version 1.3, 2007/05/21 22:11:07
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version 1.4, 2009/07/06 17:12:47
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<problem> |
<problem> |
<script type="loncapa/perl"> |
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<script type="loncapa/perl"> |
$a1 = &random(-6,6,4); |
$a1 = &random(-6,6,4); |
$a2 = &random(-6,6,4); |
$a2 = &random(-6,6,4); |
$n1 = &random(3,11,2); |
$n1 = &random(3,11,2); |
$n2 = &random(2,10,2); |
$n2 = &random(2,10,2); |
$function = "$a1*cos($n1*x)+$a2*sin($n2*x)"; |
$function = "$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>'); |
$example=&xmlparse('An example would be <m eval="on">$ (sin($n1\cdot x)+cos($n2\cdot x))/\sqrt{2} $</m>'); |
</script> |
</script> |
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<startouttext /> |
<startouttext /> |
Give an example of a function |
Give an example of a function |
<ol> |
<ol> |
<li>which is orthogonal to <algebra>$function</algebra> with respect to the |
<li>which is orthogonal to <algebra>$function</algebra> with respect to the scalar product |
scalar product |
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<m>\[<g \mid h> = \frac{1}{\pi} \int_{-\pi}^{\pi}dx g(x) \cdot h(x)\]</m> |
<m>\[<g \mid h> = \frac{1}{\pi} \int_{-\pi}^{\pi}dx g(x) \cdot h(x)\]</m> |
</li> |
</li> |
<li>whose norm is 1.</li> |
<li>whose norm is 1.</li> |
</ol> |
</ol> |
<endouttext /> |
<endouttext /> |
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<mathresponse answerdisplay="$example" cas="maxima" args="$function"> |
<mathresponse answerdisplay="$example" cas="maxima" args="$function"> |
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<answer> |
<answer> |
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overlap:integrate((RESPONSE[1])*(LONCAPALIST[1]),x,-%pi,%pi)/%pi; |
overlap:integrate((RESPONSE[1])*(LONCAPALIST[1]),x,-%pi,%pi)/%pi; |
norm:integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi; |
norm:integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi; |
is(overlap=0 and norm=1); |
is(overlap=0 and norm=1); |
</answer> |
</answer> |
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<textline readonly="no" size="50" /> |
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<hintgroup showoncorrect="no"> |
<textline readonly="no" size="50" /> |
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<mathhint name="ortho" args="$function" cas="maxima"> |
<hintgroup showoncorrect="no"> |
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<mathhint name="ortho" args="$function" cas="maxima"> |
<answer> |
<answer> |
overlap: integrate((LONCAPALIST[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi; |
overlap: integrate((LONCAPALIST[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi; |
is(not overlap = 0); |
is(not overlap = 0); |
</answer> |
</answer> |
</mathhint> |
</mathhint> |
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<mathhint name="norm" args="$function" cas="maxima"> |
<mathhint name="norm" args="$function" cas="maxima"> |
<answer> |
<answer> |
norm: integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi; |
norm: integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi; |
is(not norm = 1); |
is(not norm = 1); |
</answer> |
</answer> |
</mathhint> |
</mathhint> |
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<hintpart on="norm"> |
<hintpart on="norm"> |
<startouttext /> |
<startouttext /> |
The function you have provided does not have a norm of one. |
The function you have provided does not have a norm of one. |
<endouttext /> |
<endouttext /> |
</hintpart> |
</hintpart> |
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<hintpart on="ortho"> |
<hintpart on="ortho"> |
<startouttext /> |
<startouttext /> |
The function you have provided is not orthogonal. |
The function you have provided is not orthogonal. |
<endouttext /> |
<endouttext /> |
</hintpart> |
</hintpart> |
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</hintgroup> |
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</mathresponse> |
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<postanswerdate> |
</hintgroup> |
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</mathresponse> |
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<startouttext /> |
<postanswerdate> |
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<startouttext /> |
<p> |
<p> |
Note that with respect to the above norm, <m>$\cos(nx)$</m> is |
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 $</m>. |
perpendicular to <m>$\sin(nx)$</m> and perpendicular to <m>$\cos(mx)$</m> |
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for <m>$n\ne m$</m>. |
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</p> |
</p> |
<endouttext /> |
<endouttext /> |
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</postanswerdate> |
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</postanswerdate> |
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</problem> |
</problem> |