qu.1.topic=Section 1.3.1: Basics@

qu.1.1.mode=Sketch@
qu.1.1.name=1.3.1.1 a@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=-3.5+rint(10)*0.1;
$b=-1+rint(10)*0.1;
$c=1.5+rint(10)*0.1;
$d=4.5-rint(10)*0.1;@
qu.1.1.uid=351217b8-0478-4bad-850a-aa72fbe3a777@
qu.1.1.question=<p class="noindent">Sketch the graph of a function which is increasing on the interval
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
decreasing on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
    <mo>,</mo>
    <mn>${d}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.1.1.example=${a},-1 ${b},2 ${c},2 ${d},-1@
qu.1.1.answer=check(( increasing(restriction($1,${a},${b})) ) && ( decreasing(restriction($1,${c},${d})) ))@
qu.1.1.axes=-5,5,-5,5@
qu.1.1.gridlines=5@
qu.1.1.axes.labeled=true@

qu.1.2.mode=Multipart@
qu.1.2.name=1.3.1.6@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$x=int(rint(4)+2);
$ansb=decimal(0,8+10*($x)/3-65*($x)^2/36+7*($x)^3/36);
$x1=int(rint(3)+1);@
qu.1.2.uid=9d9f179a-bffe-4d0c-ba06-1d7aee690dd1@
qu.1.2.question=<p class="noindent">Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>v</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>t</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
shown below.
<br class="newline"/>
 <br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="8+10*x/3-65*x^2/36+7*x^3/36"/>
 <param name="gridLines" value="6"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="12"/>
</applet>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math> is plotted on the
horizontal axis, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>v</mi>
 </math>
on the vertical axis. </p>@
qu.1.2.weighting=1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.extra=@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.question=<p class="noindent">What is the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>v</mi>
 </math>
when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>?
<1></p>@
qu.1.2.part.1.blank.1=8%3f0.25@
qu.1.2.part.1.grader.1=formula@
qu.1.2.part.1.mode=Blanks@
qu.1.2.part.1.comment=<p class="noindent">At <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>, we see
that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>v</mi>
  <mo>=</mo>
  <mn>8</mn>
 </math>.</p>@
qu.1.2.part.2.extra=@
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.question=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>v</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>=<1> to the nearest integer. </p>@
qu.1.2.part.2.blank.1=%24%7bansb%7d %3f%0a0.25@
qu.1.2.part.2.grader.1=formula@
qu.1.2.part.2.mode=Blanks@
qu.1.2.part.2.comment=<p class="noindent">When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>${x}</mn>
 </math>, we
see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>v</mi>
  <mo>=</mo>
  <mn>${ansb}</mn>
 </math>.</p>@

qu.1.3.mode=Sketch@
qu.1.3.name=1.3.1.1 d@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=-4.5+rint(4)*0.2;
$b=-3+rint(4)*0.2;
$c=-1+rint(4)*0.2;
$d=2+rint(4)*0.2;
$f=4;
$g=5-rint(3)*0.25;@
qu.1.3.uid=95911179-7b28-4873-aad3-a0ffa25b4dd1@
qu.1.3.question=<p class="noindent">Sketch the graph of a function which is increasing on the intervals
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${f}</mn>
    <mo>,</mo>
    <mn>${g}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> but
decreasing on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
    <mo>,</mo>
    <mn>${d}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.1.3.example=${a},-4 ${b},-2 ${c},-2 ${d},-4, ${f},-5 ${g},-2@
qu.1.3.answer=check(( increasing(restriction($1,${a},${b})) ) && ( decreasing(restriction($1,${c},${d})) ) && ( increasing(restriction($1,${f},${g})) ))@
qu.1.3.axes=-5,5,-5,5@
qu.1.3.gridlines=5@
qu.1.3.axes.labeled=true@

qu.1.4.mode=Plain Number@
qu.1.4.name=1.3.1.2@
qu.1.4.comment=<p class="noindent">Looking at the graph, we see that the point on the graph with
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>-coordinate
of ${b} has a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
coordinate of ${f}. Thus

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>f</mi>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${b}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${f}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=int(rint(3)+1);
$b=int(rint(5)+1);
$f=decimal(2,$a/$b);@
qu.1.4.uid=76d7e93e-ddf2-4346-876a-b788c359b2e6@
qu.1.4.question=<p class="noindent">Given the following graph for a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
 </math>
find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${b}</mn>
   </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="${a}/x"/>
 <param name="gridLines" value="10"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="10"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="10"/>
</applet>@
qu.1.4.answer=
${f} ? 0.05@

qu.1.5.mode=Sketch@
qu.1.5.name=1.3.1.3@
qu.1.5.comment=<p class="noindent">The contamination is probably greatest right at the tank, at a depth of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>6</mn>
 </math>
meters. Contamination probably goes down as the distance from the tank increases. If we
assume that the gas spreads both up and down, the graph should look similar to the one
shown.</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=@
qu.1.5.uid=01f0d5c7-bd63-44d0-9f2a-619b849e1649@
qu.1.5.question=<p class="noindent">A gas tank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>6</mn>
 </math>
meters underground springs a leak. Gas seeps out and contaminates the soil around
it.
<br class="newline"/>
 <br class="newline"/>Graph the amount of contamination as a function of the depth (in meters) below ground.
<br class="newline"/>
 <br class="newline"/>Contamination, as a percentage, is plotted on the vertical axis, distance below ground on the
horizontal axis.)</p>@
qu.1.5.example=0,0 1,10 3,50 6,100 9,50 11,10 12,0@
qu.1.5.answer=check(( increasing(restriction($1,0,5.5)) ) && ( decreasing(restriction($1,6.5,12)) ) && ( concave_up(restriction($1,0,3)) ))@
qu.1.5.axes=0,12,0,100@
qu.1.5.gridlines=4@
qu.1.5.axes.labeled=true@

qu.1.6.mode=Multipart@
qu.1.6.name=1.3.1.4@
qu.1.6.comment=@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$x=int(rint(3)+1);
$a=int(rint(7)+2);
$ansb=$a+10*($x)/3-65*($x)^2/36+7*($x)^3/36;
$x1=int(rint(3)+1);@
qu.1.6.uid=79a942fb-910d-49de-a7e6-a2cc3e5f4113@
qu.1.6.question=<p class="noindent">Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>r</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>p</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
show below.
<br class="newline"/>
 <br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="${a}+10*x/3-65*x^2/36+7*x^3/36"/>
 <param name="gridLines" value="6"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="12"/>
</applet>
<p class="noindent">The independent variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>p</mi>
 </math> is
plotted on the horizontal axis, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>r</mi>
 </math>
is plotted on the vertical axis. </p>@
qu.1.6.weighting=1,1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.question=<p class="noindent">What is the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>r</mi>
 </math>
when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>p</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>?
</p>@
qu.1.6.part.1.answer=
${a}@
qu.1.6.part.1.mode=Plain Number@
qu.1.6.part.1.comment=<p class="noindent">At <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>p</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>, we see
that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>r</mi>
  <mo>=</mo>
  <mn>${a}</mn>
 </math>.</p>@
qu.1.6.part.2.editing=useHTML@
qu.1.6.part.2.question=<p class="noindent">What is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>?
</p>@
qu.1.6.part.2.answer=
${ansb} ? 0.1@
qu.1.6.part.2.mode=Plain Number@
qu.1.6.part.2.comment=<p class="noindent">When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>p</mi>
  <mo>=</mo>
  <mn>${x}</mn>
 </math>, we
see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>r</mi>
  <mo>=</mo>
  <mn>${ansb}</mn>
 </math>.</p>@

qu.1.7.mode=Multipart@
qu.1.7.name=1.3.1.5@
qu.1.7.comment=@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$a=int(rint(2)+2);
$b=int(rint(3)+1);
$x=int(rint(3)+1);
$ansb=int($a*$x*($x)+ $b);
$xc=int($x+1);
$xcm=int(-$xc);
$c1=int($xc*$xc);
$ansc=int($a*($xc)*($xc) + $b);
$qd=int($b-1);
$xmax=int(2*$xc);
$ymax=int($a*$xmax*$xmax + $b + 2);@
qu.1.7.uid=884317eb-c448-4618-aa40-b5f0596a163c@
qu.1.7.question=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
  <mo>+</mo>
  <mn>${b}</mn>
 </math>.
</p>@
qu.1.7.weighting=1,1,1,1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.question=<p class="noindent">Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
is zero. </p>@
qu.1.7.part.1.answer=
${b}? 0.05@
qu.1.7.part.1.mode=Plain Number@
qu.1.7.part.1.comment=<p class="noindent">We are asked for the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math> is zero. That
is, we are asked for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>0</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.
Plugging in we get
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>f</mi>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>0</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${a}</mn>
 <msup>
  <mrow>
   <mrow>
    <mo stretchy="false">(</mo>
    <mrow>
     <mn>0</mn>
    </mrow>
    <mo stretchy="false">)</mo>
   </mrow>
  </mrow>
  <mrow>
   <mn>2</mn>
  </mrow>
 </msup>
 <mo>+</mo>
 <mn>${b}</mn>
 <mo>=</mo>
 <mn>0</mn>
 <mo>+</mo>
 <mn>${b}</mn>
 <mo>=</mo>
 <mn>${b}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.7.part.2.editing=useHTML@
qu.1.7.part.2.question=<p class="noindent">What is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>?
</p>@
qu.1.7.part.2.answer=
${ansb} ? 0.05@
qu.1.7.part.2.mode=Plain Number@
qu.1.7.part.2.comment=<p class="noindent">Substituting we get

