qu.1.topic=Section 2.1: Trig Functions@

qu.1.1.mode=Blanks@
qu.1.1.name=1.5.3@
qu.1.1.comment=
     
     <p class="noindent">The average value is approximately 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${s}</mn>
     </math>. The graph is a sine curve with period 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mfrac>
       <mrow>
        <mn>2</mn>
        <mi>&pi;</mi>
       </mrow>
       <mrow>
        <mn>${f}</mn>
       </mrow>
      </mfrac>
     </math>and amplitude 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${a}</mn>
     </math>. This gives a frequency 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${f}</mn>
      <mo>,</mo>
     </math>so a possible formula is 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>f</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
      <mo>=</mo>
      <mn>${s}</mn>
      <mo>+</mo>
      <mn>${a}</mn>
      <mo>&#8290;</mo>
      <mo>sin</mo>
      
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mn>${f}</mn>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>.</p>
    @
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=int(rint(3)+1); $f=int(rint(2)+1);
    $p=decimal(2,2/$f); $s=int((rint(3)+2)); $ymax=int($a+1+$s);
    $ymin=-int($a+1+$s);@
qu.1.1.uid=f0fc9948-7443-40cd-b550-c2c6163b7720@
qu.1.1.question=
    
    <p class="noindent">Consider the graph shown below.</p>
    <applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
     <param name="y1" value="${s}+${a}*sin(${f}*x)"></param>
     <param name="gridLines" value="10"></param>
     <param name="xMin" value="0"></param>
     <param name="xMax" value="10"></param>
     <param name="yMin" value="${ymin}"></param>
     <param name="yMax" value="${ymax} "></param>
    </applet>
    <br class="newline">
    <br class="newline">The average value of this function is 
    <1>. 
    <br class="newline">
    <br class="newline">The amplitude of this function is 
    <2>. 
    <br class="newline">
    <br class="newline">The frequency of this function is 
    <3>. 
    <br class="newline">
    <br class="newline">The period of this function is 
    <4>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>&pi;</mi>
    </math>.@
qu.1.1.blank.1=%24%7bs%7d%3f0.2@
qu.1.1.blank.2=%24%7ba%7d%3f0.2@
qu.1.1.blank.3=%24%7bf%7d%3f0.2@
qu.1.1.blank.4=%24%7bp%7d%3f0.2@
qu.1.1.grader.1=formula@
qu.1.1.grader.2=formula@
qu.1.1.grader.3=formula@
qu.1.1.grader.4=formula@
qu.1.1.extra=@

qu.1.2.mode=Blanks@
qu.1.2.name=1.5.1@
qu.1.2.comment=
     
     <p class="noindent">The graph is a sine curve with period 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mfrac>
       <mrow>
        <mn>2</mn>
        <mi>&pi;</mi>
       </mrow>
       <mrow>
        <mn>${f}</mn>
       </mrow>
      </mfrac>
     </math>and amplitude 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${a}</mn>
     </math>. This gives a frequency 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${f}</mn>
      <mo>,</mo>
     </math>so a possible formula is 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>f</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
      <mo>=</mo>
      <mn>${a}</mn>
      <mo>&#8290;</mo>
      <mo>sin</mo>
      
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mn>${f}</mn>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>.</p>
    @
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=int(rint(5)+3); $f=int(rint(2)+1);
    $p=decimal(4,2/$f); $ymax=int($a+1);
    $ymin=-int($ymax);@
qu.1.2.uid=b9639f5c-e704-460e-9b44-27b3773a472a@
qu.1.2.question=
    
