="s">\)</span> for the input 0 and a line should go to state <span class="s">\(</span><span class="nb">q_</span><span class="m">1</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L284" class="blob-num js-line-number" data-line-number="284"></td>
        <td id="LC284" class="blob-code js-file-line">    for input 1. Continuing this gives us the Figure~<span class="k">\ref</span><span class="nb">{</span>fig:multiple5<span class="nb">}</span>.</td>
      </tr>
      <tr>
        <td id="L285" class="blob-num js-line-number" data-line-number="285"></td>
        <td id="LC285" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L286" class="blob-num js-line-number" data-line-number="286"></td>
        <td id="LC286" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L287" class="blob-num js-line-number" data-line-number="287"></td>
        <td id="LC287" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L288" class="blob-num js-line-number" data-line-number="288"></td>
        <td id="LC288" class="blob-code js-file-line">    Write part of <span class="k">\alg</span><span class="nb">{</span>Quick-Sort(<span class="s">$</span><span class="nb">list, start, end</span><span class="s">$</span>)<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L289" class="blob-num js-line-number" data-line-number="289"></td>
        <td id="LC289" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L290" class="blob-num js-line-number" data-line-number="290"></td>
        <td id="LC290" class="blob-code js-file-line">    <span class="k">\begin</span><span class="nb">{</span>algorithm<span class="nb">}</span>[]</td>
      </tr>
      <tr>
        <td id="L291" class="blob-num js-line-number" data-line-number="291"></td>
        <td id="LC291" class="blob-code js-file-line">        <span class="k">\begin</span><span class="nb">{</span>algorithmic<span class="nb">}</span>[1]</td>
      </tr>
      <tr>
        <td id="L292" class="blob-num js-line-number" data-line-number="292"></td>
        <td id="LC292" class="blob-code js-file-line">            <span class="k">\Function</span><span class="nb">{</span>Quick-Sort<span class="nb">}{</span><span class="s">$</span><span class="nb">list, start, end</span><span class="s">$</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L293" class="blob-num js-line-number" data-line-number="293"></td>
        <td id="LC293" class="blob-code js-file-line">                <span class="k">\If</span><span class="nb">{</span><span class="s">$</span><span class="nb">start </span><span class="nv">\geq</span><span class="nb"> end</span><span class="s">$</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L294" class="blob-num js-line-number" data-line-number="294"></td>
        <td id="LC294" class="blob-code js-file-line">                    <span class="k">\State</span><span class="nb">{}</span> <span class="k">\Return</span><span class="nb">{}</span></td>
      </tr>
      <tr>
        <td id="L295" class="blob-num js-line-number" data-line-number="295"></td>
        <td id="LC295" class="blob-code js-file-line">                <span class="k">\EndIf</span><span class="nb">{}</span></td>
      </tr>
      <tr>
        <td id="L296" class="blob-num js-line-number" data-line-number="296"></td>
        <td id="LC296" class="blob-code js-file-line">                <span class="k">\State</span><span class="nb">{}</span> <span class="s">$</span><span class="nb">mid </span><span class="nv">\gets</span><span class="nb"> </span><span class="nv">\Call</span><span class="nb">{Partition}{list, start, end}</span><span class="s">$</span></td>
      </tr>
      <tr>
        <td id="L297" class="blob-num js-line-number" data-line-number="297"></td>
        <td id="LC297" class="blob-code js-file-line">                <span class="k">\State</span><span class="nb">{}</span> <span class="k">\Call</span><span class="nb">{</span>Quick-Sort<span class="nb">}{</span><span class="s">$</span><span class="nb">list, start, mid </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="s">$</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L298" class="blob-num js-line-number" data-line-number="298"></td>
        <td id="LC298" class="blob-code js-file-line">                <span class="k">\State</span><span class="nb">{}</span> <span class="k">\Call</span><span class="nb">{</span>Quick-Sort<span class="nb">}{</span><span class="s">$</span><span class="nb">list, mid </span><span class="o">+</span><span class="nb"> </span><span class="m">1</span><span class="nb">, end</span><span class="s">$</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L299" class="blob-num js-line-number" data-line-number="299"></td>
        <td id="LC299" class="blob-code js-file-line">            <span class="k">\EndFunction</span><span class="nb">{}</span></td>
      </tr>
      <tr>
        <td id="L300" class="blob-num js-line-number" data-line-number="300"></td>
        <td id="LC300" class="blob-code js-file-line">        <span class="k">\end</span><span class="nb">{</span>algorithmic<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L301" class="blob-num js-line-number" data-line-number="301"></td>
        <td id="LC301" class="blob-code js-file-line">        <span class="k">\caption</span><span class="nb">{</span>Start of QuickSort<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L302" class="blob-num js-line-number" data-line-number="302"></td>
        <td id="LC302" class="blob-code js-file-line">    <span class="k">\end</span><span class="nb">{</span>algorithm<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L303" class="blob-num js-line-number" data-line-number="303"></td>
        <td id="LC303" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L304" class="blob-num js-line-number" data-line-number="304"></td>
        <td id="LC304" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L305" class="blob-num js-line-number" data-line-number="305"></td>
        <td id="LC305" class="blob-code js-file-line"><span class="k">\pagebreak</span></td>
      </tr>
      <tr>
        <td id="L306" class="blob-num js-line-number" data-line-number="306"></td>
        <td id="LC306" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L307" class="blob-num js-line-number" data-line-number="307"></td>
        <td id="LC307" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L308" class="blob-num js-line-number" data-line-number="308"></td>
        <td id="LC308" class="blob-code js-file-line">    Suppose we would like to fit a straight line through the origin, i.e.,</td>
      </tr>
      <tr>
        <td id="L309" class="blob-num js-line-number" data-line-number="309"></td>
        <td id="LC309" class="blob-code js-file-line">    <span class="s">\(</span><span class="nb">Y_i </span><span class="o">=</span><span class="nb"> </span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb"> x_i </span><span class="o">+</span><span class="nb"> e_i</span><span class="s">\)</span> with <span class="s">\(</span><span class="nb">i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">, </span><span class="nv">\ldots</span><span class="nb">, n</span><span class="s">\)</span>, <span class="s">\(</span><span class="nv">\E</span><span class="nb"> </span><span class="o">[</span><span class="nb">e_i</span><span class="o">]</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">0</span><span class="s">\)</span>,</td>
      </tr>
      <tr>
        <td id="L310" class="blob-num js-line-number" data-line-number="310"></td>
