qu.1.topic=1-instantaneous velocity@

qu.1.1.mode=Inline@
qu.1.1.editing=useHTML@
qu.1.1.algorithm=$b=range(5,8);
$c=range(2,5);
$k=$c/$b;
$t=range(1,3);
$tt=$t+0.1;
$tu=$t+0.01;
$tv=$t+0.001;
$answer1=decimal(4,sqrt($b)/(sqrt($tt+$k)+sqrt($t+$k)));
$answer2=decimal(4,sqrt($b)/(sqrt($tu+$k)+sqrt($t+$k)));
$answer3=decimal(4,sqrt($b)/(sqrt($tv+$k)+sqrt($t+$k)));
$answer4=decimal(2,.5*sqrt($b)/sqrt($t+$k));@
qu.1.1.weighting=1,1,1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.answer.units=@
qu.1.1.part.1.numStyle=   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.err=1.0E-4@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=toler_abs@
qu.1.1.part.1.negStyle=minus@
qu.1.1.part.1.answer.num=$answer1@
qu.1.1.part.2.answer.units=@
qu.1.1.part.2.numStyle=   @
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.showUnits=false@
qu.1.1.part.2.err=1.0E-4@
qu.1.1.part.2.question=(Unset)@
qu.1.1.part.2.mode=Numeric@
qu.1.1.part.2.grading=toler_abs@
qu.1.1.part.2.negStyle=minus@
qu.1.1.part.2.answer.num=$answer2@
qu.1.1.part.3.answer.units=@
qu.1.1.part.3.numStyle=   @
qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.showUnits=false@
qu.1.1.part.3.err=1.0E-4@
qu.1.1.part.3.question=(Unset)@
qu.1.1.part.3.mode=Numeric@
qu.1.1.part.3.grading=toler_abs@
qu.1.1.part.3.negStyle=minus@
qu.1.1.part.3.answer.num=$answer3@
qu.1.1.part.4.answer.units=@
qu.1.1.part.4.numStyle=   @
qu.1.1.part.4.editing=useHTML@
qu.1.1.part.4.showUnits=false@
qu.1.1.part.4.err=0.01@
qu.1.1.part.4.question=(Unset)@
qu.1.1.part.4.mode=Numeric@
qu.1.1.part.4.grading=toler_abs@
qu.1.1.part.4.negStyle=minus@
qu.1.1.part.4.answer.num=$answer4@
qu.1.1.question=A car moves along the 
<math><mi>x</mi></math>
axis with its position given by 
<math>
  <mi>x</mi>
  <mo>=</mo>
  <msqrt>
    <mrow>
      <mi>$b</mi>
      <mo>&InvisibleTimes;</mo>
      <mi>t</mi>
      <mo>+</mo>
      <mi>$c</mi>
    </mrow>
  </msqrt>
</math>.
<br>
Our goal is to estimate the instantaneous velocity (speedometer reading) at the time 
<math>
 
  <mi>t</mi>
  <mo>=</mo>
  <mi>$t</mi>
</math>.
<br>
<br>
<hr>
<br>
Determine the average velocity of the car during the period from time 
<math><mi>$t</mi></math>
to time 
<math><mi>$tt</mi></math>.
<br>
Report your answer with 4 decimal digits. 
<1> &#160;
<br>
<br>
Determine the average velocity of the car during the period from time 
<math><mi>$t</mi></math>
to time 
<math><mi>$tu</mi></math>.
<br>
Report your answer with 4 decimal digits. 
<2> &#160;
<br>
<br>
Determine the average velocity of the car during the period from time 
<math><mi>$t</mi></math>
to time 
<math><mi>$tv</mi></math>.
<br>
Report your answer with 4 decimal digits. 
<3> &#160;
<br>
<br>
Based on your first three answers, determine the instantaneous velocity 
of the car at time 
<math><mi>$t</mi></math>.
<br>
Report your answer with 2 decimal digits. 
<4> &#160;@