    <p class="noindent">Consider the function shown below 
    <br class="newline"></p>
    <applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
     <param name="y1" value="${a}*sin(${f}*x)"></param>
     <param name="gridLines" value="10"></param>
     <param name="xMin" value="0"></param>
     <param name="xMax" value="10"></param>
     <param name="yMin" value="-8"></param>
     <param name="yMax" value="8 "></param>
    </applet>
    <br class="newline">
    <br class="newline">The amplitude of this function is 
    <1>. 
    <br class="newline">
    <br class="newline">The frequency of this function is 
    <2>. 
    <br class="newline">
    <br class="newline">The period of this function is 
    <3>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>&pi;</mi>
    </math>.@
qu.1.2.blank.1=%24%7ba%7d%3f0.2@
qu.1.2.blank.2=%24%7bf%7d%3f0.2@
qu.1.2.blank.3=%24%7bp%7d%3f0.2@
qu.1.2.grader.1=formula@
qu.1.2.grader.2=formula@
qu.1.2.grader.3=formula@
qu.1.2.extra=@

qu.1.3.mode=Blanks@
qu.1.3.name=1.5.4@
qu.1.3.comment=
     
     <p class="noindent">The average value is approximately 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${s}</mn>
     </math>. The graph is a sine curve with period 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mfrac>
       <mrow>
        <mn>2</mn>
        <mi>&pi;</mi>
       </mrow>
       <mrow>
        <mn>${f}</mn>
       </mrow>
      </mfrac>
     </math>and amplitude 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${a}</mn>
     </math>. This gives a frequency 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${f}</mn>
      <mo>,</mo>
     </math>so a possible formula is 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>f</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
      <mo>=</mo>
      <mn>${s}</mn>
      <mo>+</mo>
      <mn>${a}</mn>
      <mo>&#8290;</mo>
      <mo>cos</mo>
      
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mn>${f}</mn>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>.</p>
    @
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=int(rint(3)+1); $f=int(rint(2)+1);
    $p=decimal(2,2/$f); $s=int((rint(3)+2)); $ymax=int($a+1+$s);
    $ymin=-int($a+1+$s);@
qu.1.3.uid=8db6faa5-3cef-49e0-96c5-06fc9991912e@
qu.1.3.question=
    
    <p class="noindent">Consider the graph shown below.</p>
    <applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
     <param name="y1" value="${s}+${a}*cos(${f}*x)"></param>
     <param name="gridLines" value="10"></param>
     <param name="xMin" value="0"></param>
     <param name="xMax" value="10"></param>
     <param name="yMin" value="${ymin}"></param>
     <param name="yMax" value="${ymax} "></param>
    </applet>
    <br class="newline">
    <br class="newline">The average value of this function is 
    <1>. 
    <br class="newline">
    <br class="newline">The amplitude of this function is 
    <2>. 
    <br class="newline">
    <br class="newline">The frequency of this function is 
    <3>. 
    <br class="newline">
    <br class="newline">The period of this function is 
    <4>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>&pi;</mi>
    </math>.@
qu.1.3.blank.1=%24%7bs%7d%3f0.2@
qu.1.3.blank.2=%24%7ba%7d%3f0.2@
qu.1.3.blank.3=%24%7bf%7d%3f0.2@
qu.1.3.blank.4=%24%7bp%7d%3f0.2@
qu.1.3.grader.1=formula@
qu.1.3.grader.2=formula@
qu.1.3.grader.3=formula@
qu.1.3.grader.4=formula@
qu.1.3.extra=@

qu.1.4.mode=Blanks@
qu.1.4.name=1.5.2@
qu.1.4.comment=
     
     <p class="noindent">The graph is a sine curve with period 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mfrac>
       <mrow>
        <mn>2</mn>
        <mi>&pi;</mi>
       </mrow>
       <mrow>
        <mn>${f}</mn>
       </mrow>
      </mfrac>
     </math>and amplitude 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${a}</mn>
     </math>. This gives a frequency 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mn>${f}</mn>
      <mo>,</mo>
     </math>so a possible formula is 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>f</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
      <mo>=</mo>
      <mn>${a}</mn>
      <mo>&#8290;</mo>
      <mo>cos</mo>
      
