qu.1.topic=Max/min/inflection@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=vertical tangentPtofInf@
qu.1.1.comment=<p>$plot</p>
<p>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a vertical tangent point of inflection. Note that the derivative $Fp does not change sign<strong><font size="3"><font size="2"> </font></font></strong><font size="3"><font size="2">as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>passes through <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. The function either increases on both sides of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> or decreases on both sides of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. So it can't be a max or min!</font></font></p>
<p><font size="3"><font size="2">Notice that the exponent on $X is between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>1</mn></mrow></math>. If it had been greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>1</mn></mrow></math>, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>would have been a vertical asymptote.</font></font></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(-5,5);
$n=rint(3,7,2);
$f1="(x-($a))^(1/$n)";
$f2="-($f1)";
$z=rint(2);
$f=switch($z,"$f1","$f2");
$fp=switch($z, "(1/$n)*(x-($a))^((1/$n-1))", "-(1/$n)*(x-($a))^((1/$n)-1)");
$M=maple("
MathML[ExportPresentation](x-($a)),
MathML[ExportPresentation](y=$f),
MathML[ExportPresentation](dy/dx=$fp)
");
$X=switch(0,$M);
$F=switch(1,$M);
$Fp=switch(2,$M);


$plot=plotmaple("plot(surd(x-($a),$n),x=-abs($a)-2..abs($a)+2,y=-5..5,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.1.uid=f58f4f06-e5b3-4d43-a339-e7a18687f22a@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Max/Min/Pt of Inflection;
  Difficulty=Easy;
@
qu.1.1.question=<p>The function $F has derivative $Fp. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a</p>@
qu.1.1.answer=1@
qu.1.1.choice.1=point of inflection with a vertical tangent.@
qu.1.1.choice.2=point of inflection with a horizontal tangent.@
qu.1.1.choice.3=cusp point (ie., vertical tangent) maximum.@
qu.1.1.choice.4=cusp point (ie., vertical tangent) minimum.@
qu.1.1.choice.5=horizontal tangent maximum.@
qu.1.1.choice.6=horizontal tangent minimum.@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=HorTanPtofInf@
qu.1.2.comment=<p>$plot</p>
<p>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a horizontal tangent point of inflection. Note that the derivative $Fp does not change sign<strong><font size="3"><font size="2"> </font></font></strong><font size="3"><font size="2">as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>passes through <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. The function either increases on both sides of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> or decreases on both sides of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. So it can't be a max or min!</font></font></p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(-5,5);
$n=rint(3,7,2);
$f1="(x-($a))^$n";
$f2="-($f1)";
$z=rint(2);
$f=switch($z,"$f1","$f2");
$fp=switch($z, "$n*(x-($a))^($n-1)", "-($n)*(x-($a))^($n-1)");
$M=maple("
MathML[ExportPresentation](y=$f),
MathML[ExportPresentation](dy/dx=$fp)
");
$F=switch(0,$M);
$Fp=switch(1,$M);
$plot=plotmaple("plot($f,x=-abs($a)-2..abs($a)+2,y=-5..5,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.2.uid=6b1c1afb-e264-4529-86a0-a48e1b89804f@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficutly=Easy;
  Topic=Max/Min/Pt of Inflection;
  Course=Introduction to Calculus I;
@
qu.1.2.question=<p>The function $F has derivative $Fp. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a</p>@
qu.1.2.answer=2@
qu.1.2.choice.1=point of inflection with a vertical tangent.@
qu.1.2.choice.2=point of inflection with a horizontal tangent.@
qu.1.2.choice.3=cusp point (ie., vertical tangent) maximum.@
qu.1.2.choice.4=cusp point (ie., vertical tangent) minimum.@
qu.1.2.choice.5=horizontal tangent maximum.@
qu.1.2.choice.6=horizontal tangent minimum.@
qu.1.2.fixed=@