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>f</mi>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${x}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${a}</mn>
 <msup>
  <mrow>
   <mrow>
    <mo stretchy="false">(</mo>
    <mrow>
     <mn>${x}</mn>
    </mrow>
    <mo stretchy="false">)</mo>
   </mrow>
  </mrow>
  <mrow>
   <mn>2</mn>
  </mrow>
 </msup>
 <mo>+</mo>
 <mn>${b}</mn>
 <mo>=</mo>
 <mn>${ansb}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.7.part.3.editing=useHTML@
qu.1.7.part.3.question=<p class="noindent">Which values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
give <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math> a
value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${ansc}</mn>
 </math>?
</p>@
qu.1.7.part.3.answer=
${xc};${xcm} @
qu.1.7.part.3.mode=Multi Formula@
qu.1.7.part.3.comment=<p class="noindent">Asking what values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
give a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-value
of 11 is the same as solving
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtable columnspacing="0" columnalign="right center left">
  <mtr>
   <mtd>
    <mi>y</mi>
    <mo>=</mo>
    <mn>${ansc}</mn>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${a}</mn>
    <msup>
     <mrow>
      <mi>x</mi>
     </mrow>
     <mrow>
      <mn>2</mn>
     </mrow>
    </msup>
    <mo>+</mo>
    <mn>${b}</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <msup>
     <mrow>
      <mi>x</mi>
     </mrow>
     <mrow>
      <mn>2</mn>
     </mrow>
    </msup>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${c1}</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mi>x</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mo>&plusmn;</mo>
    <mn>${xc}</mn>
    <mo>.</mo>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
 </mtable>
</math>
<p class="nopar">
We can also solve this problem graphically. Looking at the graph of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
  <mo>+</mo>
  <mn>${b}</mn>
 </math>
below
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules/">
 <param name="y1" value="${a}*x*x+${b}"/>
 <param name="gridLines" value="1"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="${xmax}"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="${ymax}"/>
</applet>
<p class="noindent">We see that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
intersects the line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${ansc}</mn>
 </math>
at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${xc}</mn>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mo>&minus;</mo>
  <mn>${xc}</mn>
 </math>. Thus,
when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${xc}</mn>
 </math> or
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math> equals
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mo>&minus;</mo>
  <mn>${xc}</mn>
 </math> we
have <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>${ansc}</mn>
 </math>.</p>@
qu.1.7.part.4.extra=@
qu.1.7.part.4.editing=useHTML@
qu.1.7.part.4.question=<p class="noindent">Are there any values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
that give <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math> a
value of ${qd}?
<br class="newline"/><1></p>@
qu.1.7.part.4.blank.1=No, yes@
qu.1.7.part.4.grader.1=menu@
qu.1.7.part.4.mode=Blanks@
qu.1.7.part.4.comment=<p class="noindent">No. No matter what, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
 </math> is
greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
 </math>,
so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
  <mo>+</mo>
  <mn>2</mn>
 </math> is greater
than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>2</mn>
 </math>.</p>@

qu.1.8.mode=Sketch@
qu.1.8.name=1.3.1.1 c@
qu.1.8.comment=@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$a=-4.5+rint(4)*0.2;
$b=-3+rint(4)*0.2;
$c=-1+rint(4)*0.2;
$d=1+rint(4)*0.2;
$f=3;
$g=5-rint(3)*0.25;@
qu.1.8.uid=6a7e3bd6-e04c-4960-8914-cfd8a39275a8@
qu.1.8.question=<p class="noindent">Sketch the graph of a function which is decreasing on the intervals
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${f}</mn>
    <mo>,</mo>
    <mn>${g}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> but increasing
on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
    <mo>,</mo>
    <mn>${d}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.1.8.example=${a},4 ${b},2 ${c},2 ${d},4, ${f},5 ${g},2@
qu.1.8.answer=check(( decreasing(restriction($1,${a},${b})) ) && ( increasing(restriction($1,${c},${d})) ) && ( decreasing(restriction($1,${f},${g})) ))@
qu.1.8.axes=-5,5,-5,5@
qu.1.8.gridlines=5@
qu.1.8.axes.labeled=true@

qu.1.9.mode=Sketch@
qu.1.9.name=1.3.1.1 b@
qu.1.9.comment=@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$a=-3.5+rint(10)*0.1;
$b=-1+rint(10)*0.1;
$c=1.5+rint(10)*0.1;
$d=4.5-rint(10)*0.1;@
qu.1.9.uid=fd5e5647-6404-4d0e-a12f-d9432f17b816@
qu.1.9.question=<p class="noindent">Sketch the graph of a function which is decreasing on the interval
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and increasing
on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
    <mo>,</mo>
    <mn>${d}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.1.9.example=${a},4 ${b},2 ${c},2 ${d},4@
qu.1.9.answer=check(( decreasing(restriction($1,${a},${b})) ) && ( increasing(restriction($1,${c},${d})) ))@
qu.1.9.axes=-5,5,-5,5@
qu.1.9.gridlines=5@
qu.1.9.axes.labeled=true@

qu.2.topic=Section 1.3.2: Linear functions@

qu.2.1.mode=Blanks@
qu.2.1.name=1.3.2.2@
qu.2.1.comment=<p class="noindent">A graph 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>
is shown below.
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules/">
 <param name="y1" value="${m}*(x-${b})*(x-${b})"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet>
<p class="noindent">This is not a straight line.
</p>
<p class="noindent">The slope at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${b}</mn>
 </math> is zero
and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-intercept
is at ${c}.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$m=switch(rint(6),-4,-3,-2,2,3,4,5);
$b=rint(4)+1;
$c=$m*$b^2;@
qu.2.1.uid=ec44eda1-df77-4c2d-9cd9-e313c94a22a1@
qu.2.1.question=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${m}</mn>
  <msup>
   <mrow>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mi>x</mi>
      <mo>&minus;</mo>
      <mn>${b}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
 </math>
 <br class="newline"/>
 <br class="newline"/>
</p>
<p class="noindent">This is <1> line.

<br class="newline"/>
 <br class="newline"/>
The slope at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${b}</mn>
 </math> is
<2> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-intercept
<3>.
</p>@
qu.2.1.blank.1=not a straight, a straight@
qu.2.1.blank.2=0@
qu.2.1.blank.3=%24%7bc%7d%3f0.5@
qu.2.1.grader.1=menu@
qu.2.1.grader.2=formula@
qu.2.1.grader.3=formula@
qu.2.1.extra=@

qu.2.2.mode=Blanks@
qu.2.2.name=1.3.2.3@
qu.2.2.comment=<p class="noindent">A graph 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>
is shown below.
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules/">
 <param name="y1" value="${m}*(x+${b})*(x+${b})"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet>
<p class="noindent">This is not a straight line.
</p>
<p class="noindent">The slope at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mo>&minus;</mo>
  <mn>${b}</mn>
 </math> is zero
and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-intercept
is at ${c}.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$m=switch(rint(6),-4,-3,-2,2,3,4,5);
$b=rint(4)+1;
$c=$m*$b^2;@
qu.2.2.uid=d725a9dd-fd18-44e3-a4c3-1b82fc4bf464@
qu.2.2.question=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${m}</mn>
  <msup>
   <mrow>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mi>x</mi>
      <mo>+</mo>
      <mn>${b}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
 </math>
 <br class="newline"/>
 <br class="newline"/>
</p>
<p class="noindent">This is <1> line.
<br class="newline"/>
 <br class="newline"/>
The slope at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mo>&minus;</mo>
  <mn>${b}</mn>
 </math> is
<2> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-intercept
<3>.
</p>@
qu.2.2.blank.1=not a straight, a straight@
qu.2.2.blank.2=0@
qu.2.2.blank.3=%24%7bc%7d%3f0.5@
qu.2.2.grader.1=menu@
qu.2.2.grader.2=formula@
qu.2.2.grader.3=formula@
qu.2.2.extra=@

qu.2.3.mode=Multipart@
qu.2.3.name=1.3.2.1@
qu.2.3.comment=@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$m=(switch(rint(4),1.5,2,3,1.25,2));
$b=(switch(rint(4),1.5,2,3,4));
$f1=decimal(2,-4*$m+$b);
$f2=decimal(2,4*$m+$b);@
qu.2.3.uid=29bba1e3-92c1-44eb-9b38-665d20be0003@
qu.2.3.question=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${m}</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>${b}</mn>
 </math>
</p>@
qu.2.3.weighting=1,1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.blank.3=%24%7bb%7d@
qu.2.3.part.1.blank.2=%24%7bm%7d@
qu.2.3.part.1.blank.1=a straight, not a straight@
qu.2.3.part.1.extra=@
qu.2.3.part.1.editing=useHTML@
qu.2.3.part.1.question=<p class="noindent">It is <1> line.
It is an example of the general form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>m</mi>
  <mi>x</mi>
  <mo>+</mo>
  <mi>c</mi>
 </math>,
with slope <2> and intercept <3>.</p>@
qu.2.3.part.1.comment=<p class="noindent">It is a straight line because it is an example of the general form
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>m</mi>
  <mi>x</mi>
  <mo>+</mo>
  <mi>c</mi>
 </math>, with
slope <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
  <mo>=</mo>
  <mn>${m}</mn>
 </math> and
intercept <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>b</mi>
  <mo>=</mo>
  <mn>${b}</mn>
 </math>.</p>@
qu.2.3.part.1.mode=Blanks@
qu.2.3.part.1.grader.3=formula@
qu.2.3.part.1.grader.2=formula@
qu.2.3.part.1.grader.1=menu@
qu.2.3.part.2.example=-4,${f1} 4,${f2}@
qu.2.3.part.2.editing=useHTML@
qu.2.3.part.2.question=<p class="noindent">Sketch the graph of the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.</p>@
qu.2.3.part.2.axes.labeled=true@
qu.2.3.part.2.answer=check(( linear($1) ) && ( increasing($1) ) && ( goes_through($1,0,${b}) ))@
qu.2.3.part.2.axes=-4,4,-4,4@
qu.2.3.part.2.gridlines=4@
qu.2.3.part.2.mode=Sketch@

qu.3.topic=Section 1.3.3: Slopes and equations@

qu.3.1.mode=Equation@
qu.3.1.name=1.3.3.2@
qu.3.1.comment=<p class="noindent">The equation of a straight line is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>m</mi>
  <mi>x</mi>
  <mo>+</mo>
  <mi>b</mi>
  <mo>,</mo>
 </math>
where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math> is the
slope and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>b</mi>
 </math> is the
intercept. We have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
  <mo>=</mo>
  <mn>${m}</mn>
 </math>
so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${m}</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mi>b</mi>
 </math>.
Substituting, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${a}</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${c}</mn>
 </math>
gives
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mn>${c}</mn>
 <mo>=</mo>
 <mn>${m}</mn>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${a}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>+</mo>
 <mi>b</mi>
</math>
<p class="nopar"> so