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mn>${f}</mn>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>.</p>
    @
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=int(rint(5)+3); $f=int(rint(2)+1);
    $p=decimal(4,2/$f); $ymax=int($a+1);
    $ymin=-int($ymax);@
qu.1.4.uid=b9df93a6-bfb7-44b2-a3a0-30888f111352@
qu.1.4.question=
    
    <p class="noindent">Consider the function shown below 
    <br class="newline"></p>
    <applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
     <param name="y1" value="${a}*cos(${f}*x)"></param>
     <param name="gridLines" value="10"></param>
     <param name="xMin" value="0"></param>
     <param name="xMax" value="10"></param>
     <param name="yMin" value="-8"></param>
     <param name="yMax" value="8 "></param>
    </applet>
    <br class="newline">
    <br class="newline">The amplitude of this function is 
    <1>. 
    <br class="newline">
    <br class="newline">The frequency of this function is 
    <2>. 
    <br class="newline">
    <br class="newline">The period of this function is 
    <3>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>&pi;</mi>
    </math>.@
qu.1.4.blank.1=%24%7ba%7d%3f0.2@
qu.1.4.blank.2=%24%7bf%7d%3f0.2@
qu.1.4.blank.3=%24%7bp%7d%3f0.2@
qu.1.4.grader.1=formula@
qu.1.4.grader.2=formula@
qu.1.4.grader.3=formula@
qu.1.4.extra=@

qu.1.5.mode=Plain Number@
qu.1.5.name=1.R.7@
qu.1.5.comment=
     
     <p class="noindent">
      
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
       <mi>f</mi>
       <mrow>
        <mo stretchy="false">(</mo>
        <mrow>
         <mi>g</mi>
         <mrow>
          <mo stretchy="false">(</mo>
          <mrow>
           <mn>${x}</mn>
          </mrow>
          <mo stretchy="false">)</mo>
         </mrow>
        </mrow>
        <mo stretchy="false">)</mo>
       </mrow>
       <mo>&#8776;</mo>
       <mi>f</mi>
       <mrow>
        <mo stretchy="false">(</mo>
        <mrow>
         <mn>${gx}</mn>
        </mrow>
        <mo stretchy="false">)</mo>
       </mrow>
       <mo>&#8776;</mo>
       <mn>${fgx}</mn>
       <mo>.</mo>
      </math>
     </p>
    @
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$x=int(rint(5)-3); $fx=decimal(2,3*sin($x));
    $gx=decimal(2,3*cos($x)); $fgx=decimal(2,3*sin($gx));
    $gfx=decimal(2,3*cos($fx)); $ffx=decimal(2,3*sin($fx));
    $ggx=decimal(2,3*cos($gx));@
qu.1.5.uid=2f2c9a51-d711-417b-aac9-d2feba96585e@
qu.1.5.question=
     
     <p class="noindent">Given the graphs of 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>f</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>and 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>g</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>shown below, estimate 
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>f</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>g</mi>
        <mrow>
         <mo stretchy="false">(</mo>
         <mrow>
          <mn>${x}</mn>
         </mrow>
         <mo stretchy="false">)</mo>
        </mrow>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>. 
     <br class="newline"></p>
     <applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
      <param name="y1" value="3*sin(x)"></param>
      <param name="gridLines" value="12"></param>
      <param name="xMin" value="-3"></param>
      <param name="xMax" value="3"></param>
      <param name="yMin" value="-3"></param>
      <param name="yMax" value="3"></param>
     </applet>
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>f</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>
     <applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
      <param name="y1" value="3*cos(x)"></param>
      <param name="gridLines" value="12"></param>
      <param name="xMin" value="-3"></param>
      <param name="xMax" value="3"></param>
      <param name="yMin" value="-3"></param>
      <param name="yMax" value="3"></param>
     </applet>
     
     <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
      <mi>g</mi>
      <mrow>
       <mo stretchy="false">(</mo>
       <mrow>
        <mi>x</mi>
       </mrow>
       <mo stretchy="false">)</mo>
      </mrow>
     </math>
    @
qu.1.5.answer=${fgx} ? 0.2@