qu.1.2.question=Determine the instantaneous velocity at time 
<math>
  <mi>t</mi>
  <mo>=</mo>
  <mn>$t</mn>
</math> 
for motion given by 
<math>
  <mi>x</mi>
  <mo>=</mo>
</math>
${mathml($a*t^2+$b*t+$c)}.
<br>
Hint: look at the average velocities during intervals from 
<math><mn>$t</mn></math> 
to 
<math>
  <mn>$t</mn>
  <mo>+</mo>
  <mi>h</mi>
</math> ,
where 
<math>
  <mi>h</mi>
  <mo>=</mo>
  <mn>0.01</mn>
</math>, 
<math><mn>0.001</mn></math>.@
qu.1.2.answer.num=$answer@
qu.1.2.answer.units=@
qu.1.2.showUnits=false@
qu.1.2.grading=exact_value@
qu.1.2.negStyle=minus@
qu.1.2.numStyle=thousands scientific dollars arithmetic@
qu.1.2.mode=Numeric@
qu.1.2.comment=The answer is $answer.@
qu.1.2.editing=useHTML@
qu.1.2.algorithm=$a=range(2,4);
$b=range(-5,5);
$c=range(1,5);
$t=range(2,4);
$answer=2*$a*$t+$b;@



qu.2.topic=2-the derivative by the definition@

qu.2.1.mode=Inline@
qu.2.1.comment=
<math>
  <mi>f</mi>
  <mo>(</mo>
  <mn>$c</mn>
  <mo>+</mo>
  <mi>h</mi>
  <mo>)=</mo>
</math>
${mathml($answer1)}
<math><mo>=</mo></math>
${mathml(($c^2+$b*$c)+(2*$c+$b)*h+h^2)}.
<br> 
The secant slope is ${mathml($answer2)}, and 
<math>
  <msup>
    <mrow>
      <mi>f</mi>
    </mrow>
    <mrow>
      <mo>&prime;</mo>
    </mrow>
  </msup>
  <mo>(</mo>
  <mn>$c</mn>
  <mo>)=</mo>
  <mn>$answer3</mn>
</math>.@
qu.2.1.editing=useHTML@
qu.2.1.algorithm=$b=range(-5,5);
$c=range(-3,3);
condition:not(eq($b*$c*($b+2*$c),0));
$answer1=($c+h)^2+$b*($c+h);
$answer2=2*$c+$b+h;
$answer3=2*$c+$b;@
qu.2.1.weighting=1,1,1@
qu.2.1.numbering=alpha@
qu.2.1.part.1.editing=useHTML@
qu.2.1.part.1.question=(Unset)@
qu.2.1.part.1.answer=$answer1@
qu.2.1.part.1.mode=Formula@
qu.2.1.part.2.editing=useHTML@
qu.2.1.part.2.question=(Unset)@
qu.2.1.part.2.answer=$answer2@
qu.2.1.part.2.mode=Formula@
qu.2.1.part.3.editing=useHTML@
qu.2.1.part.3.question=(Unset)@
qu.2.1.part.3.answer=$answer3@
qu.2.1.part.3.mode=Formula@
qu.2.1.question=Let 
<math>
  <mi>f</mi>
  <mo>(</mo>
  <mi>x</mi>
  <mo>)</mo>
  <mo>=</mo>
</math> 
${mathml(x^2+$b*x)}.
<br>
<br>
What is 
<math>
  <mi>f</mi>
  <mo>(</mo>
  <mn>$c</mn>
  <mo>+</mo>
  <mi>h</mi>
  <mo>)</mo>
</math>?
<1> &#160;
<br>
<br>
What is the slope of the secant line connecting the points at 
<math>
  <mi>x</mi>
  <mo>=</mo>
  <mi>$c</mi>
</math>
and 
<math>
  <mi>x</mi>
  <mo>=</mo>
  <mi>$c</mi>
  <mo>+</mo>
  <mi>h</mi>
</math>?
<2> &#160;
<br>
<br>
What is 
<math>
  <msup>
    <mrow>
      <mi>f</mi>
    </mrow>
    <mrow>
      <mo>&prime;</mo>
    </mrow>
  </msup>
  <mo>(</mo>
  <mi>$c</mi>
  <mo>)</mo>
</math>?
<3> &#160;@



