﻿<?xml version="1.0" encoding="utf-8" standalone="yes"?>
<?application progid="SMath Studio Desktop" version="1.0.8348.30405"?>
<worksheet xmlns="http://smath.info/schemas/worksheet/1.0">
  <settings ppi="144">
    <identity>
      <id>c98c9033-1bb0-4f70-a1c1-cb043c71bb89</id>
      <revision>22</revision>
    </identity>
    <calculation>
      <precision>3</precision>
      <exponentialThreshold>5</exponentialThreshold>
      <trailingZeros>false</trailingZeros>
      <significantDigitsMode>false</significantDigitsMode>
      <roundingMode>0</roundingMode>
      <approximateEqualAccuracy>0</approximateEqualAccuracy>
      <fractions>decimal</fractions>
    </calculation>
    <pageModel active="false" viewMode="2" printGrid="false" printAreas="true" simpleEqualsOnly="true" printBackgroundImages="true">
      <paper id="9" orientation="Portrait" width="827" height="1169" />
      <margins left="39" right="39" top="49" bottom="49" />
      <header alignment="Center" color="#a9a9a9">&amp;[DATE] &amp;[TIME] - &amp;[FILENAME]</header>
      <footer alignment="Center" color="#a9a9a9">&amp;[PAGENUM] / &amp;[COUNT]</footer>
      <backgrounds />
    </pageModel>
    <dependencies>
      <assembly name="SMath Studio Desktop" version="1.0.8348.30405" guid="a37cba83-b69c-4c71-9992-55ff666763bd" />
      <assembly name="MathRegion" version="1.11.8348.30405" guid="02f1ab51-215b-466e-a74d-5d8b1cf85e8d" />
      <assembly name="Custom Functions" version="1.1.8051.30522" guid="18dadffd-79a3-4cf9-aee1-d66deb0ea720" />
      <assembly name="SpecialFunctions" version="1.12.8348.30405" guid="2814e667-4e12-48b1-8d51-194e480eabc5" />
      <assembly name="TextRegion" version="1.11.8348.30405" guid="485d28c5-349a-48b6-93be-12a35a1c1e39" />
      <assembly name="X-Y Plot Region (JXCharts)" version="0.2.8209.38826" guid="c12231ec-4873-43c1-a7d0-a167ebd17066" />
    </dependencies>
  </settings>
  <regions type="content">
    <region left="0" top="0" width="408" height="84" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">f.n</e>
          <e type="operand">k</e>
          <e type="operand">Q</e>
          <e type="function" args="3">M</e>
          <e type="operand">k</e>
          <e type="operand">f.n</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">1</e>
          <e type="operand">k</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operand">f.n</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">1</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operand">Q</e>
          <e type="operand">k</e>
          <e type="operator" args="2">*</e>
          <e type="operand">f.n</e>
          <e type="operator" args="2">*</e>
          <e type="operand">f.n</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operand">1</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="function" args="1">sqrt</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="450" top="27" width="62" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">Q</e>
          <e type="operand">0.5</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="531" top="27" width="46" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">k</e>
          <e type="operand">5</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="0" top="90" width="391" height="219" color="#000000" bgColor="#ffffff" fontSize="10">
      <xyplot width="381" height="211" points="100" name="XYPlot">
        <chartstyle backcolor="White" bordercolor="Black" />
        <propertiessource index="1" sourcetype="PropertyGrid" />
        <grid gridcolor="LightGray" gridpattern="Dash" gridthickness="1" isxgrid="true" isygrid="true" isy2grid="false" />
        <xaxes xmin="0.0306167" xmax="1.304985" xtick="0.2" visible="True" decimalplaces="3" numberformat="General" />
        <yaxes ymin="-0.0161464" ymax="1.627733" ytick="0.4" visible="True" decimalplaces="3" numberformat="General" />
        <y2axes isy2axis="false" y2min="-1.44" y2max="1.44" y2tick="1" visible="True" decimalplaces="3" numberformat="General" />
        <title2d title="" titlefont="Arial, 10pt" titlefontcolor="Black" />
        <xylabel labelfont="Arial, 10pt" labelfontcolor="Black" tickfont="Arial, 8pt" tickfontcolor="Black" xlabel="fn" ylabel="M" y2label="y2" />
        <legend isbordervisible="true" islegendvisible="false" legendbackcolor="White" legendbordercolor="Black" legendfont="Arial, 8pt" legendposition="NorthEast" textcolor="Black" />
        <traces>
          <trace seriesname="" isy2data="false" isvisible="true" plotmethod="Lines" lineantialias="true" linecolor="Blue" linethickness="1" linepattern="Solid" symbolantialias="true" symbolsize="8" symboltype="None" symbolborderthickness="1" symbolbordercolor="Black" symbolfillcolor="White" />
        </traces>
        <input>
          <e type="operand">f.n</e>
          <e type="operand">k</e>
          <e type="operand">Q</e>