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>b</mi>
 <mo>=</mo>
 <mn>${c}</mn>
 <mn>${sign}</mn>
 <mn>${ma1}</mn>
 <mo>.</mo>
</math>
<p class="nopar"> The equation of the line with slope <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${m}</mn>
 </math>
passing through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${c}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
is
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>y</mi>
 <mo>=</mo>
 <mn>${m}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${c}</mn>
 <mn>${sign}</mn>
 <mn>${ma}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$m=int(2+rint(5));
$a=int(switch(rint(5),4,2,1,1,2,3));
$ma=int($a*$m);
$ma1=if(gt($ma,0),$ma,-$ma);
$sign=if(gt($ma,0),"-","+");
$c=switch(rint(3),"A","L","c","g");
$b=$c-int($m*$a);@
qu.3.1.uid=6c34533d-3244-4ea5-ac1e-21ed28129097@
qu.3.1.question=<p class="noindent">Find the equation of the line with slope <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${m}</mn>
 </math>
through the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${c}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.
</p>@
qu.3.1.answer=
y=${m}x+${b} @

qu.3.2.mode=Equation@
qu.3.2.name=1.3.3.4@
qu.3.2.comment=<p class="noindent">The equation of a straight line is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>m</mi>
  <mi>x</mi>
  <mo>+</mo>
  <mi>b</mi>
  <mo>,</mo>
 </math>
where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math> is the
slope and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>b</mi>
 </math>
is the intercept.
</p>
<p class="noindent">The slope is given by
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
  <mo>=</mo>
  <mfrac>
   <mrow>
    <mn>${y1}</mn>
    <mo>&minus;</mo>
    <mn>${y0}</mn>
   </mrow>
   <mrow>
    <mn>${x1}</mn>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>${m}</mn>
  <mo>.</mo>
 </math>
</p>
<p class="noindent">The equation of the line is given by
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>&minus;</mo>
  <mn>${y0}</mn>
  <mo>=</mo>
  <mn>${m}</mn>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
</p>
<p class="noindent">or
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${m}</mn>
  <mi>x</mi>
  <mo>&minus;</mo>
  <mn>${b}</mn>
  <mo>.</mo>
 </math>
</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$m=int(2+rint(5));
$b=rint(5);
$x0=int(rint(3));
$y0=int($m*$x0-$b);
$x1=int($x0+1+rint(3));
$y1=int($m*$x1-$b);@
qu.3.2.uid=e172e7e5-6eb9-4d51-bbc5-e72fc9c857fd@
qu.3.2.question=<p class="noindent">Find the equation of the line passing through the points
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x0}</mn>
    <mo>,</mo>
    <mn>${y0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.
</p>@
qu.3.2.answer=
y=${m}x-${b} @

qu.3.3.mode=Sketch@
qu.3.3.name=1.3.3.8@
qu.3.3.comment=@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$a=-4.5+rint(4)*0.2;
$b=-3+rint(4)*0.2;
$c=-1+rint(4)*0.2;
$d=1+rint(4)*0.2;
$f=3;
$g=5-rint(3)*0.25;@
qu.3.3.uid=c7f9a411-c0a5-4760-b3ae-00414ae78123@
qu.3.3.question=<p class="noindent">Sketch the graph of a function which is decreasing on the intervals
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${f}</mn>
    <mo>,</mo>
    <mn>${g}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> but increasing
on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
    <mo>,</mo>
    <mn>${d}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.3.3.example=${a},4 ${b},2 ${c},2 ${d},4, ${f},5 ${g},2@
qu.3.3.answer=check(( decreasing(restriction($1,${a},${b})) ) && ( increasing(restriction($1,${c},${d})) ) && ( decreasing(restriction($1,${f},${g})) ))@
qu.3.3.axes=-5,5,-5,5@
qu.3.3.gridlines=5@
qu.3.3.axes.labeled=true@

qu.3.4.mode=Equation@
qu.3.4.name=1.3.3.3@
qu.3.4.comment=<p class="noindent">The equation of a straight line is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>m</mi>
  <mi>x</mi>
  <mo>+</mo>
  <mi>b</mi>
  <mo>,</mo>
 </math>
where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math> is the
slope and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>b</mi>
 </math>

is the intercept.
</p>
<p class="noindent">The slope is given by
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
  <mo>=</mo>
  <mfrac>
   <mrow>
    <mn>${y1}</mn>
    <mo>&minus;</mo>
    <mn>${y0}</mn>
   </mrow>
   <mrow>
    <mn>${x1}</mn>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>${m}</mn>
  <mo>.</mo>
 </math>
</p>
<p class="noindent">The equation of the line is given by
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>&minus;</mo>
  <mn>${y0}</mn>
  <mo>=</mo>
  <mn>${m}</mn>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
</p>
<p class="noindent">or
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${m}</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>${b}</mn>
  <mo>.</mo>
 </math>
</p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$m=int(2+rint(5));
$b=rint(5);
$x0=int(rint(3));
$y0=int($m*$x0+$b);
$x1=int($x0+1+rint(3));
$y1=int($m*$x1+$b);@
qu.3.4.uid=9e8e8ee1-3b2a-4bbd-89d6-4abcfc974010@
qu.3.4.question=<p class="noindent">Find the equation of the line passing through the points
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x0}</mn>
    <mo>,</mo>
    <mn>${y0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.
</p>@
qu.3.4.answer=
y=${m}x+${b} @

qu.3.5.mode=Sketch@
qu.3.5.name=1.3.3.7@
qu.3.5.comment=@
qu.3.5.editing=useHTML@
qu.3.5.solution=@
qu.3.5.algorithm=$a=-3.5+rint(10)*0.1;
$b=-1+rint(10)*0.1;
$c=1.5+rint(10)*0.1;
$d=4.5-rint(10)*0.1;@
qu.3.5.uid=7d3fe330-d144-41c6-ad68-856ad14f269f@
qu.3.5.question=<p class="noindent">Sketch the graph of a function which is decreasing on the interval
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and increasing
on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
    <mo>,</mo>
    <mn>${d}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.3.5.example=${a},4 ${b},2 ${c},2 ${d},4@
qu.3.5.answer=check(( decreasing(restriction($1,${a},${b})) ) && ( increasing(restriction($1,${c},${d})) ))@
qu.3.5.axes=-5,5,-5,5@
qu.3.5.gridlines=5@
qu.3.5.axes.labeled=true@

qu.3.6.mode=Blanks@
qu.3.6.name=1.3.3.1@
qu.3.6.comment=<p class="noindent">Rewriting the equation as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${a}</mn>
  <mi>y</mi>
  <mo>=</mo>
  <mo>&minus;</mo>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${b}</mn>
    <mi>x</mi>
    <mo>+</mo>
    <mn>${c}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
and then dividing both sides by ${a} gives

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>y</mi>
 <mo>=</mo>
 <mo>&minus;</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${b}</mn>
   </mrow>
   <mrow>
    <mn>${a}</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mi>x</mi>
 <mo>&minus;</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${c}</mn>
   </mrow>
   <mrow>
    <mn>${a}</mn>
   </mrow>
  </mfrac>
 </mstyle>
</math>
<p class="nopar"> Therefore, the slope is ${slope} and the vertical intercept is ${cross}. </p>@
qu.3.6.editing=useHTML@
qu.3.6.solution=@
qu.3.6.algorithm=$a=int(2+rint(4));
$b=int(2+rint(4));
$c=int(2+rint(4));
$slope=decimal(4,-$b/$a);
$cross=decimal(4,-$c/$a);@
qu.3.6.uid=71ef68b6-2bf0-40ad-a4b2-1156fa6280b6@
qu.3.6.question=<p class="noindent">Find the slope and vertical intercept of the line whose equation is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${a}</mn>
  <mi>y</mi>
  <mo>+</mo>
  <mn>${b}</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>${c}</mn>
  <mo>=</mo>
  <mn>0</mn>
 </math>.
<br class="newline"/>
 <br class="newline"/>Slope=<1>.
<br class="newline"/>
 <br class="newline"/>Intercept=<2>. </p>@
qu.3.6.blank.1=%24%7bslope%7d%3f0.0005@
qu.3.6.blank.2=%24%7bcross%7d%3f0.0005@
qu.3.6.grader.1=formula@
qu.3.6.grader.2=formula@
qu.3.6.extra=@

qu.3.7.mode=Sketch@
qu.3.7.name=1.3.3.9@
qu.3.7.comment=@
qu.3.7.editing=useHTML@
qu.3.7.solution=@
qu.3.7.algorithm=$a=-4.5+rint(4)*0.2;
$b=-3+rint(4)*0.2;
$c=-1+rint(4)*0.2;
$d=2+rint(4)*0.2;
$f=4;
$g=5-rint(3)*0.25;@
qu.3.7.uid=94dd0d23-1da5-4de4-a1bc-d0fd8f21ff3e@
qu.3.7.question=<p class="noindent">Sketch the graph of a function which is increasing on the intervals
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${f}</mn>
    <mo>,</mo>
    <mn>${g}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> but
decreasing on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
    <mo>,</mo>
    <mn>${d}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.3.7.example=${a},-4 ${b},-2 ${c},-2 ${d},-4, ${f},-5 ${g},-2@
qu.3.7.answer=check(( increasing(restriction($1,${a},${b})) ) && ( decreasing(restriction($1,${c},${d})) ) && ( increasing(restriction($1,${f},${g})) ))@
qu.3.7.axes=-5,5,-5,5@
qu.3.7.gridlines=5@
qu.3.7.axes.labeled=true@

