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    <name>
      <text>20.1.2.2 Calculus with casual function</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>If \(f(t)=(\text{e}^{-t}+t)u(t)\) then find \(\displaystyle {df\over dt}\) and \(\displaystyle \int_{-3}^4 f(t)\,dt\).</p>
<ol class="HELM_parts">
<li><p>Find the derivative first: </p>
<p>\(\displaystyle \frac{\mathrm{d}f}{\mathrm{d}t}=\) [[input:ans1]] [[validation:ans1]][[feedback:prt1]]</p></li>
<li><p>Now obtain another integral representing \(\displaystyle \int_{-3}^4 f(t)\,dt\):</p>
<p>
<table>
<tbody>
<tr>
  <td>\(I = \displaystyle \int_{-3}^4 f(t)\,dt = \displaystyle  \int\)</td>
  <td><table><tbody>
      <tr><td> [[input:ans2]][[validation:ans2]]</td></tr>
      <tr><td>[[input:ans3]][[validation:ans3]]</td></tr>
  </tbody></table></td>
<td>[[input:ans4]][[validation:ans4]] \(\mathrm{d}t\)</td>
</tr></tbody></table>[[feedback:prt2]]</p></li>
<li><p>Now complete the integration: (Give your answer correct to 4 d.p.)</p>
<p>\(I =\) [[input:ans5]] [[validation:ans5]][[feedback:prt3]]</p></li>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text><![CDATA[<ol class="HELM_parts">
<li>\[\displaystyle {\mathrm{d}f\over \mathrm{d}t}=(-\text{e}^{-t}+1)u(t)\]</li>
<li>You should obtain \(\displaystyle \int_0^4 (\text{e}^{-t}+t)\,dt\) since
\[\displaystyle \int_{-3}^4f(t)\,dt=\int_{-3}^4 (\text{e}^{-t}+t)u(t)\,dt=\int_0^4(\text{e}^{-t}+t)\,dt.\]
This follows because in the range \(t=-3\) to \(t=0\) the step function \(u(t)=0\) and so that part of the integral is zero.  In the other part of the integral \(u(t)=1\).</li>
<li>You should obtain 8.9817 (to 4 d.p.) since
\[\begin{align}\displaystyle \int_0^4(\text{e}^{-t}+t)\,dt &= \left[-\text{e}^{-t}+\displaystyle {t^2\over 2}\right]_0^4 \\ &= (-\text{e}^{-4}+8)-(-1) \\ &=-\text{e}^{-4}+9 \\ &\approx 8.9817.\end{align}\]</li>
</ol>]]></text>
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      <text>2023010400</text>
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    <questionvariables>
      <text>ta1: (-e^(-t) + 1)*u(t);
ta2: 4;
ta3: 0;
ta4: (e^(-t) + t);
ta5: 8.9817;</text>
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