qu.3.topic=3-derivatives and graphs@

qu.3.1.mode=Multiple Selection@
qu.3.1.comment=Remember that "
<math>
  <msup>
    <mrow>
      <mi>f</mi>
    </mrow>
    <mrow>
      <mo>&prime;</mo>
    </mrow>
  </msup>
</math> 
is increasing" 
means that the slope of 
<math><mi>f</mi></math> 
is getting more positive or less negative.@
qu.3.1.editing=useHTML@
qu.3.1.algorithm=$j=rint(6);
$s=range(-1,1,2);
$r=range(-1,1,2);
$case=4*$j+(1-$r)+(1-$s)/2;
$fnc=.83*$s*(.05*$r*x^3+(.12*$j-.3)*x^2+.1*(1-$j)*$r*x+5-2*$j);
$pta=switch($j,-4*$r,2*$r,-2*$r,2*$r,-2*$r,4*$r);
$ptb=switch($j,2*$r,4*$r,-2*$r,-3*$r,2*$r,2*$r);
$ptc=switch($j,-2*$r,4*$r,4*$r,0,2*$r,0);
$ptd=switch($j,-2*$r,-2*$r,0,-3*$r,-4*$r,2*$r);
$pte=switch($j,4*$r,-2*$r,4*$r,2*$r,-4*$r,0);
$f="<math><mi>f</mi></math>";
$fp="<math><msup><mi>f</mi><mo>&prime;</mo></msup></math>";
$fpp="<math><msup><mi>f</mi><mrow><mo>&prime;</mo><mo>&prime;</mo></mrow></msup></math>";
$fnca=switch($j,"$fp","$f","$fp","$f","$f","$fp");
$fncb="$fp";
$fncc=switch($j,"$fp","$fpp","$fpp","$fp","$fpp","$fpp");
$fncd=if(eq($j,0),"$f","$fp");
$fnce=switch($j,"$fpp","$fp","$f","$fpp","$fp","$f");
$desca=switch($case,
"positive","negative","negative","positive", 
"decreasing","increasing","increasing","decreasing", 
"positive","negative","negative","positive", 
"increasing","decreasing","decreaasing","increasing", 
"decreasing","increasing","increasing","decreasing", 
"positive","negative","negative","positive");
$descb=switch($case,
"negative","positive","positive","negative", 
"positive","negative","negative","positive", 
"decreasing","increasing","decreasing","increasing", 
"positive","negative","negative","positive", 
"positive","negative","negative","positive", 
"increasing","decreasing","increasing","decreasing");
$descc=switch($case,
"decreasing","increasing","decreasing","increasing", 
"positive","negative","positive","negative", 
"positive","negative","positive","negative", 
"negative","positive","positive","negative", 
"positive","negative","positive","negative", 
"positive","negative","positive","negative");
$descd=switch($case,
"decreasing","increasing","increasing","decreasing", 
"negative","positive","positive","negative", 
"positive","negative","negative","positive", 
"increasing","decreasing","increasing","decreasing", 
"negative","positive","positive","negative", 
"negative","positive","positive","negative");
$desce=switch($case,
"negative","positive","negative","positive", 
"increasing","decreasing","increasing","decreasing", 
"decreasing","increasing","increasing","decreasing", 
"negative","positive","negative","positive", 
"increasing","decreasing","increasing","decreasing", 
"increasing","decreasing","decreaasing","increasing");@
qu.3.1.question=Mark all those statements that are true for the function
<math><mi>f</mi></math> 
whose graph is depicted below.
<br>
  <applet code="applets.grapher.Graph" width="250" height="250">
    <param name="xMin" value="-5">
    <param name="xMax" value="5">
    <param name="yMin" value="-5">
    <param name="gridLines" value="10">
    <param name="yMax" value="5">
    <param name="y1" value="$fnc">
  </applet>
<br>
@
qu.3.1.answer=1,2,3@
qu.3.1.choice.1=$fnca is $desca at 
<math><mi>x</mi><mo>=</mo><mn>$pta</mn></math>@
qu.3.1.choice.2=$fncb is $descb at 
<math><mi>x</mi><mo>=</mo><mn>$ptb</mn></math>@
qu.3.1.choice.3=$fncc is $descc at 
<math><mi>x</mi><mo>=</mo><mn>$ptc</mn></math>@
qu.3.1.choice.4=$fncd is $descd at 
<math><mi>x</mi><mo>=</mo><mn>$ptd</mn></math>@
qu.3.1.choice.5=$fnce is $desce at 
<math><mi>x</mi><mo>=</mo><mn>$pte</mn></math>@