qu.3.8.mode=Equation@
qu.3.8.name=1.3.3.5@
qu.3.8.comment=<p class="noindent">The equation of a straight line is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>m</mi>
  <mi>x</mi>
  <mo>+</mo>
  <mi>b</mi>
  <mo>,</mo>
 </math>
where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math> is the
slope and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>b</mi>
 </math>
is the intercept.
</p>
<p class="noindent">The slope is given by
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
  <mo>=</mo>
  <mfrac>
   <mrow>
    <mn>${y1}</mn>
    <mo>&minus;</mo>
    <mn>${y0}</mn>
   </mrow>
   <mrow>
    <mn>${x1}</mn>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
  </mfrac>
  <mo>=</mo>
  <mn>${m}</mn>
  <mo>.</mo>
 </math>
</p>
<p class="noindent">The equation of the line is given by
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>&minus;</mo>
  <mn>${y0}</mn>
  <mo>=</mo>
  <mn>${m}</mn>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
</p>
<p class="noindent">or
</p>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${m}</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>${b}</mn>
  <mo>.</mo>
 </math>
</p>@
qu.3.8.editing=useHTML@
qu.3.8.solution=@
qu.3.8.algorithm=$m=int(-2-rint(5));
$b=rint(5);
$x0=int(rint(3));
$y0=int($m*$x0+$b);
$x1=int($x0+1+rint(3));
$y1=int($m*$x1+$b);@
qu.3.8.uid=0f3317fd-9d86-43a7-abea-044978fcfa33@
qu.3.8.question=<p class="noindent">Find the equation of the line passing through the points
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x0}</mn>
    <mo>,</mo>
    <mn>${y0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.
</p>@
qu.3.8.answer=

y=${m}x+${b} @

qu.3.9.mode=Sketch@
qu.3.9.name=1.3.3.6@
qu.3.9.comment=@
qu.3.9.editing=useHTML@
qu.3.9.solution=@
qu.3.9.algorithm=$a=-3.5+rint(10)*0.1;
$b=-1+rint(10)*0.1;
$c=1.5+rint(10)*0.1;
$d=4.5-rint(10)*0.1;@
qu.3.9.uid=262fcf86-ffba-4882-a6f3-711a64cbd93d@
qu.3.9.question=<p class="noindent">Sketch the graph of a function which is increasing on the interval
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
decreasing on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
    <mo>,</mo>
    <mn>${d}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.3.9.example=${a},-1 ${b},2 ${c},2 ${d},-1@
qu.3.9.answer=check(( increasing(restriction($1,${a},${b})) ) && ( decreasing(restriction($1,${c},${d})) ))@
qu.3.9.axes=-5,5,-5,5@
qu.3.9.gridlines=5@
qu.3.9.axes.labeled=true@

qu.4.topic=Section 1.3.4: Concavity@

qu.4.1.mode=Sketch@
qu.4.1.name=1.3.4.1 b@
qu.4.1.comment=@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=-3.5+rint(3)*0.5;
$b=4.5-rint(3)*0.5;
$ab=($a+$b)/2;@
qu.4.1.uid=6f1bc9c0-8987-4009-8b44-e05a98e7286c@
qu.4.1.question=<p class="noindent">Sketch the graph of a function which is concave down on the interval
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.4.1.example=${a},1 ${ab},3 ${b},1@
qu.4.1.answer=check(( concave_down(restriction($1,${a},${b})) ))@
qu.4.1.axes=-5,5,-5,5@
qu.4.1.gridlines=5@
qu.4.1.axes.labeled=true@

qu.4.2.mode=Sketch@
qu.4.2.name=1.3.4.1 a@
qu.4.2.comment=@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=-3.5+rint(3)*0.5;
$b=2+rint(3)*0.5;
$ab=($a+$b)/2;@
qu.4.2.uid=2b6626c0-8730-4f64-bbc9-24499aea7080@
qu.4.2.question=<p class="noindent">Sketch the graph of a function which is concave up on the interval
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${a}</mn>
    <mo>,</mo>
    <mn>${b}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.4.2.example=${a},-1 ${ab},-3 ${b},-1@
qu.4.2.answer=check(( concave_up(restriction($1,${a},${b})) ))@
qu.4.2.axes=-5,5,-5,5@
qu.4.2.gridlines=5@
qu.4.2.axes.labeled=true@

qu.5.topic=Section 1.3.5: Factoring quadratics@

qu.5.1.mode=Inline@
qu.5.1.name=1.3.5.2@
qu.5.1.comment=<p class="noindent">The curve cuts the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
axis at ${a}, ${b} and ${c}. These are the intercepts.</p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$a=int(rint(2));
$b=$a+1+rint(2);
$c=-($b+1+rint(2));@
qu.5.1.uid=57a86e40-430d-4ede-9760-fc8e23c75428@
qu.5.1.weighting=1,1,1@
qu.5.1.numbering=alpha@
qu.5.1.part.1.answer.units=@
qu.5.1.part.1.numStyle=thousands scientific dollars arithmetic@
qu.5.1.part.1.editing=useHTML@
qu.5.1.part.1.showUnits=false@
qu.5.1.part.1.err=0.1@
qu.5.1.part.1.question=xx@
qu.5.1.part.1.mode=Numeric@
qu.5.1.part.1.grading=toler_abs@
qu.5.1.part.1.negStyle=minus@
qu.5.1.part.1.answer.num=${a}@
qu.5.1.part.2.answer.units=@
qu.5.1.part.2.numStyle=thousands scientific dollars arithmetic@
qu.5.1.part.2.editing=useHTML@
qu.5.1.part.2.showUnits=false@
qu.5.1.part.2.err=0.1@
qu.5.1.part.2.question=xx@
qu.5.1.part.2.mode=Numeric@
qu.5.1.part.2.grading=toler_abs@
qu.5.1.part.2.negStyle=minus@
qu.5.1.part.2.answer.num=${b}@
qu.5.1.part.3.answer.units=@
qu.5.1.part.3.numStyle=thousands scientific dollars arithmetic@
qu.5.1.part.3.editing=useHTML@
qu.5.1.part.3.showUnits=false@
qu.5.1.part.3.err=0.1@
qu.5.1.part.3.question=xx@
qu.5.1.part.3.mode=Numeric@
qu.5.1.part.3.grading=toler_abs@
qu.5.1.part.3.negStyle=minus@
qu.5.1.part.3.answer.num=${c}@
qu.5.1.question=<p class="noindent">What are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>-intercepts
of the graph shown?
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="(x-${a})(x-${b})(x-${c})"/>
 <param name="gridLines" value="10"/>
 <param name="xMin" value="-10"/>
 <param name="xMax" value="10"/>
 <param name="yMin" value="-10"/>
 <param name="yMax" value="10"/>
</applet>
<p class="noindent">The intercept, in increasing order of magnitude, are
<1><2><3></p>@

qu.5.2.mode=Inline@
qu.5.2.name=1.3.5.3@
qu.5.2.comment=<p class="noindent">The curve cuts the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
axis at ${a}, this is the intercept.
</p>
<p class="noindent">The slope of the line is approximately ${b}.</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$a=int(rint(2));
$b=switch(rint(2),rint(5)+1,int(-1-rint(4)));@
qu.5.2.uid=efb47509-d461-4303-ae63-82baf185822a@
qu.5.2.weighting=1,1@
qu.5.2.numbering=alpha@
qu.5.2.part.1.answer.units=@
qu.5.2.part.1.numStyle=thousands scientific dollars arithmetic@
qu.5.2.part.1.editing=useHTML@
qu.5.2.part.1.showUnits=false@
qu.5.2.part.1.err=0.1@
qu.5.2.part.1.question=xx@
qu.5.2.part.1.mode=Numeric@
qu.5.2.part.1.grading=toler_abs@
qu.5.2.part.1.negStyle=minus@
qu.5.2.part.1.answer.num=${a}@
qu.5.2.part.2.answer.units=@
qu.5.2.part.2.numStyle=thousands scientific dollars arithmetic@
qu.5.2.part.2.editing=useHTML@
qu.5.2.part.2.showUnits=false@
qu.5.2.part.2.err=0.1@
qu.5.2.part.2.question=xx@
qu.5.2.part.2.mode=Numeric@
qu.5.2.part.2.grading=toler_abs@
qu.5.2.part.2.negStyle=minus@
qu.5.2.part.2.answer.num=${b}@
qu.5.2.question=<p class="noindent">What is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>-intercept
of the graph shown?
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="${b}(x-${a})"/>
 <param name="gridLines" value="10"/>
 <param name="xMin" value="-10"/>
 <param name="xMax" value="10"/>
 <param name="yMin" value="-10"/>
 <param name="yMax" value="10"/>
</applet>
<p class="noindent">The intercept is at <1>
The slope is <2></p>@

qu.5.3.mode=Inline@
qu.5.3.name=1.3.5.1@
qu.5.3.comment=<p class="noindent">The curve cuts the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
axis at ${a} and ${b}. These are the intercepts.</p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$a=int(rint(5)-4);
$b=$a+4+rint(2);@
qu.5.3.uid=e83784de-9af5-4762-bd92-8517dd27adc1@
qu.5.3.weighting=1,1@
qu.5.3.numbering=alpha@
qu.5.3.part.1.answer.units=@
qu.5.3.part.1.numStyle=thousands scientific dollars arithmetic@
qu.5.3.part.1.editing=useHTML@
qu.5.3.part.1.showUnits=false@
qu.5.3.part.1.err=0.1@
qu.5.3.part.1.question=xx@
qu.5.3.part.1.mode=Numeric@
qu.5.3.part.1.grading=toler_abs@
qu.5.3.part.1.negStyle=minus@
qu.5.3.part.1.answer.num=${a}@
qu.5.3.part.2.answer.units=@
qu.5.3.part.2.numStyle=thousands scientific dollars arithmetic@
qu.5.3.part.2.editing=useHTML@
qu.5.3.part.2.showUnits=false@
qu.5.3.part.2.err=0.01@
qu.5.3.part.2.question=xx@
qu.5.3.part.2.mode=Numeric@
qu.5.3.part.2.grading=toler_abs@
qu.5.3.part.2.negStyle=minus@
qu.5.3.part.2.answer.num=${b}@
qu.5.3.question=<p class="noindent">What are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>-intercepts
of the graph shown?
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="(x-${a})(x-${b})"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet>
<p class="noindent">The negative intercept is at <1>
The other intercept is at <2></p>@