        <td id="LC310" class="blob-code js-file-line">    and <span class="s">\(</span><span class="nv">\Var</span><span class="nb"> </span><span class="o">[</span><span class="nb">e_i</span><span class="o">]</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="nv">\sigma</span><span class="nb">^</span><span class="m">2</span><span class="nb">_e</span><span class="s">\)</span> and <span class="s">\(</span><span class="nv">\Cov</span><span class="o">[</span><span class="nb">e_i, e_j</span><span class="o">]</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">0</span><span class="nb">, </span><span class="nv">\forall</span><span class="nb"> i </span><span class="nv">\neq</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L311" class="blob-num js-line-number" data-line-number="311"></td>
        <td id="LC311" class="blob-code js-file-line"><span class="nb">    j</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L312" class="blob-num js-line-number" data-line-number="312"></td>
        <td id="LC312" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L313" class="blob-num js-line-number" data-line-number="313"></td>
        <td id="LC313" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L314" class="blob-num js-line-number" data-line-number="314"></td>
        <td id="LC314" class="blob-code js-file-line">    <span class="k">\part</span></td>
      </tr>
      <tr>
        <td id="L315" class="blob-num js-line-number" data-line-number="315"></td>
        <td id="LC315" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L316" class="blob-num js-line-number" data-line-number="316"></td>
        <td id="LC316" class="blob-code js-file-line">    Find the least squares esimator for <span class="s">\(</span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span><span class="s">\)</span> for the slope</td>
      </tr>
      <tr>
        <td id="L317" class="blob-num js-line-number" data-line-number="317"></td>
        <td id="LC317" class="blob-code js-file-line">    <span class="s">\(</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L318" class="blob-num js-line-number" data-line-number="318"></td>
        <td id="LC318" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L319" class="blob-num js-line-number" data-line-number="319"></td>
        <td id="LC319" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L320" class="blob-num js-line-number" data-line-number="320"></td>
        <td id="LC320" class="blob-code js-file-line">    <span class="k">\solution</span></td>
      </tr>
      <tr>
        <td id="L321" class="blob-num js-line-number" data-line-number="321"></td>
        <td id="LC321" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L322" class="blob-num js-line-number" data-line-number="322"></td>
        <td id="LC322" class="blob-code js-file-line">    To find the least squares estimator, we should minimize our Residual Sum</td>
      </tr>
      <tr>
        <td id="L323" class="blob-num js-line-number" data-line-number="323"></td>
        <td id="LC323" class="blob-code js-file-line">    of Squares, RSS:</td>
      </tr>
      <tr>
        <td id="L324" class="blob-num js-line-number" data-line-number="324"></td>
        <td id="LC324" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L325" class="blob-num js-line-number" data-line-number="325"></td>
        <td id="LC325" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L326" class="blob-num js-line-number" data-line-number="326"></td>
        <td id="LC326" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L327" class="blob-num js-line-number" data-line-number="327"></td>
        <td id="LC327" class="blob-code js-file-line"><span class="nb">            RSS &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\sum</span><span class="nb">_{i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">}^{n} {</span><span class="o">(</span><span class="nb">Y_i </span><span class="o">-</span><span class="nb"> </span><span class="nv">\hat</span><span class="nb">{Y_i}</span><span class="o">)</span><span class="nb">}^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L328" class="blob-num js-line-number" data-line-number="328"></td>
        <td id="LC328" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L329" class="blob-num js-line-number" data-line-number="329"></td>
        <td id="LC329" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\sum</span><span class="nb">_{i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">}^{n} {</span><span class="o">(</span><span class="nb">Y_i </span><span class="o">-</span><span class="nb"> </span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">} x_i</span><span class="o">)</span><span class="nb">}^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L330" class="blob-num js-line-number" data-line-number="330"></td>
        <td id="LC330" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L331" class="blob-num js-line-number" data-line-number="331"></td>
        <td id="LC331" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L332" class="blob-num js-line-number" data-line-number="332"></td>
        <td id="LC332" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L333" class="blob-num js-line-number" data-line-number="333"></td>
        <td id="LC333" class="blob-code js-file-line">    By taking the partial derivative in respect to <span class="s">\(</span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span><span class="s">\)</span>, we get:</td>
      </tr>
      <tr>
        <td id="L334" class="blob-num js-line-number" data-line-number="334"></td>
        <td id="LC334" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L335" class="blob-num js-line-number" data-line-number="335"></td>
        <td id="LC335" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L336" class="blob-num js-line-number" data-line-number="336"></td>
        <td id="LC336" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\pderiv</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L337" class="blob-num js-line-number" data-line-number="337"></td>
        <td id="LC337" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L338" class="blob-num js-line-number" data-line-number="338"></td>
        <td id="LC338" class="blob-code js-file-line"><span class="nb">        }{RSS}</span></td>
      </tr>
      <tr>
        <td id="L339" class="blob-num js-line-number" data-line-number="339"></td>
        <td id="LC339" class="blob-code js-file-line"><span class="nb">        </span><span class="o">=</span><span class="nb"> </span><span class="o">-</span><span class="m">2</span><span class="nb"> </span><span class="nv">\sum</span><span class="nb">_{i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">}^{n} {x_i </span><span class="o">(</span><span class="nb">Y_i </span><span class="o">-</span><span class="nb"> </span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">} x_i</span><span class="o">)</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L340" class="blob-num js-line-number" data-line-number="340"></td>
        <td id="LC340" class="blob-code js-file-line"><span class="nb">        </span><span class="o">=</span><span class="nb"> </span><span class="m">0</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L341" class="blob-num js-line-number" data-line-number="341"></td>
        <td id="LC341" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L342" class="blob-num js-line-number" data-line-number="342"></td>
        <td id="LC342" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L343" class="blob-num js-line-number" data-line-number="343"></td>