qu.4.topic=4-tangent lines and linear approximation@

qu.4.1.mode=Equation@
qu.4.1.comment=The answer is ${mathml(y)}=${mathml($answer)}.@
qu.4.1.editing=useHTML@
qu.4.1.algorithm=$a=range(2,4);
$b=range(1,5);
$d=range(1,5);
$c=range(-2,2);
condition:not(eq($c,0));
$m=2*$a*$c+$b;
condition:not(eq($m*($m-1),0));
$answer=if(gt($m,0),($a*$c^2+$b*$c+$d)+($m)*(x-$c),($a*$c^2+$b*$c+$d)-(-$m)*(x-$c));@
qu.4.1.question=Find the equation of the line that is tangent to the graph of 
<br>
<math>
  <mi>y</mi>
  <mo>=</mo>
</math>
${mathml($a*x^2+$b*x+$d)} 
at &#160; 
<math>
  <mi>x</mi>
  <mo>=</mo>
  <mi>$c</mi>
</math>.@
qu.4.1.answer=y=$answer@

qu.4.2.mode=Formula@
qu.4.2.comment=The answer is ${mathml("$answer")}.@
qu.4.2.editing=useHTML@
qu.4.2.algorithm=$a=range(2,9);
$k=range(1,9);
$answer="$a/$k"*x;@
qu.4.2.question=Find the best linear approximation for ${mathml(($a*x)/($k+x))} near
<math>
  <mi>x</mi>
  <mo>=</mo>
  <mn>0</mn>
</math>.@
qu.4.2.answer=$answer@

qu.4.3.mode=Formula@
qu.4.3.comment=The answer is ${mathml("$answer")}.@
qu.4.3.editing=useHTML@
qu.4.3.algorithm=$a=range(2,9);
$c=range(1,4);
$k=range(1,9);
$dsq=($k+$c)^2;
$m="($a*$k)/($dsq)";
$b="($a*$c)/($k+$c)";
$answer="$b+$m*(x-$c)";@
qu.4.3.question=Find the best linear approximation for ${mathml(($a*x)/($k+x))} near
<math>
  <mi>x</mi>
  <mo>=</mo>
  <mn>$c</mn>
</math>.@
qu.4.3.answer=$answer@

