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- adding some design docs

    1: <problem>
    2: 
    3: <script type="loncapa/perl">
    4: $a1 = random(-5,5,2);
    5: $a2 = random(-5,5,2);
    6: $n1 = random(1,10,1);
    7: $n2 = random(1,10,1);
    8: $function = "$a1*cos($n1*x)+$a2*sin($n2*x)";
    9: @cas_list = ($function);
   10: </script>
   11: 
   12: <startouttext />
   13: Give an example of a function which is
   14: (1) orthogonal to <algebra>$function</algebra> with respect to the
   15: scalar product
   16: <m>$$\langle g \mid h \rangle =
   17: \frac{1}{\pi} \int_{-\pi}^{\pi} g(x) \cdot h(x)$$</m>
   18: and
   19: (2) whose norm is 1.
   20: <endouttext />
   21: 
   22: <mathresponse cas="maxima" args="@cas_list">
   23:    <answer>
   24:      f(x):= LONCAPALIST[1];
   25:      overlap: integrate(f(x)*RESPONSE[1],x,-%pi,%pi)/%pi;
   26:      norm: integrate(RESPONSE[1]*RESPONSE[1],x,-%pi,%pi)/%pi;
   27:      grade: is(overlap=0 and norm=1);
   28:      [grade];
   29:    </answer>
   30: 
   31:    <textline />
   32: 
   33:    <hintgroup showoncorrect="no">
   34:      <mathhint cas="maxima" args="@cas_list" name="ortho">
   35:        <answer>
   36:          f(x):= LONCAPALIST[1];
   37:          overlap: integrate(f(x)*RESPONSE[1],x,-%pi,%pi)/%pi;
   38: 	 [is(overlap != 0)];
   39:        </answer>
   40:      </mathhint>
   41:      <mathhint cas="maxima" args="@cas_list" name="norm">
   42:        <answer>
   43:          f(x):= LONCAPALIST[1];
   44:          norm: integrate(RESPONSE[1]*RESPONSE[1],x,-%pi,%pi)/%pi;
   45:          [is(norm != 1)];
   46:        </answer>
   47:      </mathhint>
   48: 
   49: 
   50:      <hintpart on="norm">
   51:        <script type="loncapa/perl">
   52:          $response = &EXT('user.resource.resource.0.submissiom');
   53: 	 $norm = &cas('maxima',...); # get the norm
   54: 	 $norm_display = &cas_tex_display($norm);
   55:        </script>
   56:        <startouttext />
   57:          The function you have provided is not normal to
   58:          <algebra>$function</algebra> its norm is
   59:           <m eval="on">$norm_display</m>
   60:        <endouttext />
   61:      </hintpart>
   62:      <hintpart on="ortho">
   63:        <startouttext />
   64:          The function you have provided is not orthognal.
   65:        <endouttext />
   66:      </hintpart>
   67: 
   68:    </hintgroup>
   69: </mathresponse>
   70: 
   71: </problem>