        <td id="LC343" class="blob-code js-file-line">    This gives us:</td>
      </tr>
      <tr>
        <td id="L344" class="blob-num js-line-number" data-line-number="344"></td>
        <td id="LC344" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L345" class="blob-num js-line-number" data-line-number="345"></td>
        <td id="LC345" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L346" class="blob-num js-line-number" data-line-number="346"></td>
        <td id="LC346" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L347" class="blob-num js-line-number" data-line-number="347"></td>
        <td id="LC347" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\sum</span><span class="nb">_{i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">}^{n} {x_i </span><span class="o">(</span><span class="nb">Y_i </span><span class="o">-</span><span class="nb"> </span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">} x_i</span><span class="o">)</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L348" class="blob-num js-line-number" data-line-number="348"></td>
        <td id="LC348" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\sum</span><span class="nb">_{i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">}^{n} {x_i Y_i} </span><span class="o">-</span><span class="nb"> </span><span class="nv">\sum</span><span class="nb">_{i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">}^{n} </span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">} x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L349" class="blob-num js-line-number" data-line-number="349"></td>
        <td id="LC349" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L350" class="blob-num js-line-number" data-line-number="350"></td>
        <td id="LC350" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\sum</span><span class="nb">_{i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">}^{n} {x_i Y_i} </span><span class="o">-</span><span class="nb"> </span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span><span class="nv">\sum</span><span class="nb">_{i </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="nb">}^{n} x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L351" class="blob-num js-line-number" data-line-number="351"></td>
        <td id="LC351" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L352" class="blob-num js-line-number" data-line-number="352"></td>
        <td id="LC352" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L353" class="blob-num js-line-number" data-line-number="353"></td>
        <td id="LC353" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L354" class="blob-num js-line-number" data-line-number="354"></td>
        <td id="LC354" class="blob-code js-file-line">    Solving for <span class="s">\(</span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span><span class="s">\)</span> gives the final estimator for <span class="s">\(</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="s">\)</span>:</td>
      </tr>
      <tr>
        <td id="L355" class="blob-num js-line-number" data-line-number="355"></td>
        <td id="LC355" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L356" class="blob-num js-line-number" data-line-number="356"></td>
        <td id="LC356" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L357" class="blob-num js-line-number" data-line-number="357"></td>
        <td id="LC357" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L358" class="blob-num js-line-number" data-line-number="358"></td>
        <td id="LC358" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L359" class="blob-num js-line-number" data-line-number="359"></td>
        <td id="LC359" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L360" class="blob-num js-line-number" data-line-number="360"></td>
        <td id="LC360" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i Y_i}</span></td>
      </tr>
      <tr>
        <td id="L361" class="blob-num js-line-number" data-line-number="361"></td>
        <td id="LC361" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L362" class="blob-num js-line-number" data-line-number="362"></td>
        <td id="LC362" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L363" class="blob-num js-line-number" data-line-number="363"></td>
        <td id="LC363" class="blob-code js-file-line"><span class="nb">            }</span></td>
      </tr>
      <tr>
        <td id="L364" class="blob-num js-line-number" data-line-number="364"></td>
        <td id="LC364" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L365" class="blob-num js-line-number" data-line-number="365"></td>
        <td id="LC365" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L366" class="blob-num js-line-number" data-line-number="366"></td>
        <td id="LC366" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L367" class="blob-num js-line-number" data-line-number="367"></td>
        <td id="LC367" class="blob-code js-file-line">    <span class="k">\pagebreak</span></td>
      </tr>
      <tr>
        <td id="L368" class="blob-num js-line-number" data-line-number="368"></td>
        <td id="LC368" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L369" class="blob-num js-line-number" data-line-number="369"></td>
        <td id="LC369" class="blob-code js-file-line">    <span class="k">\part</span></td>
      </tr>
      <tr>
        <td id="L370" class="blob-num js-line-number" data-line-number="370"></td>
        <td id="LC370" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L371" class="blob-num js-line-number" data-line-number="371"></td>
        <td id="LC371" class="blob-code js-file-line">    Calculate the bias and the variance for the estimated slope</td>
      </tr>
      <tr>
        <td id="L372" class="blob-num js-line-number" data-line-number="372"></td>
        <td id="LC372" class="blob-code js-file-line">    <span class="s">\(</span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L373" class="blob-num js-line-number" data-line-number="373"></td>
        <td id="LC373" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L374" class="blob-num js-line-number" data-line-number="374"></td>
        <td id="LC374" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L375" class="blob-num js-line-number" data-line-number="375"></td>
        <td id="LC375" class="blob-code js-file-line">    <span class="k">\solution</span></td>
      </tr>
      <tr>
        <td id="L376" class="blob-num js-line-number" data-line-number="376"></td>
        <td id="LC376" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L377" class="blob-num js-line-number" data-line-number="377"></td>
        <td id="LC377" class="blob-code js-file-line">    For the bias, we need to calculate the expected value</td>
      </tr>
      <tr>
        <td id="L378" class="blob-num js-line-number" data-line-number="378"></td>
        <td id="LC378" class="blob-code js-file-line">    <span class="s">\(</span><span class="nv">\E</span><span class="o">[</span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span><span class="o">]</span><span class="s">\)</span>:</td>
      </tr>
      <tr>
        <td id="L379" class="blob-num js-line-number" data-line-number="379"></td>
        <td id="LC379" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L380" class="blob-num js-line-number" data-line-number="380"></td>
        <td id="LC380" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L381" class="blob-num js-line-number" data-line-number="381"></td>