qu.4.4.mode=Inline@
qu.4.4.comment=The best linear approximation is ${mathml($a+x/$c)}.
<br>
Using this approximation, 
<math>
  <msqrt>
    <mrow>
      <mn>$sq</mn>
    </mrow>
  </msqrt>
</math> 
is approximately 
<math>
  <mn>$a</mn>
  <mo>$ch</mo>
  <mfrac>
    <mrow>
      <mn>1</mn>
    </mrow>
    <mrow>
      <mn>$d</mn>
    </mrow>
  </mfrac>
</math>
.@
qu.4.4.editing=useHTML@
qu.4.4.algorithm=$a=range(3,9);
$c=2*$a;
$asq=$a^2;
$r=rint(4);
$b=switch($r,1,1,2,2);
$sign=switch($r,1,-1,1,-1);
$ch=switch($r,"+","-","+","-");
$d=switch($r,$c,$c,$a,$a);
$sq=$asq+$sign*$b;
$answer=switch($r,"$a+1/$c","$a-1/$c","$a+1/$a","$a-1/$a");@
qu.4.4.weighting=1,1@
qu.4.4.numbering=alpha@
qu.4.4.part.1.editing=useHTML@
qu.4.4.part.1.question=(Unset)@
qu.4.4.part.1.answer=$a+x/$c@
qu.4.4.part.1.mode=Formula@
qu.4.4.part.2.answer.units=@
qu.4.4.part.2.numStyle=thousands scientific  arithmetic@
qu.4.4.part.2.editing=useHTML@
qu.4.4.part.2.showUnits=false@
qu.4.4.part.2.question=(Unset)@
qu.4.4.part.2.mode=Numeric@
qu.4.4.part.2.grading=exact_value@
qu.4.4.part.2.negStyle=minus@
qu.4.4.part.2.answer.num=$answer@
qu.4.4.question=Find the best linear approximation for 
<math>
  <msqrt>
    <mrow>
      <mi>$asq</mi>
      <mo>+</mo>
      <mi>x</mi>
    </mrow>
  </msqrt>
</math>
near
<math>
  <mi>x</mi>
  <mo>=</mo>
  <mn>0</mn>
</math>.
<br>
<1> &#160;
<br>
<br>
Use your answer to approximate 
<math>
  <msqrt>
    <mrow>
      <mi>$sq</mi>
    </mrow>
  </msqrt>
</math>.
<br>
(Your answer must be <strong>exact</strong>. Do not use a numerical approximation.)
<br>
<2> &#160;@



qu.5.topic=5-L'Hopital's rule@

qu.5.1.mode=Inline@
qu.5.1.comment=
<math>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
${mathml($function)}
<math>
  <mo>=</mo>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
${mathml($answer1)}
<math>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>$aa</mn>
    </mrow>
    <mrow>
      <mn>$cc</mn>
    </mrow>
  </mfrac>
</math>.@
qu.5.1.editing=useHTML@
qu.5.1.algorithm=$a=range(-9,9);
$b=range(2,9);
$c=range(2,9);
condition:not(eq($a*($a-1),0));
$function=($a*x+$b*x^2)/sin($c*x);
$answer1=($a+2*$b*x)/($c*cos($c*x));
$aa=$a/gcd($a,$c);
$cc=$c/gcd($a,$c);@
qu.5.1.weighting=1,1@
qu.5.1.numbering=alpha@
qu.5.1.part.1.editing=useHTML@
qu.5.1.part.1.question=(Unset)@
qu.5.1.part.1.answer=$answer1@
qu.5.1.part.1.mode=Formula@
qu.5.1.part.2.editing=useHTML@
qu.5.1.part.2.question=(Unset)@
qu.5.1.part.2.answer=$aa/$cc@
qu.5.1.part.2.mode=Formula@
qu.5.1.question=Fill in the blanks, using L'Hopital's rule:
<br>
<br>
<math>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
${mathml($function)}
<math>
  <mo>=</mo>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
<1> &#160;
<math><mo>=</mo></math>
<2> &#160;@

