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Mathresponse Example

<problem>
<script type="loncapa/perl">
$a1 = random(-6,6,4);
$a2 = random(-6,6,4);
$n1 = random(3,11,2);
$n2 = random(2,10,2);
$function = "$a1*cos($n1*x)+$a2*sin($n2*x)";
$example=&xmlparse('An example would be <m eval="on">$(sin($n1\cdot x)+cos($n2\cdot x))/\sqrt{2}$</m>');</script>
<startouttext />
Give an example of a function
<ol>
<li>which is orthogonal to <algebra>$function</algebra> with respect to the
scalar product
<m>\[<g \mid h> =
\frac{1}{\pi} \int_{-\pi}^{\pi}dx g(x) \cdot h(x)\]</m>
</li>
<li>whose norm is 1.</li>
</ol><endouttext />
<mathresponse answerdisplay="$example" cas="maxima" args="$function">
<answer>overlap:integrate((RESPONSE[1])*(LONCAPALIST[1]),x,-%pi,%pi)/%pi;
norm:integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(overlap=0 and norm=1);</answer>
<textline readonly="no" size="50" />
<hintgroup showoncorrect="no">
<mathhint name="ortho" args="$function" cas="maxima">
<answer>overlap: integrate((LONCAPALIST[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(not overlap = 0);</answer>
</mathhint>
<mathhint name="norm" args="$function" cas="maxima">
<answer>norm: integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(not norm = 1);</answer>
</mathhint> <hintpart on="norm">
<startouttext />
The function you have provided does not have a norm of one.<endouttext />
</hintpart>
<hintpart on="ortho">
<startouttext />
The function you have provided is not orthogonal.<endouttext />
</hintpart>
</hintgroup>
</mathresponse>
<postanswerdate><startouttext /><p>Note that with respect to the above norm, <m>$\cos(nx)$</m> is perpendicular to <m>$\sin(nx)$</m> and perpendicular to <m>$\cos(mx)$</m> for <m>$n\ne m$</m>.</p><endouttext />
</postanswerdate>
</problem>

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Site maintained by Gerd Kortemeyer.
Supported by the National Science Foundation under NSF-ITR 0085921, NSF-CCLI-ASA 0243126, and NSF-CCLI 0717790. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Initial funding for CAPA has been provided by the Alfred. P. Sloan Foundation and the Andrew W. Mellon Foundation.