<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>14.2.2.4 Finding r.m.s. of trig functions (long copy)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Calculate the r.m.s. value of the following, to 4 decimal places where appropriate:</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>
    <li>
        <p>\(f(t)={@f4@}\) across \([{@a4@},{@b4@}]\)</p>
        <p>r.m.s. value \(=\) [[input:ans4]] [[validation:ans4]] [[feedback:prt4]]</p>
    </li>
    <li>
        <p>\(f(t)={@f5@}\) across \([{@a5@},{@b5@}]\)</p>
        <p>r.m.s. value \(=\) [[input:ans5]] [[validation:ans5]] [[feedback:prt5]]</p>
    </li>
    <li>
        <p>\(f(t)={@f6@}\) across \([{@a6@},{@b6@}]\)</p>
        <p>r.m.s. value \(=\) [[input:ans6]] [[validation:ans6]] [[feedback:prt6]]</p>
    </li>
    <li>
        <p>\(f(t)={@f7@}\) across \([{@a7@},{@b7@}]\)</p>
        <p>r.m.s. value \(=\) [[input:ans7]] [[validation:ans7]] [[feedback:prt7]]</p>
    </li>
</ol>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p><strong>WORKED SOLUTION</strong></p><br>

<p>Remember the Key Point, r.m.s. value \(\displaystyle = \sqrt{\frac{1}{b-a} \int_a^b [f(t)]^2 dt}\).</p><br>

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

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

    <li>Using the Key Point with \(a={@a3@}\), \(b={@b3@}\) and \(f(t)={@f3@}\) we have: \[\begin{aligned}\text{r.m.s. value }&amp;= \sqrt{\frac{1}{{@b3@}-{@a3@}} \int_{@a3@}^{@b3@} \left[{@f3@}\right]^2 dt}\\ &amp;= \sqrt{{@1/(b3-a3)@}\left[{@integrate(f3^2,t)@}\right]_{@a3@}^{@b3@}}\\
        &amp;= \sqrt{{@1/(b3-a3)@}\left[{@subst(t=b3,integrate(f3^2,t))@}-{@subst(t=a3,integrate(f3^2,t))@}\right]}\\ &amp;\approx {@ta3@} \end{aligned}\]</li>

    <li>Using the Key Point with \(a={@a4@}\), \(b={@b4@}\) and \(f(t)={@f4@}\) we have: \[\begin{aligned}\text{r.m.s. value }&amp;= \sqrt{\frac{1}{{@b4@}-{@a4@}} \int_{@a4@}^{@b4@} \left[{@f4@}\right]^2 dt}\\ &amp;= \sqrt{{@1/(b4-a4)@}\left[{@integrate(f4^2,t)@}\right]_{@a4@}^{@b4@}}\\
        &amp;= \sqrt{{@1/(b4-a4)@}\left[{@subst(t=b4,integrate(f4^2,t))@}-{@subst(t=a4,integrate(f4^2,t))@}\right]}\\ &amp;\approx {@ta4@} \end{aligned}\]</li>

    <li>Using the Key Point with \(a={@a5@}\), \(b={@b5@}\) and \(f(t)={@f5@}\) we have: \[\begin{aligned}\text{r.m.s. value }&amp;= \sqrt{\frac{1}{{@b5@}-{@a5@}} \int_{@a5@}^{@b5@} \left[{@f5@}\right]^2 ~dt}\\ &amp;= \sqrt{{@1/(b5-a5)@}\left[{@integrate(f5^2,t)@}\right]_{@a5@}^{@b5@}}\\
        &amp;= \sqrt{{@1/(b5-a5)@}\left[{@subst(t=b5,integrate(f5^2,t))@}-{@subst(t=a5,integrate(f5^2,t))@}\right]}\\ &amp;\approx {@ta5@} \end{aligned}\]</li>