qu.5.2.mode=Inline@
qu.5.2.comment=
<math>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
${mathml($function)}
<math>
  <mo>=</mo>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
${mathml($answerone)}
<math>
  <mo>=</mo>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
${mathml($answertwo)}
<br>
<math>
  <mo>=</mo>
  <mfrac>
    <mrow>
      <mn>$cc</mn>
    </mrow>
    <mrow>
      <mn>$aa</mn>
    </mrow>
  </mfrac>
</math>.@
qu.5.2.editing=useHTML@
qu.5.2.algorithm=$a=range(2,9);
$c=range(2,9);
$function=(1-cos($c*x))/($a*x^2);
$answerone=($c*sin($c*x))/(2*$a*x);
$answertwo=($c^2*cos($c*x))/(2*$a);
$aa=(2*$a)/gcd(2*$a,$c^2);
$cc=$c^2/gcd(2*$a,$c^2);@
qu.5.2.weighting=1,1,1@
qu.5.2.numbering=alpha@
qu.5.2.part.1.editing=useHTML@
qu.5.2.part.1.question=(Unset)@
qu.5.2.part.1.answer=$answerone@
qu.5.2.part.1.mode=Formula@
qu.5.2.part.2.editing=useHTML@
qu.5.2.part.2.question=(Unset)@
qu.5.2.part.2.answer=$answertwo@
qu.5.2.part.2.mode=Formula@
qu.5.2.part.3.editing=useHTML@
qu.5.2.part.3.question=(Unset)@
qu.5.2.part.3.answer=$cc/$aa@
qu.5.2.part.3.mode=Formula@
qu.5.2.question=Fill in the blanks, using L'Hopital's rule:
<br>
<br>
<math>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
${mathml($function)}
<math>
  <mo>=</mo>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
<1> &#160;
<br>
<math>
  <mo>=</mo>
  <munder>
    <mrow>
      <mi>lim</mi>
    </mrow>
    <mrow>
      <mi>x</mi>
      <mo>&rarr;</mo>
      <mn>0</mn>
    </mrow>
  </munder>
</math>
<2> &#160;
<br>
<math><mo>=</mo></math>
<3> &#160;@



qu.6.topic=6-critical points@

qu.6.1.mode=Multi Formula@
qu.6.1.comment=The answer is 
<math>
  <msup>
    <mrow>
      <mi>e</mi>
    </mrow>
    <mrow>
      <mo>$sign</mo>
      <mn>1</mn>
      <mo>/</mo>
      <mi>$p</mi>
    </mrow>
  </msup>
</math>@
qu.6.1.editing=useHTML@
qu.6.1.algorithm=$p=range(2,8);
$r=rint(2);
$signum=switch($r,1,-1);
$sign=switch($r,"-","");
$answer="e"^(-$signum*"1/$p");@
qu.6.1.question=Find the critical point(s) of ${mathml(x^($signum*$p)*ln(x))}.  
Enter the <strong>exact</strong> answer(s), 
<strong>separated by semicolons</strong>.@
qu.6.1.answer=$answer@

qu.6.2.mode=Multi Formula@
qu.6.2.comment=The answer is
<math>
  <mi>$k</mi>
  <mo>-</mo>
  <msqrt>
    <mrow>
      <mi>$d</mi>
    </mrow>
  </msqrt>
  <mo>;</mo>
  <mtext>&ThinSpace;</mtext>
  <mi>$k</mi>
  <mo>+</mo>
  <msqrt>
    <mrow>
      <mi>$d</mi>
    </mrow>
  </msqrt>
</math>.@
qu.6.2.editing=useHTML@
qu.6.2.algorithm=$b=range(2,5);
$a=range(-5,5);
$c=range(-5,5);
$k=-$c;
$d=$c^2-$a*$c+$b;
condition:not(eq($a*$c,0));
condition:gt($d,0);
$answerone=-$c+"sqrt($d)";
$answertwo=-$c-"sqrt($d)";@
qu.6.2.question=Find the critical point(s) of ${mathml((x+$c)/(x^2+$a*x+$b))}.  
Enter the <strong>exact</strong> answer(s), 
<strong>separated by semicolons</strong>.@
qu.6.2.answer=$answerone;$answertwo@