        <td id="LC381" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L382" class="blob-num js-line-number" data-line-number="382"></td>
        <td id="LC382" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\E</span><span class="o">[</span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span><span class="o">]</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L383" class="blob-num js-line-number" data-line-number="383"></td>
        <td id="LC383" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\E</span><span class="nb"> </span><span class="nv">\left</span><span class="o">[</span><span class="nb"> </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L384" class="blob-num js-line-number" data-line-number="384"></td>
        <td id="LC384" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i Y_i}</span></td>
      </tr>
      <tr>
        <td id="L385" class="blob-num js-line-number" data-line-number="385"></td>
        <td id="LC385" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L386" class="blob-num js-line-number" data-line-number="386"></td>
        <td id="LC386" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L387" class="blob-num js-line-number" data-line-number="387"></td>
        <td id="LC387" class="blob-code js-file-line"><span class="nb">            }</span><span class="nv">\right</span><span class="o">]</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L388" class="blob-num js-line-number" data-line-number="388"></td>
        <td id="LC388" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L389" class="blob-num js-line-number" data-line-number="389"></td>
        <td id="LC389" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L390" class="blob-num js-line-number" data-line-number="390"></td>
        <td id="LC390" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i </span><span class="nv">\E</span><span class="o">[</span><span class="nb">Y_i</span><span class="o">]</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L391" class="blob-num js-line-number" data-line-number="391"></td>
        <td id="LC391" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L392" class="blob-num js-line-number" data-line-number="392"></td>
        <td id="LC392" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L393" class="blob-num js-line-number" data-line-number="393"></td>
        <td id="LC393" class="blob-code js-file-line"><span class="nb">            }</span></td>
      </tr>
      <tr>
        <td id="L394" class="blob-num js-line-number" data-line-number="394"></td>
        <td id="LC394" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L395" class="blob-num js-line-number" data-line-number="395"></td>
        <td id="LC395" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L396" class="blob-num js-line-number" data-line-number="396"></td>
        <td id="LC396" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i </span><span class="o">(</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb"> x_i</span><span class="o">)</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L397" class="blob-num js-line-number" data-line-number="397"></td>
        <td id="LC397" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L398" class="blob-num js-line-number" data-line-number="398"></td>
        <td id="LC398" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L399" class="blob-num js-line-number" data-line-number="399"></td>
        <td id="LC399" class="blob-code js-file-line"><span class="nb">            }</span></td>
      </tr>
      <tr>
        <td id="L400" class="blob-num js-line-number" data-line-number="400"></td>
        <td id="LC400" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L401" class="blob-num js-line-number" data-line-number="401"></td>
        <td id="LC401" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L402" class="blob-num js-line-number" data-line-number="402"></td>
        <td id="LC402" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i^</span><span class="m">2</span><span class="nb"> </span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L403" class="blob-num js-line-number" data-line-number="403"></td>
        <td id="LC403" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L404" class="blob-num js-line-number" data-line-number="404"></td>
        <td id="LC404" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L405" class="blob-num js-line-number" data-line-number="405"></td>
        <td id="LC405" class="blob-code js-file-line"><span class="nb">            }</span></td>
      </tr>
      <tr>
        <td id="L406" class="blob-num js-line-number" data-line-number="406"></td>
        <td id="LC406" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L407" class="blob-num js-line-number" data-line-number="407"></td>
        <td id="LC407" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb"> </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L408" class="blob-num js-line-number" data-line-number="408"></td>
        <td id="LC408" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i^</span><span class="m">2</span><span class="nb"> </span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L409" class="blob-num js-line-number" data-line-number="409"></td>
        <td id="LC409" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L410" class="blob-num js-line-number" data-line-number="410"></td>
        <td id="LC410" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L411" class="blob-num js-line-number" data-line-number="411"></td>
        <td id="LC411" class="blob-code js-file-line"><span class="nb">            }</span></td>
      </tr>
      <tr>
        <td id="L412" class="blob-num js-line-number" data-line-number="412"></td>
        <td id="LC412" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L413" class="blob-num js-line-number" data-line-number="413"></td>
        <td id="LC413" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L414" class="blob-num js-line-number" data-line-number="414"></td>
        <td id="LC414" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L415" class="blob-num js-line-number" data-line-number="415"></td>
        <td id="LC415" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L416" class="blob-num js-line-number" data-line-number="416"></td>
        <td id="LC416" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L417" class="blob-num js-line-number" data-line-number="417"></td>
        <td id="LC417" class="blob-code js-file-line">    Thus since our estimator&#39;s expected value is <span class="s">\(</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="s">\)</span>, we can conclude</td>
      </tr>
      <tr>
        <td id="L418" class="blob-num js-line-number" data-line-number="418"></td>
        <td id="LC418" class="blob-code js-file-line">    that the bias of our estimator is 0.</td>
      </tr>
      <tr>
        <td id="L419" class="blob-num js-line-number" data-line-number="419"></td>
        <td id="LC419" class="blob-code js-file-line">    <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L420" class="blob-num js-line-number" data-line-number="420"></td>
        <td id="LC420" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L421" class="blob-num js-line-number" data-line-number="421"></td>
        <td id="LC421" class="blob-code js-file-line">    For the variance:</td>
      </tr>
      <tr>
        <td id="L422" class="blob-num js-line-number" data-line-number="422"></td>