    <li>Using the Key Point with \(a={@a6@}\), \(b={@b6@}\) and \(f(t)={@f6@}\) we have: \[\begin{aligned}\text{r.m.s. value }&amp;= \sqrt{\frac{1}{{@b6@}-{@a6@}} \int_{@a6@}^{@b6@} \left[{@f6@}\right]^2 ~dt}\\ &amp;= \sqrt{{@1/(b6-a6)@}\left[{@integrate(f6^2,t)@}\right]_{@a6@}^{@b6@}}\\
        &amp;= \sqrt{{@1/(b6-a6)@}\left[{@subst(t=b6,integrate(f6^2,t))@}-{@subst(t=a6,integrate(f6^2,t))@}\right]}\\ &amp;\approx {@ta6@} \end{aligned}\]</li>

    <li>Using the Key Point with \(a={@a7@}\), \(b={@b7@}\) and \(f(t)={@f7@}\) we have: \[\begin{aligned}\text{r.m.s. value }&amp;= \sqrt{\frac{1}{{@b7@}-{@a7@}} \int_{@a7@}^{@b7@} \left[{@f7@}\right]^2 ~dt}\\ &amp;= \sqrt{{@1/(b7-a7)@}\left[{@integrate(f7^2,t)@}\right]_{@a7@}^{@b7@}}\\
        &amp;= \sqrt{{@1/(b7-a7)@}\left[{@subst(t=b7,integrate(f7^2,t))@}-\left({@subst(t=a7,integrate(f7^2,t))@}\right)\right]}\\ &amp;\approx {@ta7@} \end{aligned}\]</li>
</ol>]]></text>
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    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text>omega: 1/2;
f1: sin(t);
a1: 0;
b1: pi/2;
ta1: sqrt(1/(b1-a1)*integrate(f1^2,t,a1,b1));
ta1: decimalplaces(float(ta1),4);

f2: sin(t);
a2: 0;
b2: pi;
ta2: sqrt(1/(b2-a2)*integrate(f2^2,t,a2,b2));
ta2: decimalplaces(float(ta2),4);

f3: sin(omega*t);
a3: 0;
b3: pi;
ta3: sqrt(1/(b3-a3)*integrate(f3^2,t,a3,b3));
ta3: decimalplaces(float(ta3),4);

f4: cos(t);
a4: 0;
b4: pi/2;
ta4: sqrt(1/(b4-a4)*integrate(f4^2,t,a4,b4));
ta4: decimalplaces(float(ta4),4);

f5: cos(t);
a5: 0;
b5: pi;
ta5: sqrt(1/(b5-a5)*integrate(f5^2,t,a5,b5));
ta5: decimalplaces(float(ta5),4);

f6: cos(omega*t);
a6: 0;
b6: pi;
ta6: sqrt(1/(b6-a6)*integrate(f6^2,t,a6,b6));
ta6: decimalplaces(float(ta6),4);

f7: sin(omega*t)+ cos(omega*t);
a7: 0;
b7: 1;
ta7: sqrt(1/(b7-a7)*integrate(f7^2,t,a7,b7));
ta7: decimalplaces(float(ta7),4);</text>
    </questionvariables>
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    <questionnote format="moodle_auto_format">
      <text>({@f1@} , [{@a1@},{@b1@}]), ({@f2@} , [{@a2@},{@b2@}]), ({@f3@} , [{@a3@},{@b3@}]), ({@f4@} , [{@a4@},{@b4@}]), ({@f5@} , [{@a5@},{@b5@}])</text>
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    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
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    <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>
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    <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>
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    <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>
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        <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>
      <expected>
        <name>prt4</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt4-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt5-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt6</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt6-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt7</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt7-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1+1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2+1</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ta3+1</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value>ta4+1</value>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value>ta5+1</value>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value>ta6+1</value>
      </testinput>
      <testinput>
        <name>ans7</name>
        <value>ta7+1</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>
      <expected>
        <name>prt4</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt4-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt5-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt6</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt6-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt7</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt7-1-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