          <e type="function" args="3">M</e>
        </input>
      </xyplot>
    </region>
    <region left="0" top="342" width="666" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Arial" fontSize="10">
        <content>
          <p>You need to keep identifier of the variable for differentiation separate from the identifier of the solution variable,</p>
        </content>
      </text>
    </region>
    <region left="0" top="378" width="439" height="85" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">sln</e>
          <e type="operand">f.n</e>
          <e type="operand">k</e>
          <e type="operand">Q</e>
          <e type="function" args="3">M</e>
          <e type="operand">f.n</e>
          <e type="function" args="2">diff</e>
          <e type="operand">f.n</e>
          <e type="operand">f.n0</e>
          <e type="operator" args="2">≡</e>
          <e type="function" args="2">at</e>
          <e type="operand">f.n0</e>
          <e type="operand">0</e>
          <e type="operand">1</e>
          <e type="function" args="4">solve</e>
          <e type="operator" args="2">:</e>
        </input>
        <result action="numeric">
          <e type="operand">0</e>
          <e type="operand">0.546</e>
          <e type="operand">2</e>
          <e type="operand">1</e>
          <e type="function" args="4">mat</e>
        </result>
      </math>
    </region>
    <region left="477" top="405" width="164" height="35" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">sln</e>
          <e type="operand">2</e>
          <e type="function" args="2">el</e>
          <e type="operand">k</e>
          <e type="operand">Q</e>
          <e type="function" args="3">M</e>
        </input>
        <result action="numeric">
          <e type="operand">1.31</e>
        </result>
      </math>
    </region>
    <region left="0" top="459" width="376" height="85" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">f.n</e>
          <e type="operand">k</e>
          <e type="operand">Q</e>
          <e type="function" args="3">M</e>
          <e type="operand">f.n</e>
          <e type="function" args="2">diff</e>
          <e type="operand">f.n</e>
          <e type="operand">f.n0</e>
          <e type="operator" args="2">≡</e>
          <e type="function" args="2">at</e>
          <e type="operand">f.n0</e>
          <e type="operand">0.5</e>
          <e type="function" args="3">roots</e>
        </input>
        <result action="numeric">
          <e type="operand">0.546</e>
        </result>
      </math>
    </region>
    <region left="414" top="477" width="33" height="51" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">#</e>
          <e type="operand">#</e>
          <e type="function" args="2">at</e>
        </input>
      </math>
    </region>
    <region left="459" top="477" width="255" height="53" color="#000000" fontSize="10">
      <text lang="eng" width="255" fontFamily="Arial" fontSize="10">
        <content>
          <p>Use the substitution function at() to first perform the derivative and then find it's zero.</p>
        </content>
      </text>
    </region>
    <region left="0" top="540" width="487" height="23" color="#000000" fontSize="10">
      <text lang="eng" width="487" fontFamily="Arial" fontSize="10">
        <content>
          <p>alternatively first define the derivative as a function and then solve</p>
        </content>
      </text>
    </region>
    <region left="0" top="576" width="167" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M'</e>
          <e type="operand">f.n</e>
          <e type="operand">k</e>
          <e type="operand">Q</e>
          <e type="function" args="3">M</e>
          <e type="operand">f.n</e>
          <e type="function" args="2">diff</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="0" top="630" width="267" height="45" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">sln</e>
          <e type="operand">M'</e>
          <e type="operand">f.n</e>
          <e type="operand">0</e>
          <e type="operand">1</e>
          <e type="function" args="4">solve</e>
          <e type="operator" args="2">:</e>
        </input>
        <result action="numeric">
          <e type="operand">0</e>
          <e type="operand">0.546</e>
          <e type="operand">2</e>
          <e type="operand">1</e>
          <e type="function" args="4">mat</e>
        </result>
      </math>
    </region>
    <region left="0" top="684" width="204" height="32" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M'</e>
          <e type="operand">f.n</e>
          <e type="operand">0.5</e>
          <e type="function" args="3">roots</e>
        </input>
        <result action="numeric">
          <e type="operand">0.546</e>
        </result>
      </math>
    </region>
  </regions>
</worksheet>