qu.1.3.mode=Multiple Choice@
qu.1.3.name=vertical tangentMax@
qu.1.3.comment=<p>$plot</p>
<p>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a vertical tangent maximum. Note that the derivative $Fp <font size="2">changes sign </font><font size="3"><font size="2">from positive to negative as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>passes through <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. The function increases and then decreases, which is a maximum.<br />
</font></font></p>
<p><font size="3"><font size="2">Notice that the exponent on $X is between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>1</mn></mrow></math>. If it had been greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>1</mn></mrow></math>, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> would have been a VERTICAL ASYMPTOTE.<br />
</font></font></p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=rint(-5,5);
$n=rint(3,7,2);
$f="-(x-($a))^(2/$n)";
$fp="-(2/$n)*(x-($a))^(2/$n-1)";
$M=maple("
MathML[ExportPresentation](x-($a)),
MathML[ExportPresentation](y=$f),
MathML[ExportPresentation](dy/dx=$fp)
");
$X=switch(0,$M);
$F=switch(1,$M);
$Fp=switch(2,$M);
$plot=plotmaple("plot(-surd((x-($a))^2,$n),x=-abs($a)-2..abs($a)+2,y=-5..5,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.3.uid=6c58745d-30f4-4339-b30b-c2f5a674b5ca@
qu.1.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Max/Min/Pt of Inflection;
@
qu.1.3.question=<p>The function $F has derivative $Fp. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a</p>@
qu.1.3.answer=3@
qu.1.3.choice.1=point of inflection with a vertical tangent.@
qu.1.3.choice.2=point of inflection with a horizontal tangent.@
qu.1.3.choice.3=cusp point (ie., vertical tangent) maximum.@
qu.1.3.choice.4=cusp point (ie., vertical tangent) minimum.@
qu.1.3.choice.5=horizontal tangent maximum.@
qu.1.3.choice.6=horizontal tangent minimum.@
qu.1.3.fixed=@

qu.1.4.mode=Multiple Choice@
qu.1.4.name=HorTangMin@
qu.1.4.comment=<p>$plot</p>
<p>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a horizontal tangent minimum. Note that the derivative $Fp <font size="3"><font size="2">changes sign</font></font><strong><font size="3"><font size="2"> </font></font></strong><font size="3"><font size="2">from negative to positive as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>passes through <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. The function decreases and then increases, which is a minimum.<br />
</font></font></p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=rint(-5,5);
$n=rint(2,6,2);
$f="(x-($a))^($n)";
$fp="$n*(x-($a))^($n-1)";
$M=maple("
MathML[ExportPresentation](y=$f),
MathML[ExportPresentation](dy/dx=$fp)
");
$F=switch(0,$M);
$Fp=switch(1,$M);
$plot=plotmaple("plot($f,x=-abs($a)-2..abs($a)+2,y=-5..5,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.4.uid=0bd6a27d-4668-45f3-91b9-8a76f4202b91@
qu.1.4.info=  Course=Introduction to Calculus I;
  Topic=Max/Min/Pt of Inflection;
  Difficulty=Easy;
  Author=Jack Weiner, Gord Clement;
@
qu.1.4.question=<p>The function $F has derivative $Fp. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a</p>@
qu.1.4.answer=6@
qu.1.4.choice.1=point of inflection with a vertical tangent.@
qu.1.4.choice.2=point of inflection with a horizontal tangent.@
qu.1.4.choice.3=cusp point (ie., vertical tangent) maximum.@
qu.1.4.choice.4=cusp point (ie., vertical tangent) minimum.@
qu.1.4.choice.5=horizontal tangent maximum.@
qu.1.4.choice.6=horizontal tangent minimum.@
qu.1.4.fixed=@

qu.1.5.mode=Multiple Choice@
qu.1.5.name=HorTangMax@
qu.1.5.comment=<p>$plot</p>
<p>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a horizontal tangent maximum. Note that the derivative $Fp <font size="3"><font size="2">changes sign</font></font><strong><font size="3"><font size="2"> </font></font></strong><font size="3"><font size="2">from positive to negative as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>passes through <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. The function increases and then decreases, which is a maximum.<br />
</font></font></p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$a=rint(-5,5);
$n=rint(2,6,2);
$f="-(x-($a))^($n)";
$fp="-($n)*(x-($a))^(($n)-1)";
$M=maple("
MathML[ExportPresentation](y=$f),
MathML[ExportPresentation](dy/dx=$fp)
");
$F=switch(0,$M);
$Fp=switch(1,$M);
$plot=plotmaple("plot($f,x=-abs($a)-2..abs($a)+2,y=-5..5,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.5.uid=89eb947e-6f0e-4912-9ad9-715e31fafd4d@
qu.1.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Max/Min/Pt of Inflection;
@
qu.1.5.question=<p>The function $F has derivative $Fp. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a</p>@
qu.1.5.answer=5@
qu.1.5.choice.1=point of inflection with a vertical tangent.@
qu.1.5.choice.2=point of inflection with a horizontal tangent.@
qu.1.5.choice.3=cusp point (ie., vertical tangent) maximum.@
qu.1.5.choice.4=cusp point (ie., vertical tangent) minimum.@
qu.1.5.choice.5=horizontal tangent maximum.@
qu.1.5.choice.6=horizontal tangent minimum.@
qu.1.5.fixed=@