qu.6.topic=Section 1.3.6: Turning points@

qu.6.1.mode=Inline@
qu.6.1.name=1.3.6.1@
qu.6.1.comment=<p class="noindent">The curve cuts the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
axis at ${a} and ${b}. These are the intercepts.</p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$a=switch(rint(4),-5,-4,-3,-2,-1);
$b=$a+4+rint(2);@
qu.6.1.uid=d92fa730-6613-4935-a4b1-1befae924121@
qu.6.1.weighting=1,1@
qu.6.1.numbering=alpha@
qu.6.1.part.1.answer.units=@
qu.6.1.part.1.numStyle=thousands scientific dollars arithmetic@
qu.6.1.part.1.editing=useHTML@
qu.6.1.part.1.showUnits=false@
qu.6.1.part.1.err=0.1@
qu.6.1.part.1.question=xx@
qu.6.1.part.1.mode=Numeric@
qu.6.1.part.1.grading=toler_abs@
qu.6.1.part.1.negStyle=minus@
qu.6.1.part.1.answer.num=${a}@
qu.6.1.part.2.answer.units=@
qu.6.1.part.2.numStyle=thousands scientific dollars arithmetic@
qu.6.1.part.2.editing=useHTML@
qu.6.1.part.2.showUnits=false@
qu.6.1.part.2.err=0.1@
qu.6.1.part.2.question=xx@
qu.6.1.part.2.mode=Numeric@
qu.6.1.part.2.grading=toler_abs@
qu.6.1.part.2.negStyle=minus@
qu.6.1.part.2.answer.num=${b}@
qu.6.1.question=<p class="noindent">What are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>-intercepts
of the graph shown?
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="(x-${a})(x-${b})"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet>
<p class="noindent">The negative intercept is at <1>
The other intercept is at <2></p>@

qu.6.2.mode=Blanks@
qu.6.2.name=1.3.6.3@
qu.6.2.comment=<p class="noindent">A graph 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>
is shown below.
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules/">
 <param name="y1" value="x*x-$a2*x+$c"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet>
<p class="noindent">The turning point is at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${a}</mn>
 </math>.</p>@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=$a=rint(4)+1;
$b=rint(4)+1;
$a2=int(2*$a);
$c=int($a^2+$b);@
qu.6.2.uid=1701eb86-5d18-490f-a064-a24fae5b2e9d@
qu.6.2.question=<p class="noindent">Determine the turning point of the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
  <mo>&minus;</mo>
  <mn>${a2}</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>${c}</mn>
  <mo>.</mo>
 </math>
 <br class="newline"/>
 <br class="newline"/>
</p>
<p class="noindent">The turning point is at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
 </math><1> .
</p>@
qu.6.2.blank.1=%24%7ba%7d@
qu.6.2.grader.1=formula@
qu.6.2.extra=@

qu.6.3.mode=Blanks@
qu.6.3.name=1.3.6.4@
qu.6.3.comment=<p class="noindent">A graph 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>
is shown below.
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules/">
 <param name="y1" value="x*x*x/3-($a+$b)*x*x/2+$a*$b*x+$c"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet>
<p class="noindent">The turning points are at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${a}</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${b}</mn>
 </math>.</p>@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=$a=switch(rint(3),-2,-1,1,2);
$b=(-1)^(rint(1))*(abs($a)+1+rint(2));
$c=rint(3)+1;
$a2=($a+$b)/2;
$d=$a*$b;@
qu.6.3.uid=92f74746-88ba-4ac6-a09e-566e6561524b@
qu.6.3.question=<p class="noindent">Determine the turning point of the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mfrac>
   <mrow>
    <mn>1</mn>
   </mrow>
   <mrow>
    <mn>3</mn>
   </mrow>
  </mfrac>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>3</mn>
   </mrow>
  </msup>
  <mo>&minus;</mo>
  <mn>${a2}</mn>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
  <mo>+</mo>
  <mn>${d}</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mn>${c}</mn>
  <mo>.</mo>
 </math>
</p>
<p class="noindent">Give your answer in increasing order of magnitude.
<br class="newline"/>
 <br class="newline"/>
</p>
<p class="noindent">The turning points are at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
 </math><1> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
 </math><2>.
</p>@
qu.6.3.blank.1=%24%7ba%7d@
qu.6.3.blank.2=%24%7bb%7d@
qu.6.3.grader.1=formula@
qu.6.3.grader.2=formula@
qu.6.3.extra=@

qu.6.4.mode=Blanks@
qu.6.4.name=1.3.6.5@
qu.6.4.comment=<p class="noindent">The turning points are at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${a}</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${b}</mn>
 </math>.</p>@
qu.6.4.editing=useHTML@
qu.6.4.solution=@
qu.6.4.algorithm=$a=switch(rint(3),-2,-1,1,2);
$b=(-1)^(rint(1))*(abs($a)+1+rint(2));
$c=rint(3);
$a2=($a+$b)/2;
$d=$a*$b;@
qu.6.4.uid=b143b862-8d6f-4d5f-9b79-3929a2f723b5@
qu.6.4.question=<p class="noindent">Determine the turning point of the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
shown below.
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="(x*x*x/3-($a+$b)*x*x/2+$a*$b*x+$c)/2"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet>
<p class="noindent">Give your answer in increasing order of magnitude.
<br class="newline"/>
 <br class="newline"/>
</p>
<p class="noindent">The turning points are at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
 </math><1> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
 </math><2>.
</p>@
qu.6.4.blank.1=%24%7ba%7d@
qu.6.4.blank.2=%24%7bb%7d@
qu.6.4.grader.1=formula@
qu.6.4.grader.2=formula@
qu.6.4.extra=@

qu.6.5.mode=Blanks@
qu.6.5.name=1.3.6.6@
qu.6.5.comment=<p class="noindent">The turning point is at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${a}</mn>
 </math>.</p>@
qu.6.5.editing=useHTML@
qu.6.5.solution=@
qu.6.5.algorithm=$a=switch(rint(8),-2,-4,-3,-2,-1,1,2,3,4,0);
$b=rint(4)+1;
$a2=int(2*$a);
$c=int($a^2+$b);@
qu.6.5.uid=2517c7af-f87e-43fd-b88d-3bf3dbcd24ca@
qu.6.5.question=<p class="noindent">Determine the turning point of the function shown below.
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="x*x-$a2*x+$c"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet>
<p class="noindent">The turning point is at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
 </math><1> .
</p>@
qu.6.5.blank.1=%24%7ba%7d@
qu.6.5.grader.1=formula@
qu.6.5.extra=@

qu.6.6.mode=Inline@
qu.6.6.name=1.3.6.2@
qu.6.6.comment=<p class="noindent">The curve cuts the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
axis at ${a}, ${b} and ${c}. These are the intercepts.</p>@
qu.6.6.editing=useHTML@
qu.6.6.solution=@
qu.6.6.algorithm=$a=switch(rint(2),-1,0,1);
$b=switch(rint(4),-2,-3,1,2,3);
$c=switch(rint(3),-4,-3,3,4);@
qu.6.6.uid=3915445a-daaf-481a-a6a2-f09c4e50194e@
qu.6.6.weighting=1,1,1@
qu.6.6.numbering=alpha@
qu.6.6.part.1.answer.units=@
qu.6.6.part.1.numStyle=thousands scientific dollars arithmetic@
qu.6.6.part.1.editing=useHTML@
qu.6.6.part.1.showUnits=false@
qu.6.6.part.1.err=0.1@
qu.6.6.part.1.question=xx@
qu.6.6.part.1.mode=Numeric@
qu.6.6.part.1.grading=toler_abs@
qu.6.6.part.1.negStyle=minus@
qu.6.6.part.1.answer.num=${a}@
qu.6.6.part.2.answer.units=@
qu.6.6.part.2.numStyle=thousands scientific dollars arithmetic@
qu.6.6.part.2.editing=useHTML@
qu.6.6.part.2.showUnits=false@
qu.6.6.part.2.err=0.1@
qu.6.6.part.2.question=xx@
qu.6.6.part.2.mode=Numeric@
qu.6.6.part.2.grading=toler_abs@
qu.6.6.part.2.negStyle=minus@
qu.6.6.part.2.answer.num=${b}@
qu.6.6.part.3.answer.units=@
qu.6.6.part.3.numStyle=thousands scientific dollars arithmetic@
qu.6.6.part.3.editing=useHTML@
qu.6.6.part.3.showUnits=false@
qu.6.6.part.3.err=0.1@
qu.6.6.part.3.question=xx@
qu.6.6.part.3.mode=Numeric@
qu.6.6.part.3.grading=toler_abs@
qu.6.6.part.3.negStyle=minus@
qu.6.6.part.3.answer.num=${c}@
qu.6.6.question=<p class="noindent">What are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>-intercepts
of the graph shown?
</p>
<p class="noindent">Give your answer in increasing order of magnitude.
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="(x-${a})*(x-${b})*(x-${c})"/>
 <param name="gridLines" value="12"/>
 <param name="xMin" value="-6"/>
 <param name="xMax" value="6"/>
 <param name="yMin" value="-6"/>
 <param name="yMax" value="6"/>
</applet><1><2><3>@

