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  <question type="stack">
    <name>
      <text>14.2.1.5 Finding more means of functions</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Calculate the mean value of the given functions over the specified interval.</p><br>

<ol class="HELM_parts">
    <li>
        <p>\(f(x)={@f1@}\) across \([{@a1@},{@b1@}]\)</p>
        <p>mean value \(=\) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
    </li>
    <li>
        <p>\(f(x)=\displaystyle{@f2@}\) across \([{@a2@},{@b2@}]\)</p>
        <p>mean value \(=\) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>
    </li>
    <li>
        <p>\(f(t)={@f3@}\) across \([{@a3@},{@b3@}]\)</p>
        <p>mean value \(=\) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>
    </li>
    <li>
        <p>\(f(z)={@f4@}\) across \([{@a4@},{@b4@}]\)</p>
        <p>mean value \(=\) [[input:ans4]] [[validation:ans4]] [[feedback:prt4]]</p>
    </li>
</ol>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>Remember the Key Point, mean value \(\displaystyle = \frac{1}{b-a} \int_a^b f(t) \ \mathrm{d}t\).</p><br>

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

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

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

    <li>Using the Key Point with \(a={@a4@}\), \(b={@b4@}\) and \(f(z)={@f4@}\) we have: \[\begin{aligned}\text{mean value }&amp;= \frac{1}{{@b4@}-{@a4@}} \int_{@a4@}^{@b4@} {@f4@}\ \mathrm{d}z\\ &amp;= \frac{1}{{@b4-a4@}}\left[{@integrate(f4,z)@}\right]_{@a4@}^{@b4@}\\
        &amp;= \frac{1}{{@b4-a4@}}\left[{@subst(z=b4,integrate(f4,z))@}-{@subst(z=a4,integrate(f4,z))@}\right]\\ &amp;= {@ta4@} \end{aligned}\]</li>

</ol>]]></text>
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    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text>f1: x^3;
a1: rand(2)+1;
b1: a1+2;
ta1: 1/(b1-a1)*integrate(f1,x,a1,b1);

f2: 1/x;
a2: 1;
b2: rand(5)+1;
ta2: 1/(b2-a2)*integrate(f2,x,a2,b2);

f3: sqrt(t);
a3: 0;
b3: rand(5)+1;
ta3: 1/(b3-a3)*integrate(f3,t,a3,b3);

f4: z^3-1;
a4: rand(2);
b4: a4+rand(3)+1;
ta4: 1/(b4-a4)*integrate(f4,z,a4,b4);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\[ ({@f1@} , [{@a1@},{@b1@}], \, m = {@ta1@}), \quad ({@f2@} , [{@a2@},{@b2@}], \, m = {@ta2@}) \]
\[ ({@f3@} , [{@a3@},{@b3@}], \, m = {@ta3@}), \quad ({@f4@} , [{@a4@},{@b4@}], \, m = {@ta4@}) \]</text>
    </questionnote>
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      <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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      <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>
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