        <td id="LC422" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L423" class="blob-num js-line-number" data-line-number="423"></td>
        <td id="LC423" class="blob-code js-file-line">    <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L424" class="blob-num js-line-number" data-line-number="424"></td>
        <td id="LC424" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L425" class="blob-num js-line-number" data-line-number="425"></td>
        <td id="LC425" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\Var</span><span class="o">[</span><span class="nv">\hat</span><span class="nb">{</span><span class="nv">\beta</span><span class="nb">_</span><span class="m">1</span><span class="nb">}</span><span class="o">]</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L426" class="blob-num js-line-number" data-line-number="426"></td>
        <td id="LC426" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"> </span><span class="nv">\Var</span><span class="nb"> </span><span class="nv">\left</span><span class="o">[</span><span class="nb"> </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L427" class="blob-num js-line-number" data-line-number="427"></td>
        <td id="LC427" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i Y_i}</span></td>
      </tr>
      <tr>
        <td id="L428" class="blob-num js-line-number" data-line-number="428"></td>
        <td id="LC428" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L429" class="blob-num js-line-number" data-line-number="429"></td>
        <td id="LC429" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L430" class="blob-num js-line-number" data-line-number="430"></td>
        <td id="LC430" class="blob-code js-file-line"><span class="nb">            }</span><span class="nv">\right</span><span class="o">]</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L431" class="blob-num js-line-number" data-line-number="431"></td>
        <td id="LC431" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L432" class="blob-num js-line-number" data-line-number="432"></td>
        <td id="LC432" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L433" class="blob-num js-line-number" data-line-number="433"></td>
        <td id="LC433" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L434" class="blob-num js-line-number" data-line-number="434"></td>
        <td id="LC434" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i^</span><span class="m">2</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L435" class="blob-num js-line-number" data-line-number="435"></td>
        <td id="LC435" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L436" class="blob-num js-line-number" data-line-number="436"></td>
        <td id="LC436" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"> </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L437" class="blob-num js-line-number" data-line-number="437"></td>
        <td id="LC437" class="blob-code js-file-line"><span class="nb">            } </span><span class="nv">\Var</span><span class="o">[</span><span class="nb">Y_i</span><span class="o">]</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L438" class="blob-num js-line-number" data-line-number="438"></td>
        <td id="LC438" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L439" class="blob-num js-line-number" data-line-number="439"></td>
        <td id="LC439" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L440" class="blob-num js-line-number" data-line-number="440"></td>
        <td id="LC440" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L441" class="blob-num js-line-number" data-line-number="441"></td>
        <td id="LC441" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> {x_i^</span><span class="m">2</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L442" class="blob-num js-line-number" data-line-number="442"></td>
        <td id="LC442" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L443" class="blob-num js-line-number" data-line-number="443"></td>
        <td id="LC443" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"> </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L444" class="blob-num js-line-number" data-line-number="444"></td>
        <td id="LC444" class="blob-code js-file-line"><span class="nb">            } </span><span class="nv">\Var</span><span class="o">[</span><span class="nb">Y_i</span><span class="o">]</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L445" class="blob-num js-line-number" data-line-number="445"></td>
        <td id="LC445" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L446" class="blob-num js-line-number" data-line-number="446"></td>
        <td id="LC446" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L447" class="blob-num js-line-number" data-line-number="447"></td>
        <td id="LC447" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L448" class="blob-num js-line-number" data-line-number="448"></td>
        <td id="LC448" class="blob-code js-file-line"><span class="nb">                </span><span class="m">1</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L449" class="blob-num js-line-number" data-line-number="449"></td>
        <td id="LC449" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L450" class="blob-num js-line-number" data-line-number="450"></td>
        <td id="LC450" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L451" class="blob-num js-line-number" data-line-number="451"></td>
        <td id="LC451" class="blob-code js-file-line"><span class="nb">            } </span><span class="nv">\Var</span><span class="o">[</span><span class="nb">Y_i</span><span class="o">]</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L452" class="blob-num js-line-number" data-line-number="452"></td>
        <td id="LC452" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L453" class="blob-num js-line-number" data-line-number="453"></td>
        <td id="LC453" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L454" class="blob-num js-line-number" data-line-number="454"></td>
        <td id="LC454" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L455" class="blob-num js-line-number" data-line-number="455"></td>
        <td id="LC455" class="blob-code js-file-line"><span class="nb">                </span><span class="m">1</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L456" class="blob-num js-line-number" data-line-number="456"></td>
        <td id="LC456" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L457" class="blob-num js-line-number" data-line-number="457"></td>
        <td id="LC457" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L458" class="blob-num js-line-number" data-line-number="458"></td>
        <td id="LC458" class="blob-code js-file-line"><span class="nb">            } </span><span class="nv">\sigma</span><span class="nb">^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L459" class="blob-num js-line-number" data-line-number="459"></td>
        <td id="LC459" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L460" class="blob-num js-line-number" data-line-number="460"></td>
        <td id="LC460" class="blob-code js-file-line"><span class="nb">            &amp;</span><span class="o">=</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L461" class="blob-num js-line-number" data-line-number="461"></td>
        <td id="LC461" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\frac</span><span class="nb">{</span></td>
      </tr>
      <tr>
