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    <name>
      <text>12.1.3.2 The tangent line to a curve: exponential</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>In this exercise we will find the equation of the tangent line to the curve \(y = e^x\) at the point \(x = {@a@}\). The curve and the line are displayed in the following figure:</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12-1-T2.svg" alt="A sketch of the tangent line to the curve y=e^x
 at the point x=0" width="400" height="173" class="img-responsive atto_image_button_text-bottom" ></figure>
<ol class="HELM_parts">
    <li> We first specify \(x_0\) and \(f(x_0)\):
        <p>\(x_0=\) [[input:ans1]] [[validation:ans1]][[feedback:prt1]]</p>
        <p> \(f(x_0)=\) [[input:ans2]] [[validation:ans2]][[feedback:prt2]]</p>
    </li>
    <li>We now obtain the values of the slope at \(x_0\), i.e. \(m=\dfrac{df}{ dx} \bigg|_{ x_0}\) and the \(y\)-intercept \(f(x_0) - mx_0\):
        <p>\(\dfrac{df}{ dx} \bigg|_{ x_0}=\) [[input:ans3]] [[validation:ans3]][[feedback:prt3]] </p>
        <p>\(f(x_0) - mx_0 =\) [[input:ans4]] [[validation:ans4]][[feedback:prt4]]</p>
    </li>
    <li>Finally, we use these values to obtain the equation of the tangent line:
        <p>\(y =\) [[input:ans5]] [[validation:ans5]][[feedback:prt5]]</p>
    </li>
</ol>]]></text>
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    <generalfeedback format="html">
      <text><![CDATA[<p>We are asked to find the equation of the tangent line to the curve \(y = e^x\) at the point \(x = {@a@}\).
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12-1-T2.svg" alt="A sketch of the tangent line to the curve y=e^x
 at the point x=0" width="400" height="173" class="img-responsive atto_image_button_text-bottom" ></figure>
</p>
<p>
<ol class="HELM_parts">
    <li>
        <p>First we specify that \(x_0 = {@a@}\) and \( f(x_0) = {@ta2@}.\)</p>
    </li>
    <li>
        <p>Then we obtain the values of \(\dfrac{df}{ dx} \bigg|_{ x_0}\) and \(f(x_0) - mx_0\):</p>
        <p>\(\dfrac{df}{dx} = e^x\) therefore \(\dfrac{df}{dx}\bigg|_{@a@} = e^{{@a@}}\) and \(f({@a@}) - {@ta3@}({@a@}) = {@ta4@} .\)</p>
    </li>
    <p></p>
    <li>
        <p>Hence we obtain the equation of the tangent line: \(y= {@ta5@} .\)</p>
    </li>
</ol>
</p>]]></text>
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    <idnumber/>
    <stackversion>
      <text>2024072400</text>
    </stackversion>
    <questionvariables>
      <text>a: 0;
ta2:exp(0);
ta3:exp(a);
ta4:exp(a) - a*exp(a);
ta5:exp(a)*x + exp(a) - a*exp(a);</text>
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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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    <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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    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
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