qu.7.topic=Section 1.3.8: Applications@

qu.7.1.mode=Equation@
qu.7.1.name=1.3.8.2@
qu.7.1.comment=<p class="noindent">This looks like an exponential function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>C</mi>
  <msup>
   <mrow>
    <mi>a</mi>
   </mrow>
   <mrow>
    <mi>t</mi>
   </mrow>
  </msup>
 </math>.
The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-intercept
is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${start}</mn>
 </math> so we have
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${start}</mn>
  <msup>
   <mrow>
    <mi>a</mi>
   </mrow>
   <mrow>
    <mi>t</mi>
   </mrow>
  </msup>
 </math>. We use
the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
to find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>a</mi>
 </math>:
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtable columnspacing="0" columnalign="right center left">
  <mtr>
   <mtd>
    <mi>y</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${start}</mn>
    <msup>
     <mrow>
      <mi>a</mi>
     </mrow>
     <mrow>
      <mi>t</mi>
     </mrow>
    </msup>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mn>${y1}</mn>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${start}</mn>
    <msup>
     <mrow>
      <mi>a</mi>
     </mrow>
     <mrow>
      <mn>${x1}</mn>
     </mrow>
    </msup>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <msup>
     <mrow>
      <mi>a</mi>
     </mrow>
     <mrow>
      <mn>${x1}</mn>
     </mrow>
    </msup>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${y1}</mn>
    <mo>&#8725;</mo>
    <mn>${start}</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mi>a</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${base}</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
 </mtable>
</math>
<p class="nopar">
The formula is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${start}</mn>
  <msup>
   <mrow>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${base}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mrow>
    <mi>t</mi>
   </mrow>
  </msup>
 </math>.
</p>@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=$base=int(rint(3)+2);
$start=int((rint(3)+1));
$x1=int(rint(3)+1);
$xmax=int(1.25*$x1);
$y1=int($start*($base)^$x1);
$ymax=int($start*($base)^$xmax);@
qu.7.1.uid=3d1d472c-72a2-4462-b3ab-d1c9f6f727bd@
qu.7.1.question=<p class="noindent">Give a possible formula for the exponential function shown below given that it passes through the
points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>0</mn>
    <mo>,</mo>
    <mn>${start}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>. Use
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math> as the independent
variable and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
as the dependent variable and specify any values as integers.
<br class="newline"/>
 <br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="${start}*(${base})^x"/>
 <param name="gridLines" value="5"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="${xmax}"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="${ymax}"/>
</applet>
<br class="newline"/>
<br class="newline"/>
<p class="noindent">(The variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math> is plotted
on the horizontal axis, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
on the vertical axis.) </p>@
qu.7.1.answer=
y = ${start}(${base})^t@

qu.7.2.mode=Multipart@
qu.7.2.name=1.3.8.1 a@
qu.7.2.comment=@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=$I=1000;
$k=int(2+rint(3));
$v1=int($k*$I);
$r=5;
$r1=1+$r/100;
$w=int(4*(2+rint(4)));
$ans1=decimal(2,$I*$r1^($w));
$ans2=decimal(1,ln($k)/ln($r1));@
qu.7.2.uid=6118e01e-8e89-4149-8d67-ef7c81b382c9@
qu.7.2.question=<p class="noindent">A deposit is made into an interest-bearing account. The balance,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>B</mi>
 </math>, in the
account <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>
years later is shown in the graph below .
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="1000*(1.05)^x"/>
 <param name="gridLines" value="8"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="32"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="8000"/>
</applet>
<p class="noindent">(The balance, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>B</mi>
 </math>, is shown
on the vertical axis, time, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>,
on the horizontal axis.) </p>@
qu.7.2.weighting=1,1,1@
qu.7.2.numbering=alpha@
qu.7.2.part.1.editing=useHTML@
qu.7.2.part.1.question=<p class="noindent">What was the original deposit in dollars?</p>@
qu.7.2.part.1.answer=${I} ? 100@
qu.7.2.part.1.mode=Plain Number@
qu.7.2.part.2.editing=useHTML@
qu.7.2.part.2.question=<p class="noindent">Estimate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${w}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.</p>@
qu.7.2.part.2.answer=
${ans1} ?100@
qu.7.2.part.2.mode=Plain Number@
qu.7.2.part.3.extra=@
qu.7.2.part.3.editing=useHTML@
qu.7.2.part.3.blank.2=years, days, seconds@
qu.7.2.part.3.question=<p class="noindent">When does the balance reach \\$${v1}?
<br class="newline"/><1><2>.
<br class="newline"/>(Give your answer to the nearest year.)</p>@
qu.7.2.part.3.blank.1=%24%7bans2%7d %3f 1@
qu.7.2.part.3.grader.2=menu@
qu.7.2.part.3.grader.1=formula@
qu.7.2.part.3.mode=Blanks@

qu.7.3.mode=Multipart@
qu.7.3.name=1.3.8.1 d@
qu.7.3.comment=@
qu.7.3.editing=useHTML@
qu.7.3.solution=@
qu.7.3.algorithm=$I=1500;
$k=int(2+rint(3));
$v1=int($k*$I);
$r=6;
$r1=1+$r/100;
$w=int(4*(2+rint(4)));
$ans1=decimal(2,$I*$r1^($w));
$ans2=decimal(1,ln($k)/ln($r1));@
qu.7.3.uid=e97252de-157c-470c-8b56-27e400c9865e@
qu.7.3.question=<p class="noindent">A deposit is made into an interest-bearing account. The balance,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>B</mi>
 </math>, in the
account <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>
years later is shown in the graph below .
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="1500*(1.06)^x"/>
 <param name="gridLines" value="8"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="32"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="8000"/>
</applet>
<br class="newline"/>
<br class="newline"/>
<p class="noindent">(The balance, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>B</mi>
 </math>, is shown
on the vertical axis, time, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>,
on the horizontal axis.) </p>@
qu.7.3.weighting=1,1,1@
qu.7.3.numbering=alpha@
qu.7.3.part.1.editing=useHTML@
qu.7.3.part.1.question=<p class="noindent">What was the original deposit in dollars?</p>@
qu.7.3.part.1.answer=${I} ? 100@
qu.7.3.part.1.mode=Plain Number@
qu.7.3.part.2.editing=useHTML@
qu.7.3.part.2.question=<p class="noindent">Estimate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${w}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.</p>@
qu.7.3.part.2.answer=
${ans1} ?100@
qu.7.3.part.2.mode=Plain Number@
qu.7.3.part.3.extra=@
qu.7.3.part.3.editing=useHTML@
qu.7.3.part.3.blank.2=years, days, seconds@
qu.7.3.part.3.question=<p class="noindent">When does the balance reach \\$${v1}?
<br class="newline"/><1><2>.
<br class="newline"/>(Give your answer to the nearest year.)</p>@
qu.7.3.part.3.blank.1=%24%7bans2%7d %3f 1@
qu.7.3.part.3.grader.2=menu@
qu.7.3.part.3.grader.1=formula@
qu.7.3.part.3.mode=Blanks@

qu.7.4.mode=Multipart@
qu.7.4.name=1.3.8.6@
qu.7.4.comment=@
qu.7.4.editing=useHTML@
qu.7.4.solution=@
qu.7.4.algorithm=$s=int(rint(5)+2);
$p=int(rint(3)+2);
$N=6;
$f0=decimal($N,$s*2^($p));
$f1=decimal($N,$s*(2.1)^($p));
$fd1=decimal($N,($f1-$f0)/0.1);
$f2=decimal($N,$s*(2.01)^($p));
$fd2=decimal($N,($f2-$f0)/0.01);
$f3=decimal($N,$s*(2.001)^($p));
$fd3=decimal($N,($f3-$f0)/0.001);
$fd1d=decimal(4,$fd1);
$fd2d=decimal(4,$fd2);
$fd3d=decimal(4,$fd3);@
qu.7.4.uid=870d21b8-1207-4018-8e0f-5a03bc1c8751@
qu.7.4.question=<p class="noindent">A particle moves a distance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>s</mi>
 </math> meters
from its starting point, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>s</mi>
  <mo>=</mo>
  <mn>${s}</mn>
  <msup>
   <mrow>
    <mi>t</mi>
   </mrow>
   <mrow>
    <mn>${p}</mn>
   </mrow>
  </msup>
 </math>.</p>@
qu.7.4.weighting=1,1,1@
qu.7.4.numbering=alpha@
qu.7.4.part.1.extra=@
qu.7.4.part.1.editing=useHTML@
qu.7.4.part.1.question=<p class="noindent">Find the average velocity between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mo>+</mo>
  <mi>h</mi>
 </math> if
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>h</mi>
  <mo>=</mo>
  <mn>0.1</mn>
 </math>. Give
your answer correct to 4 decimal places.
<1></p>@
qu.7.4.part.1.blank.1=%24%7bfd1%7d %3f 0.05@
qu.7.4.part.1.grader.1=formula@
qu.7.4.part.1.mode=Blanks@
qu.7.4.part.1.comment=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>s</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>t</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>. We wish to find the
average velocity between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2.1</mn>
 </math>.
We have
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtext>Average&nbsp;velocity&nbsp;</mtext>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>2.1</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
    <mo>&minus;</mo>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>2</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mrow>
    <mn>2.1</mn>
    <mo>&minus;</mo>
    <mn>2</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${f1}</mn>
    <mo>&minus;</mo>
    <mn>${f0}</mn>
   </mrow>
   <mrow>
    <mn>0.1</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>&#8776;</mo>
 <mn>${fd1d}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.7.4.part.2.extra=@
qu.7.4.part.2.editing=useHTML@
qu.7.4.part.2.question=<p class="noindent">Find the average velocity between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mo>+</mo>
  <mi>h</mi>
 </math> if
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>h</mi>
  <mo>=</mo>
  <mn>0.01</mn>
 </math>.Give
your answer correct to 4 decimal places.
<1></p>@
qu.7.4.part.2.blank.1=%24%7bfd2%7d %3f 0.05@
qu.7.4.part.2.grader.1=formula@
qu.7.4.part.2.mode=Blanks@
qu.7.4.part.2.comment=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>s</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>t</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>. We wish to find the
average velocity between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2.01</mn>
 </math>.
We have
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtext>Average&nbsp;velocity&nbsp;</mtext>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>2.01</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
    <mo>&minus;</mo>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>2</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mrow>
    <mn>2.01</mn>
    <mo>&minus;</mo>
    <mn>2</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${f2}</mn>
    <mo>&minus;</mo>
    <mn>${f0}</mn>
   </mrow>
   <mrow>
    <mn>0.01</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>&#8776;</mo>
 <mn>${fd2d}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.7.4.part.3.extra=@
qu.7.4.part.3.editing=useHTML@
qu.7.4.part.3.question=<p class="noindent">Find the average velocity between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
  <mo>+</mo>
  <mi>h</mi>
 </math> if
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>h</mi>
  <mo>=</mo>
  <mn>0.001</mn>
 </math>. Give
your answer correct to 4 decimal places.
<1></p>@
qu.7.4.part.3.blank.1=%24%7bfd3%7d %3f 0.05@
qu.7.4.part.3.grader.1=formula@
qu.7.4.part.3.mode=Blanks@
qu.7.4.part.3.comment=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>s</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>t</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>. We wish to find the
average velocity between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>2.001</mn>
 </math>.
We have