        <td id="L462" class="blob-num js-line-number" data-line-number="462"></td>
        <td id="LC462" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sigma</span><span class="nb">^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L463" class="blob-num js-line-number" data-line-number="463"></td>
        <td id="LC463" class="blob-code js-file-line"><span class="nb">            }{</span></td>
      </tr>
      <tr>
        <td id="L464" class="blob-num js-line-number" data-line-number="464"></td>
        <td id="LC464" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\sum</span><span class="nb"> x_i^</span><span class="m">2</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L465" class="blob-num js-line-number" data-line-number="465"></td>
        <td id="LC465" class="blob-code js-file-line"><span class="nb">            }</span></td>
      </tr>
      <tr>
        <td id="L466" class="blob-num js-line-number" data-line-number="466"></td>
        <td id="LC466" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L467" class="blob-num js-line-number" data-line-number="467"></td>
        <td id="LC467" class="blob-code js-file-line"><span class="nb">    </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L468" class="blob-num js-line-number" data-line-number="468"></td>
        <td id="LC468" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L469" class="blob-num js-line-number" data-line-number="469"></td>
        <td id="LC469" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L470" class="blob-num js-line-number" data-line-number="470"></td>
        <td id="LC470" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L471" class="blob-num js-line-number" data-line-number="471"></td>
        <td id="LC471" class="blob-code js-file-line"><span class="k">\pagebreak</span></td>
      </tr>
      <tr>
        <td id="L472" class="blob-num js-line-number" data-line-number="472"></td>
        <td id="LC472" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L473" class="blob-num js-line-number" data-line-number="473"></td>
        <td id="LC473" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L474" class="blob-num js-line-number" data-line-number="474"></td>
        <td id="LC474" class="blob-code js-file-line">    Prove a polynomial of degree <span class="s">\(</span><span class="nb">k</span><span class="s">\)</span>, <span class="s">\(</span><span class="nb">a_kn^k </span><span class="o">+</span><span class="nb"> a_{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">}n^{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">} </span><span class="o">+</span><span class="nb"> </span><span class="nv">\hdots</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L475" class="blob-num js-line-number" data-line-number="475"></td>
        <td id="LC475" class="blob-code js-file-line"><span class="nb">    </span><span class="o">+</span><span class="nb"> a_</span><span class="m">1</span><span class="nb">n^</span><span class="m">1</span><span class="nb"> </span><span class="o">+</span><span class="nb"> a_</span><span class="m">0</span><span class="nb">n^</span><span class="m">0</span><span class="s">\)</span> is a member of <span class="s">\(</span><span class="nv">\Theta</span><span class="o">(</span><span class="nb">n^k</span><span class="o">)</span><span class="s">\)</span> where <span class="s">\(</span><span class="nb">a_k </span><span class="nv">\hdots</span><span class="nb"> a_</span><span class="m">0</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L476" class="blob-num js-line-number" data-line-number="476"></td>
        <td id="LC476" class="blob-code js-file-line">    are nonnegative constants.</td>
      </tr>
      <tr>
        <td id="L477" class="blob-num js-line-number" data-line-number="477"></td>
        <td id="LC477" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L478" class="blob-num js-line-number" data-line-number="478"></td>
        <td id="LC478" class="blob-code js-file-line">    <span class="k">\begin</span><span class="nb">{</span>proof<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L479" class="blob-num js-line-number" data-line-number="479"></td>
        <td id="LC479" class="blob-code js-file-line">        To prove that <span class="s">\(</span><span class="nb">a_kn^k </span><span class="o">+</span><span class="nb"> a_{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">}n^{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">} </span><span class="o">+</span><span class="nb"> </span><span class="nv">\hdots</span><span class="nb"> </span><span class="o">+</span><span class="nb"> a_</span><span class="m">1</span><span class="nb">n^</span><span class="m">1</span><span class="nb"> </span><span class="o">+</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L480" class="blob-num js-line-number" data-line-number="480"></td>
        <td id="LC480" class="blob-code js-file-line"><span class="nb">        a_</span><span class="m">0</span><span class="nb">n^</span><span class="m">0</span><span class="s">\)</span>, we must show the following:</td>
      </tr>
      <tr>
        <td id="L481" class="blob-num js-line-number" data-line-number="481"></td>
        <td id="LC481" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L482" class="blob-num js-line-number" data-line-number="482"></td>
        <td id="LC482" class="blob-code js-file-line">        <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L483" class="blob-num js-line-number" data-line-number="483"></td>
        <td id="LC483" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\exists</span><span class="nb"> c_</span><span class="m">1</span><span class="nb"> </span><span class="nv">\exists</span><span class="nb"> c_</span><span class="m">2</span><span class="nb"> </span><span class="nv">\forall</span><span class="nb"> n </span><span class="nv">\geq</span><span class="nb"> n_</span><span class="m">0</span><span class="nb">,</span><span class="nv">\ </span><span class="nb">{c_</span><span class="m">1</span><span class="nb"> </span><span class="nv">\cdot</span><span class="nb"> g</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="nv">\leq</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L484" class="blob-num js-line-number" data-line-number="484"></td>
        <td id="LC484" class="blob-code js-file-line"><span class="nb">            f</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb"> </span><span class="nv">\leq</span><span class="nb"> c_</span><span class="m">2</span><span class="nb"> </span><span class="nv">\cdot</span><span class="nb"> g</span><span class="o">(</span><span class="nb">n</span><span class="o">)</span><span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L485" class="blob-num js-line-number" data-line-number="485"></td>
        <td id="LC485" class="blob-code js-file-line"><span class="nb">        </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L486" class="blob-num js-line-number" data-line-number="486"></td>
        <td id="LC486" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L487" class="blob-num js-line-number" data-line-number="487"></td>
        <td id="LC487" class="blob-code js-file-line">        For the first inequality, it is easy to see that it holds because no</td>
      </tr>
      <tr>
        <td id="L488" class="blob-num js-line-number" data-line-number="488"></td>
