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<quiz>
  <question type="stack">
    <name>
      <text>Series_3</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>By writing \(x\) as \({@n2@}+(x-{@n2@})\) and using the Maclaurin expansion for \(\ln(1+t)\), or otherwise, find the first four non zero terms in the Taylor series expansion for</p>
<p>\[\ln({@n1@}x)\]</p>
<p>about \(x = {@n2@}\).</p>
<p>Use exact expressions, i.e. enter \(\ln(5)\) rather than \(1.609...\).</p>
<p>Answer: [[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>\[\ln({@n1@}x)\]</p>
<p>can be written as</p>
<p>\[\ln({@n1@}({@n2@}+(x-{@n2@}))) \]</p>
<p>which equals \[ \ln{@n1@} + \ln({@n2@}+(x-{@n2@})) =\ln{@n1@} + \ln\left({@n2@}\left(1+\frac{(x-{@n2@})}{@n2@}\right)\right)=\ln{@n1@} + \ln{@n2@}+\ln\left(1+\frac{(x-{@n2@})}{@n2@}\right)\]</p>
<p>As \(\ln(1+t) = t-\frac{t^2}{2}+\frac{t^3}{3}\) and \(\ln{@n1@} + \ln{@n2@} =\ln({@n1*n2@})\)</p>
<p>\[\ln({@n1@}x)= \ln({@n1*n2@})+\frac{x-{@n2@}}{@n2@}-\frac{(x-{@n2@})^2}{@2*n2^2@}+\frac{(x-{@n2@})^3}{@3*n2^3@}\]</p>
<p> </p>
<p> </p>
<p> </p>]]></text>
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    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>n1:rand(4)+2;
 n2:rand(4)+2;
 f : ln(n1*x);  
 correctanswer:ln(n1*n2)+(x-n2)/n2-((x-n2)/n2)^2/2+((x-n2)/n2)^3/3;
 wronganswer1:correctanswer-ln(n1*n2);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(\ln({@n1@}x)\) about \(x = {@n2@}\) is \({@correctanswer@}\)</text>
    </questionnote>
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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>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></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>
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    <complexno>i</complexno>
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  </question>
</quiz>