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtext>Average&nbsp;velocity&nbsp;</mtext>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>2.001</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
    <mo>&minus;</mo>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>2</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mrow>
    <mn>2.001</mn>
    <mo>&minus;</mo>
    <mn>2</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${f3}</mn>
    <mo>&minus;</mo>
    <mn>${f0}</mn>
   </mrow>
   <mrow>
    <mn>0.001</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>&#8776;</mo>
 <mn>${fd3d}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@

qu.7.5.mode=Multipart@
qu.7.5.name=1.3.8.5@
qu.7.5.comment=@
qu.7.5.editing=useHTML@
qu.7.5.solution=@
qu.7.5.algorithm=@
qu.7.5.uid=dbfc02a1-8c35-4b08-9cb9-61a93190a364@
qu.7.5.question=<p class="noindent">The volume of water in a tank over a period of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>20</mn>
 </math> weeks
is shown below.
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value=".24*x^4-7*x^3+49*x^2+500"/>
 <param name="gridLines" value="10"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="20"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="2000"/>
</applet>
<br class="newline"/>
<br class="newline"/>(The volume of water, (in cubic meters), is shown on the vertical axis; time,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
 <mi>t</mi>
</math> is
shown on the horizontal axis.) @
qu.7.5.weighting=1,1,1,1@
qu.7.5.numbering=alpha@
qu.7.5.part.1.editing=useHTML@
qu.7.5.part.1.fixed=@
qu.7.5.part.1.choice.4=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>20</mn>
 </math>
</p>@
qu.7.5.part.1.question=<p class="noindent">Is the average rate of change of volume positive over the following intervals?
<br class="newline"/>
 <br class="newline"/>Check all appropriate answers.</p>@
qu.7.5.part.1.choice.3=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>15</mn>
 </math>
</p>@
qu.7.5.part.1.choice.2=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>10</mn>
 </math>
</p>@
qu.7.5.part.1.choice.1=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>5</mn>
 </math>
</p>@
qu.7.5.part.1.comment=<p class="noindent">The average rate of change is positive if the volume of water is increasing with time and is negative
if the volume is decreasing. </p>
<ul class="itemize1">
 <li class="itemize">Since the volume is rising from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mn>500</mn>
  </math>
      to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mn>1000</mn>
  </math>
      from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mi>t</mi>
   <mo>=</mo>
   <mn>0</mn>
  </math>
      to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mi>t</mi>
   <mo>=</mo>
   <mn>5</mn>
  </math>,
      the average change is positive.
      </li>
 <li class="itemize">We can see that the volume at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mi>t</mi>
   <mo>=</mo>
   <mn>10</mn>
  </math>
      is greater than the volume at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mi>t</mi>
   <mo>=</mo>
   <mn>0</mn>
  </math>.
      Thus the average rate of change is positive.
      </li>
 <li class="itemize">We can see that the volume at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mi>t</mi>
   <mo>=</mo>
   <mn>15</mn>
  </math>
      is lower than the volume at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mi>t</mi>
   <mo>=</mo>
   <mn>0</mn>
  </math>.
      Thus the average rate of change is negative.
      </li>
 <li class="itemize">We can see that the volume at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mi>t</mi>
   <mo>=</mo>
   <mn>20</mn>
  </math>
      is greater than the volume at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
   <mi>t</mi>
   <mo>=</mo>
   <mn>0</mn>
  </math>.
      Thus the average rate of change is positive.</li>
</ul>@
qu.7.5.part.1.mode=Multiple Selection@
qu.7.5.part.1.answer=1 2 4 @
qu.7.5.part.2.editing=useHTML@
qu.7.5.part.2.question=<p class="noindent">During which of the following time intervals was the average rate of change largest?
<br class="newline"/>
 <br class="newline"/>Check all appropriate answers.</p>@
qu.7.5.part.2.fixed=@
qu.7.5.part.2.choice.2=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>10</mn>
 </math>
</p>@
qu.7.5.part.2.answer=1@
qu.7.5.part.2.choice.1=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>5</mn>
 </math>
</p>@
qu.7.5.part.2.mode=Multiple Choice@
qu.7.5.part.2.comment=<p class="noindent">The secant line between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>5</mn>
 </math> is steeper than

the secant line between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>10</mn>
 </math>, so the slope of the
secant line is greater on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>5</mn>
 </math>.
Since the average rate of change is represented by the slope of a secant line, the average rate of change on the
interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>5</mn>
 </math> is greater than
that in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>10</mn>
 </math>.</p>@
qu.7.5.part.3.editing=useHTML@
qu.7.5.part.3.question=<p class="noindent">During which of the following time intervals was the average rate of change largest?
<br class="newline"/>
 <br class="newline"/>Check all appropriate answers.</p>@
qu.7.5.part.3.fixed=@
qu.7.5.part.3.choice.2=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>20</mn>
 </math>
</p>@
qu.7.5.part.3.answer=2@
qu.7.5.part.3.choice.1=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>10</mn>
 </math>
</p>@
qu.7.5.part.3.mode=Multiple Choice@
qu.7.5.part.3.comment=<p class="noindent">The slope of secant line between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>20</mn>
 </math> is greater than the slope
of the secant line between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>20</mn>
 </math>,
so average rate of change is greater than that in the interval
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>20</mn>
 </math>.</p>@
qu.7.5.part.4.comment=<p class="noindent">The average rate of change in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>10</mn>
 </math>
is about

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>800</mn>
    <mo>&minus;</mo>
    <mn>500</mn>
   </mrow>
   <mrow>
    <mn>10</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>300</mn>
   </mrow>
   <mrow>
    <mn>10</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mn>30</mn>
 <mtext>&nbsp;cubic&nbsp;meters&nbsp;per&nbsp;week</mtext>
</math>
<p class="nopar"> This tells us that for the first ten weeks, the volume of water is growing at an average of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>30</mn>
 </math> cubic
meters per week. </p>@
qu.7.5.part.4.editing=useHTML@
qu.7.5.part.4.extra=@
qu.7.5.part.4.grader.7=formula@
qu.7.5.part.4.grader.6=menu@
qu.7.5.part.4.grader.5=menu@
qu.7.5.part.4.grader.4=menu@
qu.7.5.part.4.grader.3=menu@
qu.7.5.part.4.grader.2=menu@
qu.7.5.part.4.grader.1=formula@
qu.7.5.part.4.question=<p class="noindent">An estimate of the average rate of change between
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>10</mn>
 </math> is <1><2> per <3>.
<br class="newline"/>
 <br class="newline"/>This tells us that for the first <4> weeks, the <5> of water is <6> at an average of <7> cubic meters per
week.</p>@
qu.7.5.part.4.mode=Blanks@
qu.7.5.part.4.blank.7=30@
qu.7.5.part.4.blank.6=growing, falling@
qu.7.5.part.4.blank.5=volume, depth, area@
qu.7.5.part.4.blank.4=ten, twenty, five@
qu.7.5.part.4.blank.3=week, day, hour@
qu.7.5.part.4.blank.2=cubic meters, cubic feet, liters@
qu.7.5.part.4.blank.1=30%3f3@

qu.7.6.mode=Sketch@
qu.7.6.name=1.3.8.4@
qu.7.6.comment=@
qu.7.6.editing=useHTML@
qu.7.6.solution=@
qu.7.6.algorithm=$v0=int(40+rint(5)*10);
$v1=int(50/60*$v0);
$v2=int(30/60*$v0);
$v3=int(10/60*$v0);
$v4=int(2/60*$v0);@
qu.7.6.uid=21a7b7ec-9643-461e-a844-7d8efa1957dc@
qu.7.6.question=<p class="noindent">A driver applies his brakes at ${v0} km/hour. As he approaches an hazard he applies them harder and
harder. Shortly before he reaches the hazard, the driver releases the brakes to avoid skidding and
the car comes to a complete stop and safe stop. Graph the velocity of the car against time. </p>@
qu.7.6.example=0,${v0} 0.25,${v1} 0.5,${v2} 0.75,${v3} 1,${v4}@
qu.7.6.answer=check(( decreasing($1) ) && ( concave_down(restriction($1,0,0.5)) ) && ( concave_up(restriction($1,0.8,1)) ))@
qu.7.6.axes=0,1,0,80@
qu.7.6.gridlines=4@
qu.7.6.axes.labeled=true@