        <td id="LC488" class="blob-code js-file-line">        matter what the constants are, <span class="s">\(</span><span class="nb">n^k </span><span class="nv">\leq</span><span class="nb"> a_kn^k </span><span class="o">+</span><span class="nb"> a_{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">}n^{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">} </span><span class="o">+</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L489" class="blob-num js-line-number" data-line-number="489"></td>
        <td id="LC489" class="blob-code js-file-line"><span class="nb">        </span><span class="nv">\hdots</span><span class="nb"> </span><span class="o">+</span><span class="nb"> a_</span><span class="m">1</span><span class="nb">n^</span><span class="m">1</span><span class="nb"> </span><span class="o">+</span><span class="nb"> a_</span><span class="m">0</span><span class="nb">n^</span><span class="m">0</span><span class="s">\)</span> even if <span class="s">\(</span><span class="nb">c_</span><span class="m">1</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="s">\)</span> and <span class="s">\(</span><span class="nb">n_</span><span class="m">0</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="s">\)</span>.  This</td>
      </tr>
      <tr>
        <td id="L490" class="blob-num js-line-number" data-line-number="490"></td>
        <td id="LC490" class="blob-code js-file-line">        is because <span class="s">\(</span><span class="nb">n^k </span><span class="nv">\leq</span><span class="nb"> c_</span><span class="m">1</span><span class="nb"> </span><span class="nv">\cdot</span><span class="nb"> a_kn^k</span><span class="s">\)</span> for any nonnegative constant,</td>
      </tr>
      <tr>
        <td id="L491" class="blob-num js-line-number" data-line-number="491"></td>
        <td id="LC491" class="blob-code js-file-line">        <span class="s">\(</span><span class="nb">c_</span><span class="m">1</span><span class="s">\)</span> and <span class="s">\(</span><span class="nb">a_k</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L492" class="blob-num js-line-number" data-line-number="492"></td>
        <td id="LC492" class="blob-code js-file-line">        <span class="k">\\</span></td>
      </tr>
      <tr>
        <td id="L493" class="blob-num js-line-number" data-line-number="493"></td>
        <td id="LC493" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L494" class="blob-num js-line-number" data-line-number="494"></td>
        <td id="LC494" class="blob-code js-file-line">        Taking the second inequality, we prove it in the following way.</td>
      </tr>
      <tr>
        <td id="L495" class="blob-num js-line-number" data-line-number="495"></td>
        <td id="LC495" class="blob-code js-file-line">        By summation, <span class="s">\(</span><span class="nv">\sum\limits</span><span class="nb">_{i</span><span class="o">=</span><span class="m">0</span><span class="nb">}^k a_i</span><span class="s">\)</span> will give us a new constant,</td>
      </tr>
      <tr>
        <td id="L496" class="blob-num js-line-number" data-line-number="496"></td>
        <td id="LC496" class="blob-code js-file-line">        <span class="s">\(</span><span class="nb">A</span><span class="s">\)</span>. By taking this value of <span class="s">\(</span><span class="nb">A</span><span class="s">\)</span>, we can then do the following:</td>
      </tr>
      <tr>
        <td id="L497" class="blob-num js-line-number" data-line-number="497"></td>
        <td id="LC497" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L498" class="blob-num js-line-number" data-line-number="498"></td>
        <td id="LC498" class="blob-code js-file-line">        <span class="sb">\[</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L499" class="blob-num js-line-number" data-line-number="499"></td>
        <td id="LC499" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\begin</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L500" class="blob-num js-line-number" data-line-number="500"></td>
        <td id="LC500" class="blob-code js-file-line"><span class="nb">                a_kn^k </span><span class="o">+</span><span class="nb"> a_{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">}n^{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">} </span><span class="o">+</span><span class="nb"> </span><span class="nv">\hdots</span><span class="nb"> </span><span class="o">+</span><span class="nb"> a_</span><span class="m">1</span><span class="nb">n^</span><span class="m">1</span><span class="nb"> </span><span class="o">+</span><span class="nb"> a_</span><span class="m">0</span><span class="nb">n^</span><span class="m">0</span><span class="nb"> &amp;</span><span class="o">=</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L501" class="blob-num js-line-number" data-line-number="501"></td>
        <td id="LC501" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L502" class="blob-num js-line-number" data-line-number="502"></td>
        <td id="LC502" class="blob-code js-file-line"><span class="nb">                &amp;</span><span class="nv">\leq</span><span class="nb"> </span><span class="o">(</span><span class="nb">a_k </span><span class="o">+</span><span class="nb"> a_{k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="nb">} </span><span class="nv">\hdots</span><span class="nb"> a_</span><span class="m">1</span><span class="nb"> </span><span class="o">+</span><span class="nb"> a_</span><span class="m">0</span><span class="o">)</span><span class="nb"> </span><span class="nv">\cdot</span><span class="nb"> n^k</span></td>
      </tr>
      <tr>
        <td id="L503" class="blob-num js-line-number" data-line-number="503"></td>
        <td id="LC503" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L504" class="blob-num js-line-number" data-line-number="504"></td>
        <td id="LC504" class="blob-code js-file-line"><span class="nb">                &amp;</span><span class="o">=</span><span class="nb"> A </span><span class="nv">\cdot</span><span class="nb"> n^k</span></td>
      </tr>
      <tr>
        <td id="L505" class="blob-num js-line-number" data-line-number="505"></td>
        <td id="LC505" class="blob-code js-file-line"><span class="nb">                </span><span class="nv">\\</span><span class="nb"></span></td>
      </tr>
      <tr>
        <td id="L506" class="blob-num js-line-number" data-line-number="506"></td>
        <td id="LC506" class="blob-code js-file-line"><span class="nb">                &amp;</span><span class="nv">\leq</span><span class="nb"> c_</span><span class="m">2</span><span class="nb"> </span><span class="nv">\cdot</span><span class="nb"> n^k</span></td>
      </tr>
      <tr>
        <td id="L507" class="blob-num js-line-number" data-line-number="507"></td>
        <td id="LC507" class="blob-code js-file-line"><span class="nb">            </span><span class="nv">\end</span><span class="nb">{split}</span></td>
      </tr>
      <tr>
        <td id="L508" class="blob-num js-line-number" data-line-number="508"></td>
        <td id="LC508" class="blob-code js-file-line"><span class="nb">        </span><span class="s">\]</span></td>
      </tr>
      <tr>
        <td id="L509" class="blob-num js-line-number" data-line-number="509"></td>
        <td id="LC509" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L510" class="blob-num js-line-number" data-line-number="510"></td>