qu.7.topic=7-global extrema@

qu.7.1.mode=Formula@
qu.7.1.comment=The answer is 
<math>
  <mfrac>
    <mrow>
      <msqrt>
        <mrow>
          <mi>$d</mi>
        </mrow>
      </msqrt>
      <mo>-</mo>
      <mi>$b</mi>
    </mrow>
    <mrow>
      <mn>9</mn>
    </mrow>
  </mfrac>
</math>.@
qu.7.1.editing=useHTML@
qu.7.1.algorithm=$r=rint(13);
$b=switch($r,-1,1,-1,1,-2,-1,1,2,1,2,2,3,3);
$c=switch($r,-6,-6,-5,-5,-4,-4,-4,-4,-3,-3,-2,-2,-1);
$d=$b^2-9*$c;
$answer="(sqrt($d)-$b)/9";@
qu.7.1.question=Find the value of ${mathml(x)} that yields the 
<strong>global minimum</strong>
 of the function 
<br>
${mathml(3*x^3+$b*x^2+$c*x)} on the interval 
<math>
  <mo>[-</mo>
  <mn>1</mn>
  <mo>,</mo>
  <mn>1</mn>
  <mo>]</mo>
</math>.  
<br>
<strong>Your answer must be exact.</strong>
<br>
<script>drawMaplePlot("plot(3*x^3+($b)*x^2+($c)*x,x=-1..1,color=blue,thickness=2,labels=[`x`,`f(x)`])");</script>@
qu.7.1.answer=$answer@

qu.7.2.mode=Formula@
qu.7.2.comment=The answer is -1.@
qu.7.2.editing=useHTML@
qu.7.2.algorithm=$r=rint(9);
$b=switch($r,-3,-2,-1,-3,-2,-1,1,-4,-5);
$c=switch($r,-2,-2,-2,-1,-1,-1,-1,1,2);
$d=$b^2-9*$c;
$answer=-1;@
qu.7.2.question=Find the value of ${mathml(x)} that yields the 
<strong>global minimum</strong>
 of the function 
<br>
${mathml(3*x^3+$b*x^2+$c*x)} on the interval 
<math>
  <mo>[-</mo>
  <mn>1</mn>
  <mo>,</mo>
  <mn>1</mn>
  <mo>]</mo>
</math>.  
<br>
<strong>Your answer must be exact.</strong>
<br>
<script>drawMaplePlot("plot(3*x^3+($b)*x^2+($c)*x,x=-1..1,color=blue,thickness=2,labels=[`x`,`f(x)`])");</script>@
qu.7.2.answer=$answer@



qu.8.topic=8-finding local extrema@

qu.8.1.mode=Multi Formula@
qu.8.1.comment=The answer is
<math>
  <mo>$sign</mo>
  <msqrt>
    <mfrac>
      <mrow>
        <mn>$b</mn>
      </mrow>
      <mrow>
        <mn>$A</mn>
      </mrow>
    </mfrac>
  </msqrt>
</math> 
.@
qu.8.1.editing=useHTML@
qu.8.1.algorithm=$a=range(1,6);
$bb=range(1,6);
$c=range(1,6);
$AA=3*$a;
$b=$bb/gcd($bb,$AA);
$A=$AA/gcd($bb,$AA);
condition:not($eq($A,1));
$r=range(0,3);
$extrema=switch($r,"maxima","minima","maxima","minima");
$function=switch($r,$c+$b*x-$a*x^3,$c+$b*x-$a*x^3,$c-$b*x+$a*x^3,$c-$b*x+$a*x^3);
$sgn=switch($r,1,-1,-1,1);
$sign=switch($r,"","-","-","");
$answer=$sgn*sqrt("$b/$A");@
qu.8.1.question=Find the local 
<strong>$extrema</strong> 
of ${mathml($function)}.
<br>
Enter the <strong>exact</strong> 
<math><mi>x</mi></math> 
coordinates, 
<strong>separated by semicolons</strong>.@
qu.8.1.answer=$answer@