qu.7.7.mode=Multipart@
qu.7.7.name=1.3.8.1 c@
qu.7.7.comment=@
qu.7.7.editing=useHTML@
qu.7.7.solution=@
qu.7.7.algorithm=$I=1000;
$k=int(2+rint(3));
$v1=int($k*$I);
$r=6;
$r1=1+$r/100;
$w=int(4*(2+rint(4)));
$ans1=decimal(2,$I*$r1^($w));
$ans2=decimal(1,ln($k)/ln($r1));@
qu.7.7.uid=ff63ffb1-c35c-4240-9f8a-542d9bda570b@
qu.7.7.question=<p class="noindent">A deposit is made into an interest-bearing account. The balance,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>B</mi>
 </math>, in the
account <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>
years later is shown in the graph below .
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="1000*(1.06)^x"/>
 <param name="gridLines" value="8"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="32"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="8000"/>
</applet>
<br class="newline"/>
<br class="newline"/>
<p class="noindent">(The balance, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>B</mi>
 </math>, is shown
on the vertical axis, time, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>,
on the horizontal axis.) </p>@
qu.7.7.weighting=1,1,1@
qu.7.7.numbering=alpha@
qu.7.7.part.1.editing=useHTML@
qu.7.7.part.1.question=<p class="noindent">What was the original deposit in dollars?</p>@
qu.7.7.part.1.answer=${I} ? 100@
qu.7.7.part.1.mode=Plain Number@
qu.7.7.part.2.editing=useHTML@
qu.7.7.part.2.question=<p class="noindent">Estimate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${w}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.</p>@
qu.7.7.part.2.answer=
${ans1} ?100@
qu.7.7.part.2.mode=Plain Number@
qu.7.7.part.3.extra=@
qu.7.7.part.3.editing=useHTML@
qu.7.7.part.3.blank.2=years, days, seconds@
qu.7.7.part.3.question=<p class="noindent">When does the balance reach \\$${v1}?
<br class="newline"/><1><2>.
<br class="newline"/>(Give your answer to the nearest year.)</p>@
qu.7.7.part.3.blank.1=%24%7bans2%7d %3f 1@
qu.7.7.part.3.grader.2=menu@
qu.7.7.part.3.grader.1=formula@
qu.7.7.part.3.mode=Blanks@

qu.7.8.mode=Multipart@
qu.7.8.name=1.3.8.1 b@
qu.7.8.comment=@
qu.7.8.editing=useHTML@
qu.7.8.solution=@
qu.7.8.algorithm=$I=1500;
$k=int(2+rint(3));
$v1=int($k*$I);
$r=5;
$r1=1+$r/100;
$w=int(4*(2+rint(4)));
$ans1=decimal(2,$I*$r1^($w));
$ans2=decimal(1,ln($k)/ln($r1));@
qu.7.8.uid=457b10f0-2817-41e4-b9e5-a3616f7a4aea@
qu.7.8.question=<p class="noindent">A deposit is made into an interest-bearing account. The balance,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>B</mi>
 </math>, in the
account <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>
years later is shown in the graph below .
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="1500*(1.05)^x"/>
 <param name="gridLines" value="8"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="32"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="8000"/>
</applet>
<br class="newline"/>
<br class="newline"/>
<p class="noindent">(The balance, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>B</mi>
 </math>, is shown
on the vertical axis, time, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>,
on the horizontal axis.) </p>@
qu.7.8.weighting=1,1,1@
qu.7.8.numbering=alpha@
qu.7.8.part.1.editing=useHTML@
qu.7.8.part.1.question=<p class="noindent">What was the original deposit in dollars?</p>@
qu.7.8.part.1.answer=${I} ? 100@
qu.7.8.part.1.mode=Plain Number@
qu.7.8.part.2.editing=useHTML@
qu.7.8.part.2.question=<p class="noindent">Estimate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${w}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.</p>@
qu.7.8.part.2.answer=
${ans1} ?100@
qu.7.8.part.2.mode=Plain Number@
qu.7.8.part.3.extra=@
qu.7.8.part.3.editing=useHTML@
qu.7.8.part.3.blank.2=years, days, seconds@
qu.7.8.part.3.question=<p class="noindent">When does the balance reach \\$${v1}?
<br class="newline"/><1><2>.
<br class="newline"/>(Give your answer to the nearest year.)</p>@
qu.7.8.part.3.blank.1=%24%7bans2%7d %3f 1@
qu.7.8.part.3.grader.2=menu@
qu.7.8.part.3.grader.1=formula@
qu.7.8.part.3.mode=Blanks@

qu.7.9.mode=Sketch@
qu.7.9.name=1.3.8.7@
qu.7.9.comment=@
qu.7.9.editing=useHTML@
qu.7.9.solution=@
qu.7.9.algorithm=@
qu.7.9.uid=2f0a9d33-0a1c-4d7b-81f4-11659cae693d@
qu.7.9.question=<p class="noindent">The population of Washington DC grew from 1900 to 1950, stayed approximately constant during
the 1950s, and decreased from about 1960 to 2000. Graph the population as a function of years
since 1900.</p>@
qu.7.9.example=1900,0.5 1950,0.8 1960,0.8 2000,0.75@
qu.7.9.answer=check(( concave_down($1) ))@
qu.7.9.axes=1900,2000,0,1@
qu.7.9.gridlines=5@
qu.7.9.axes.labeled=true@

qu.7.10.mode=Multipart@
qu.7.10.name=1.3.8.8@
qu.7.10.comment=@
qu.7.10.editing=useHTML@
qu.7.10.solution=@
qu.7.10.algorithm=$a=0.2+rint(5)/10;
$t=int(rint(7)+1);
$ansa=decimal(2,$a*e^(-0.3466*$t));
$ansb=decimal(2,ln($a/0.1)/0.3466);@
qu.7.10.uid=5c65f36a-2fc9-47d8-b631-c883dfa13fc3@
qu.7.10.question=<p class="noindent">The figure shows the amount of nicotine, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>N</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>t</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>,
in mg, in a person's bloodstream as a function of time,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math>, in
hours, since the person finished smoking a cigarette.
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="${a}* e^(-0.3466*x)"/>
 <param name="gridLines" value="5"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="10"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="1"/>
</applet>@
qu.7.10.weighting=1,1,1,1,1@
qu.7.10.numbering=alpha@
qu.7.10.part.1.editing=useHTML@
qu.7.10.part.1.question=<p class="noindent">Estimate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${t}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
correct to 2 decimal places. </p>@
qu.7.10.part.1.answer= ${ansa} ? 0.05@
qu.7.10.part.1.mode=Plain Number@
qu.7.10.part.1.comment=<p class="noindent">From the graph, we see <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${t}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${ansa}</mn>
 </math>.
This means that after ${t} hours, the level of nicotine is ${ansa} mg. </p>@
qu.7.10.part.2.blank.2=%24%7bansa%7d %3f 0.05@
qu.7.10.part.2.blank.1=%24%7bt%7d@
qu.7.10.part.2.extra=@
qu.7.10.part.2.editing=useHTML@
qu.7.10.part.2.question=<p class="noindent">The result in part (a) means that after <1> hours, the level of nicotine is <2> mg </p>@
qu.7.10.part.2.comment=<p class="noindent">This means that after ${t} hours, the level of nicotine is ${ansa} mg. </p>@
qu.7.10.part.2.mode=Blanks@
qu.7.10.part.2.grader.2=formula@
qu.7.10.part.2.grader.1=formula@
qu.7.10.part.3.editing=useHTML@
qu.7.10.part.3.question=<p class="noindent">About how many hours have passed before the nicotine level is down to 0.1 mg?
<br class="newline"/>
 <br class="newline"/>(Give your answer to 1 decimal place.)</p>@
qu.7.10.part.3.answer=${ansb} ? 0.5 @
qu.7.10.part.3.mode=Plain Number@
qu.7.10.part.3.comment=<p class="noindent">From the graph, about ${ansb} hours.</p>@
qu.7.10.part.4.blank.2=right after the cigarette is smoked, one hour after the cigarette is smoked, when the cigarette is lit@
qu.7.10.part.4.blank.1=%24%7ba%7d@
qu.7.10.part.4.extra=@
qu.7.10.part.4.editing=useHTML@
qu.7.10.part.4.question=<p class="noindent">What is the vertical intercept? <1>. This represents the level of nicotine in the blood <2>.
</p>@
qu.7.10.part.4.comment=<p class="noindent">It represents the level of nicotine in the blood right after the cigarette is smoked.</p>@
qu.7.10.part.4.mode=Blanks@
qu.7.10.part.4.grader.2=menu@
qu.7.10.part.4.grader.1=formula@
qu.7.10.part.5.blank.2=all, some, half@
qu.7.10.part.5.blank.1=0@
qu.7.10.part.5.extra=@
qu.7.10.part.5.editing=useHTML@
qu.7.10.part.5.question=<p class="noindent">If this function had a horizontal intercept, it would represent the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math> when
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>N</mi>
  <mo>=</mo>
 </math><1>, or
the number of hours until <2> the nicotine is gone from the body.</p>@
qu.7.10.part.5.comment=<p class="noindent">If this function had a horizontal intercept, it would represent the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
 </math> when
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>N</mi>
  <mo>=</mo>
  <mn>0</mn>
 </math>, or the
number of hours until all the nicotine is gone from the body.</p>@
qu.7.10.part.5.mode=Blanks@
qu.7.10.part.5.grader.2=menu@
qu.7.10.part.5.grader.1=formula@

qu.7.11.mode=Blanks@
qu.7.11.name=1.3.8.3@
qu.7.11.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>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${x}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>&#8776;</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${fx}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>&#8776;</mo>
  <mn>${ffx}</mn>
  <mo>.</mo>
 </math>
 <br class="newline"/>
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>g</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${x}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>&#8776;</mo>
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${gx}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>&#8776;</mo>
  <mn>${ggx}</mn>
  <mo>.</mo>
 </math>
</p>@
qu.7.11.editing=useHTML@
qu.7.11.solution=@
qu.7.11.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.7.11.uid=caed3b9d-11f9-459d-a901-1d4beb9fff84@
qu.7.11.question=<p class="noindent">Examine 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
<br class="newline"/>
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${x}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
 </math><1><br class="newline"/>
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>g</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${x}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
 </math><2><br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" codebase="/keele/modules">
 <param name="y1" value="3*sin(x)"/>
 <param name="gridLines" value="6"/>
 <param name="xMin" value="-3"/>
 <param name="xMax" value="3"/>
 <param name="yMin" value="-3"/>
 <param name="yMax" value="3"/>
</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 name="gridLines" value="6"/>
 <param name="xMin" value="-3"/>
 <param name="xMax" value="3"/>
 <param name="yMin" value="-3"/>
 <param name="yMax" value="3"/>
</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>
<br class="newline"/>@
qu.7.11.blank.1=%24%7bffx%7d %3f%0a0.5@
qu.7.11.blank.2=%24%7bggx%7d %3f%0a0.5@
qu.7.11.grader.1=formula@
qu.7.11.grader.2=formula@
qu.7.11.extra=@

