<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>14.2.2.5 Finding more r.m.s. of more functions</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Calculate the r.m.s. value of the following to 4 decimal places:</p><br>

<ol class="HELM_parts">
    <li>
        <p>\(f(t)={@f1@}\) across \([{@a1@},{@b1@}]\)</p>
        <p>r.m.s. value \(=\) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
    </li>
    <li>
        <p>\(f(t)={@f2@}\) across \([{@a2@},{@b2@}]\)</p>
        <p>r.m.s. value \(=\) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>
    </li>
    <li>
        <p>\(f(t)={@f3@}\) across \([{@a3@},{@b3@}]\)</p>
        <p>r.m.s. value \(=\) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>
    </li>
</ol>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>Remember the Key Point, r.m.s. value \(\displaystyle = \sqrt{\frac{1}{b-a} \int_a^b [f(t)]^2 \ \mathrm{d}t}\).</p><br>

<ol class="HELM_parts">
    <li>Using the Key Point with \(a={@a1@}\), \(b={@b1@}\) and \(f(t)={@f1@}\) we have: \[\begin{aligned}\text{r.m.s. value }&amp;= \sqrt{\frac{1}{{@b1@}-{@a1@}} \int_{@a1@}^{@b1@} \left[{@f1@}\right]^2 \ \mathrm{d}t}\\ &amp;= \sqrt{\frac{1}{{@b1-a1@}}\left[{@integrate(f1^2,t)@}\right]_{@a1@}^{@b1@}}\\
        &amp;= \sqrt{\frac{1}{{@b1-a1@}}\left[{@subst(t=b1,integrate(f1^2,t))@}-{@subst(t=a1,integrate(f1^2,t))@}\right]}\\ &amp;\approx {@ta1@} \end{aligned}\]</li>

    <li>Using the Key Point with \(a={@a2@}\), \(b={@b2@}\) and \(f(t)={@f2@}\) we have: \[\begin{aligned}\text{r.m.s. value }&amp;= \sqrt{\frac{1}{{@b2@}-({@a2@})} \int_{@a2@}^{@b2@} \left[{@f2@}\right]^2 \ \mathrm{d}t}\\ &amp;= \sqrt{\frac{1}{{@b2-a2@}}\left[{@integrate(f2^2,t)@}\right]_{@a2@}^{@b2@}}\\
        &amp;= \sqrt{\frac{1}{{@b2-a2@}}\left[{@subst(t=b2,integrate(f2^2,t))@}-{@subst(t=a2,integrate(f2^2,t))@}\right]}\\ &amp;\approx {@ta2@} \end{aligned}\]</li>

    <li>Using the Key Point with \(a={@a3@}\), \(b={@b3@}\) and \(f(t)={@f3@}\) we have: \[\begin{aligned}\text{r.m.s. value }&amp;= \sqrt{\frac{1}{{@b3@}-({@a3@})} \int_{@a3@}^{@b3@} \left[{@f3@}\right]^2 \ \mathrm{d}t}\\ &amp;= \sqrt{\frac{1}{{@b3-a3@}}\left[{@integrate(f3^2,t)@}\right]_{@a3@}^{@b3@}}\\
        &amp;= \sqrt{\frac{1}{{@b3-a3@}}\left[{@subst(t=b3,integrate(f3^2,t))@}-\left({@subst(t=a3,integrate(f3^2,t))@}\right)\right]}\\ &amp;\approx {@ta3@} \end{aligned}\]</li>
</ol>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2024072400</text>
    </stackversion>
    <questionvariables>
      <text>const1: rand([1,2,3]);
const2: rand([1,2,3]);

f1: sqrt(t+const1);
a1: 0;
b1: 3;
ta1: sqrt(1/(b1-a1)*integrate(f1^2,t,a1,b1));
ta1: decimalplaces(float(ta1),4);

f2: e^(const2*t);
a2: -1;
b2: 1;
ta2: sqrt(1/(b2-a2)*integrate(f2^2,t,a2,b2));
ta2: decimalplaces(float(ta2),4);

f3: const1+e^(const2*t);
a3: -1;
b3: 1;
ta3: sqrt(1/(b3-a3)*integrate(f3^2,t,a3,b3));
ta3: decimalplaces(float(ta3),4);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\( ({@f1@} , [{@a1@},{@b1@}], \, m={@ta1@}), \quad ({@f2@} , [{@a2@},{@b2@}], \, m={@ta2@}), \quad ({@f3@} , [{@a3@},{@b3@}], \, m={@ta3@}) \)</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i> Your answer is partially correct.</span>]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>numerical</type>
      <tans>ta1</tans>
      <boxsize>5</boxsize>
      <strictsyntax>1</strictsyntax>
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      <options>mindp:4,maxdp:4</options>
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      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
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        <text/>
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      <node>
        <name>0</name>
        <description/>
        <answertest>NumDecPlaces</answertest>
        <sans>ans1</sans>
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    </prt>
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      <value>1.0000000</value>
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      <value>1.0000000</value>
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        <truenextnode>-1</truenextnode>
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    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>dispdp(ta1,4)</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>dispdp(ta2,4)</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>dispdp(ta3,4)</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt2-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt3-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>dispdp(ev(ta1+1,simp),4)</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>dispdp(ev(ta2+1,simp),4)</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>dispdp(ev(ta3+1,simp),4)</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt2-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt3-1-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>-ta1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>-ta2</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>-ta3</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt2-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt3-1-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