qu.8.2.mode=Inline@
qu.8.2.editing=useHTML@
qu.8.2.algorithm=$b=range(1,9);
$c=range(1,9);
$base=2*$b/5;
$r=range(0,1);
$function=switch($r,x^5+$b*x^2+$c,x^5-$b*x^2+$c);
$answera=switch($r,"-$base^(1/3)",0);
$answerb=switch($r,0,"$base^(1/3)");@
qu.8.2.weighting=1,1@
qu.8.2.numbering=alpha@
qu.8.2.part.1.editing=useHTML@
qu.8.2.part.1.question=(Unset)@
qu.8.2.part.1.answer=$answera@
qu.8.2.part.1.mode=Formula@
qu.8.2.part.2.editing=useHTML@
qu.8.2.part.2.question=(Unset)@
qu.8.2.part.2.answer=$answerb@
qu.8.2.part.2.mode=Formula@
qu.8.2.question=Let 
<math>
  <mi>f</mi>
  <mo>(</mo>
  <mi>x</mi>
  <mo>)</mo>
  <mo>=</mo>
</math>
${mathml($function)}.
<br>
<br>
In the questions that follow, report the <strong>exact</strong> 
<math><mi>x</mi></math>
coordinate(s), separated by semicolons.
<br>
<br>
Find the local <strong>maxima</strong> of 
<math><mi>f</mi></math>.
<br>
<1> &#160;
<br>
<br>
Find the local <strong>minima</strong> of 
<math><mi>f</mi></math>.
<br>
<2> &#160;@

qu.8.3.mode=Inline@
qu.8.3.comment=Keep in mind that the 
<math><mi>y</mi></math> axis of the graph is showing you the 
<strong>slope</strong> of the function <math><mi>f</mi></math>.@
qu.8.3.editing=useHTML@
qu.8.3.algorithm=$b=range(-4,0);
$d=range(4,6);
$c=$b+$d;
condition:lt($c,5);
$case=rint(2);
$s=2*$case-1;
$k=max((5-$b)*(5-$c),(5+$b)*(5+$c));
$a=6/$k;
$min=switch($case,$c,$b);
$max=switch($case,$b,$c);
$fnc=$s*$a*(x-$b)*(x-$c);@
qu.8.3.weighting=1,1@
qu.8.3.numbering=alpha@
qu.8.3.part.1.editing=useHTML@
qu.8.3.part.1.question=(Unset)@
qu.8.3.part.1.answer=$min@
qu.8.3.part.1.mode=Formula@
qu.8.3.part.2.editing=useHTML@
qu.8.3.part.2.question=(Unset)@
qu.8.3.part.2.answer=$max@
qu.8.3.part.2.mode=Formula@
qu.8.3.question=Use the graph of 
<math>
  <msup>
    <mrow>
      <mi>f</mi>
    </mrow>
    <mrow>
      <mo>&prime;</mo>
    </mrow>
  </msup>
</math> 
shown below to determine the local extrema of 
<math><mi>f</mi></math>.
<br>
  <applet code="applets.grapher.Graph" width="250" height="250">
    <param name="xMin" value="-5">
    <param name="xMax" value="5">
    <param name="yMin" value="-5">
    <param name="gridLines" value="10">
    <param name="yMax" value="5">
    <param name="y1" value="$fnc">
  </applet>
<br>
<br>
Find the local <strong>maxima</strong> of 
<math><mi>f</mi></math>.
<br>
<1> &#160;
<br>
<br>
Find the local <strong>minima</strong> of 
<math><mi>f</mi></math>.
<br>
<2> &#160;
<br>
<br>
[If there is more than one answer, separate them with semicolons.  
<br>
If there are no answers, enter "-10".]@