        <td id="LC510" class="blob-code js-file-line">        where <span class="s">\(</span><span class="nb">n_</span><span class="m">0</span><span class="nb"> </span><span class="o">=</span><span class="nb"> </span><span class="m">1</span><span class="s">\)</span> and <span class="s">\(</span><span class="nb">c_</span><span class="m">2</span><span class="nb"> </span><span class="o">=</span><span class="nb"> A</span><span class="s">\)</span>. <span class="s">\(</span><span class="nb">c_</span><span class="m">2</span><span class="s">\)</span> is just a constant. Thus the</td>
      </tr>
      <tr>
        <td id="L511" class="blob-num js-line-number" data-line-number="511"></td>
        <td id="LC511" class="blob-code js-file-line">        proof is complete.</td>
      </tr>
      <tr>
        <td id="L512" class="blob-num js-line-number" data-line-number="512"></td>
        <td id="LC512" class="blob-code js-file-line">    <span class="k">\end</span><span class="nb">{</span>proof<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L513" class="blob-num js-line-number" data-line-number="513"></td>
        <td id="LC513" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L514" class="blob-num js-line-number" data-line-number="514"></td>
        <td id="LC514" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L515" class="blob-num js-line-number" data-line-number="515"></td>
        <td id="LC515" class="blob-code js-file-line"><span class="k">\pagebreak</span></td>
      </tr>
      <tr>
        <td id="L516" class="blob-num js-line-number" data-line-number="516"></td>
        <td id="LC516" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L517" class="blob-num js-line-number" data-line-number="517"></td>
        <td id="LC517" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L518" class="blob-num js-line-number" data-line-number="518"></td>
        <td id="LC518" class="blob-code js-file-line"><span class="c">% Non sequential homework problems</span></td>
      </tr>
      <tr>
        <td id="L519" class="blob-num js-line-number" data-line-number="519"></td>
        <td id="LC519" class="blob-code js-file-line"><span class="c">%</span></td>
      </tr>
      <tr>
        <td id="L520" class="blob-num js-line-number" data-line-number="520"></td>
        <td id="LC520" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L521" class="blob-num js-line-number" data-line-number="521"></td>
        <td id="LC521" class="blob-code js-file-line"><span class="c">% Jump to problem 18</span></td>
      </tr>
      <tr>
        <td id="L522" class="blob-num js-line-number" data-line-number="522"></td>
        <td id="LC522" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span>[18]</td>
      </tr>
      <tr>
        <td id="L523" class="blob-num js-line-number" data-line-number="523"></td>
        <td id="LC523" class="blob-code js-file-line">    Evaluate <span class="s">\(</span><span class="nv">\sum</span><span class="nb">_{k</span><span class="o">=</span><span class="m">1</span><span class="nb">}^{</span><span class="m">5</span><span class="nb">} k^</span><span class="m">2</span><span class="s">\)</span> and <span class="s">\(</span><span class="nv">\sum</span><span class="nb">_{k</span><span class="o">=</span><span class="m">1</span><span class="nb">}^{</span><span class="m">5</span><span class="nb">} </span><span class="o">(</span><span class="nb">k </span><span class="o">-</span><span class="nb"> </span><span class="m">1</span><span class="o">)</span><span class="nb">^</span><span class="m">2</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L524" class="blob-num js-line-number" data-line-number="524"></td>
        <td id="LC524" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L525" class="blob-num js-line-number" data-line-number="525"></td>
        <td id="LC525" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L526" class="blob-num js-line-number" data-line-number="526"></td>
        <td id="LC526" class="blob-code js-file-line"><span class="c">% Continue counting to 19</span></td>
      </tr>
      <tr>
        <td id="L527" class="blob-num js-line-number" data-line-number="527"></td>
        <td id="LC527" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L528" class="blob-num js-line-number" data-line-number="528"></td>
        <td id="LC528" class="blob-code js-file-line">    Find the derivative of <span class="s">\(</span><span class="nb">f</span><span class="o">(</span><span class="nb">x</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb"> x^</span><span class="m">4</span><span class="nb"> </span><span class="o">+</span><span class="nb"> </span><span class="m">3</span><span class="nb">x^</span><span class="m">2</span><span class="nb"> </span><span class="o">-</span><span class="nb"> </span><span class="m">2</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L529" class="blob-num js-line-number" data-line-number="529"></td>
        <td id="LC529" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L530" class="blob-num js-line-number" data-line-number="530"></td>
        <td id="LC530" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L531" class="blob-num js-line-number" data-line-number="531"></td>
        <td id="LC531" class="blob-code js-file-line"><span class="c">% Go back to where we left off</span></td>
      </tr>
      <tr>
        <td id="L532" class="blob-num js-line-number" data-line-number="532"></td>
        <td id="LC532" class="blob-code js-file-line"><span class="k">\begin</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span>[6]</td>
      </tr>
      <tr>
        <td id="L533" class="blob-num js-line-number" data-line-number="533"></td>
        <td id="LC533" class="blob-code js-file-line">    Evaluate the integrals</td>
      </tr>
      <tr>
        <td id="L534" class="blob-num js-line-number" data-line-number="534"></td>
        <td id="LC534" class="blob-code js-file-line">    <span class="s">\(</span><span class="nv">\int</span><span class="nb">_</span><span class="m">0</span><span class="nb">^</span><span class="m">1</span><span class="nb"> </span><span class="o">(</span><span class="m">1</span><span class="nb"> </span><span class="o">-</span><span class="nb"> x^</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\dx</span><span class="s">\)</span></td>
      </tr>
      <tr>
        <td id="L535" class="blob-num js-line-number" data-line-number="535"></td>
        <td id="LC535" class="blob-code js-file-line">    and</td>
      </tr>
      <tr>
        <td id="L536" class="blob-num js-line-number" data-line-number="536"></td>
        <td id="LC536" class="blob-code js-file-line">    <span class="s">\(</span><span class="nv">\int</span><span class="nb">_</span><span class="m">1</span><span class="nb">^{</span><span class="nv">\infty</span><span class="nb">} </span><span class="nv">\frac</span><span class="nb">{</span><span class="m">1</span><span class="nb">}{x^</span><span class="m">2</span><span class="nb">} </span><span class="nv">\dx</span><span class="s">\)</span>.</td>
      </tr>
      <tr>
        <td id="L537" class="blob-num js-line-number" data-line-number="537"></td>
        <td id="LC537" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>homeworkProblem<span class="nb">}</span></td>
      </tr>
      <tr>
        <td id="L538" class="blob-num js-line-number" data-line-number="538"></td>
        <td id="LC538" class="blob-code js-file-line">
</td>
      </tr>
      <tr>
        <td id="L539" class="blob-num js-line-number" data-line-number="539"></td>
        <td id="LC539" class="blob-code js-file-line"><span class="k">\end</span><span class="nb">{</span>document<span class="nb">}</span></td>
      </tr>
</table>

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