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<quiz>
  <question type="stack">
    <name>
      <text>18.2.2.5 Evaluate second partial derivatives 4</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the second partial derivatives of the function</p>
\[f(x,y) = {@exp@}.\]
<p>\(\displaystyle \frac{\partial^2 f}{\partial x^2} = \) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
<p>\(\displaystyle \frac{\partial^2 f}{\partial y^2} = \) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>
<p>\(\displaystyle \frac{\partial^2 f}{\partial y \partial x} = \) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>
<p>\(\displaystyle \frac{\partial^2 f}{\partial x \partial y} = \) [[input:ans4]] [[validation:ans4]] [[feedback:prt4]]</p>
<p>Now find \(f_{xx}({@x1@}, {@y1@})\), \(f_{yy}({@x2@}, {@y2@})\) and \(f_{xy}({@x3@}, {@y3@})\).</p>
<p>\(f_{xx}({@x1@}, {@y1@}) = \) [[input:vans1]] [[validation:vans1]] [[feedback:vprt1]]</p>
<p>\(f_{yy}({@x2@}, {@y2@}) = \) [[input:vans2]] [[validation:vans2]] [[feedback:vprt2]]</p>
<p>\(f_{xy}({@x3@}, {@y3@}) = \) [[input:vans3]] [[validation:vans3]] [[feedback:vprt3]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>We are asked to find the second partial derivatives of the function</p>
\[f(x,y) = {@exp@}.\]
<p>First, we find the first partial derivatives:</p>
\[\frac{\partial f}{\partial x} = {@f_x@} \quad \text{and} \quad \frac{\partial f}{\partial y} = {@f_y@}\]
<p>We can then find the second derivatives:</p>
\[\begin{aligned}
\frac{\partial^2 f}{\partial x^2} &amp;= {@ta1@} \\
\frac{\partial^2 f}{\partial y^2} &amp;= {@ta2@} \\
\frac{\partial^2 f}{\partial y \partial x} &amp;= {@ta3@} \\
\frac{\partial^2 f}{\partial x \partial y} &amp;= {@ta4@}
\end{aligned}\]
<p>Next, we are asked to evaluate these derivatives at different points.</p>
\[\begin{aligned}
f_{xx}({@x1@}, {@y1@}) &amp;= {@val1@} \\
f_{yy}({@x2@}, {@y2@}) &amp;= {@val2@} \\
f_{xy}({@x3@}, {@y3@}) &amp;= {@val3@} \\
\end{aligned}\]]]></text>
    </generalfeedback>
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    <idnumber/>
    <stackversion>
      <text>2021120900</text>
    </stackversion>
    <questionvariables>
      <text>ordergreat(x, y);
a: rand([3, 5, 7]);
b: rand([2, 4, 6]);
exp: x^4 + x * y^a + b * x^a * y^2;

f_x: diff(exp, x);
f_y: diff(exp, y);

f_xx: diff(f_x, x);
f_yy: diff(f_y, y);
f_xy: diff(f_x, y);
f_yx: diff(f_y, x);

x1: rand_with_prohib(-2, 2, []);
x2: rand_with_prohib(-2, 2, [x1]);
x3: rand_with_prohib(-2, 2, [x1, x2]);
y1: rand_with_prohib(-2, 2, []);
y2: rand_with_prohib(-2, 2, [y1]);
y3: rand_with_prohib(-2, 2, [y1, y2]);

ta1: f_xx;
ta2: f_yy;
ta3: f_xy;
ta4: f_yx;
val1: ev(f_xx, x=x1, y=y1);
val2: ev(f_yy, x=x2, y=y2);
val3: ev(f_xy, x=x3, y=y3);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\[ f(x,y)={@exp@} \]
\[ f_{xx}({@x1@}, {@y1@}) = {@val1@}, \quad f_{yy}({@x2@}, {@y2@}) = {@val2@}, \quad f_{xy}({@x3@}, {@y3@}) = {@val3@}\]</text>
    </questionnote>
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      <text/>
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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>
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    <hint format="html">
      <text>(Hint: you need to simplify sin(t^3 + 1))</text>
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