qu.1.6.mode=Multiple Choice@
qu.1.6.name=vertical tangentMin@
qu.1.6.comment=<p>$plot</p>
<p>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, there is a vertical tangent minimum. Note that the derivative $Fp <font size="3"><font size="2">changes</font></font><strong><font size="3"><font size="2"> </font></font></strong><font size="3"><font size="2">from negative to positive as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>passes through <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. The function decreases and then increases, which is a minimum.<br />
</font></font></p>
<p><font size="3"><font size="2">Notice also the exponent on $X in the denominator is between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>1</mn></mrow></math>. If it had been greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>1</mn></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> would have been a vertical asymptote.<br />
</font></font></p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$a=rint(-5,5);
$n=rint(3,7,2);
$f="(x-($a))^(2/$n)";
$fp="(2/$n)*(x-($a))^(2/$n-1)";
$M=maple("
MathML[ExportPresentation](x-($a)),
MathML[ExportPresentation](y=$f),
MathML[ExportPresentation](dy/dx=$fp)
");
$X=switch(0,$M);
$F=switch(1,$M);
$Fp=switch(2,$M);
$plot=plotmaple("plot(surd((x-($a))^2,$n),x=-abs($a)-2..abs($a)+2,y=-5..5,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.6.uid=4e1291cc-a3de-4ab4-95e4-4643b122a9db@
qu.1.6.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Max/Min/Pt of Inflection;
@
qu.1.6.question=<p>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, the function $F has a . . .</p>
<p>(Note that the derivative of $F is $Fp.)</p>@
qu.1.6.answer=4@
qu.1.6.choice.1=point of inflection with a vertical tangent.@
qu.1.6.choice.2=point of inflection with a horizontal tangent.@
qu.1.6.choice.3=cusp point (ie., vertical tangent) maximum.@
qu.1.6.choice.4=cusp point (ie., vertical tangent) minimum.@
qu.1.6.choice.5=horizontal tangent maximum.@
qu.1.6.choice.6=horizontal tangent minimum.@
qu.1.6.fixed=@

qu.1.7.mode=Multiple Selection@
qu.1.7.name=THEORY@
qu.1.7.comment=@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=@
qu.1.7.uid=1d015f1a-4598-4304-9246-9cb7007d478a@
qu.1.7.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Max/Min/Pt of Inflection;
@
qu.1.7.question=<p>Click beside each true statement.</p>@
qu.1.7.answer=3, 4, 5@
qu.1.7.choice.1=If f '(a)=0 then f(x) has either a maximum or a minimum point at (a, f(a)).@
qu.1.7.choice.2=If y=f(x) has a maximum or minimum at the point (a, f(a)), then f '(a)=0.@
qu.1.7.choice.3=If (a, f(a)) is a corner point, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msubsup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow></mrow></msubsup><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><msub><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msub><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></math>@
qu.1.7.choice.4=If (a, f(a)) is a cusp point or a point of inflection with a vertical tangent , then |f '(a)| is infinite@
qu.1.7.choice.5=A continuous function has a maximum or minimum point at (a,f(a)) if and only if f '(a)=0 or (a,f(a)) is a corner point or (a,f(a)) is a cusp point or (a,f(a)) is an endpoint.@
qu.1.7.fixed=@

qu.2.topic=Mean Value Theorem@

qu.2.1.mode=Inline@
qu.2.1.name=MVT Cubic@
qu.2.1.comment=<p>$plot</p>
<p>The curve <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$F is black and the line joining <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></math>to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfenced></mrow></math>is red. There are two<strong> </strong>tangent lines that satisfy the MVT<strong> </strong>in this example. The blue one is tangent to&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>c</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$C1. The green line is tangent at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>c</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$C2.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$A=rint(-2,2);
condition:ne($A,0);
$a=rint(-6,-1);
$b=rint(-($a)+1,-2*($a));
$d=rint(-3,6);
condition:ne($d,0);
$D=rint(-3,6);
$f="($A)*(x^3+($d)*x+($D))";
$M=maple("
c1:=-sqrt((($a)^2+($b)^2+($a)*($b))/3):
c2:=sqrt((($a)^2+($b)^2+($a)*($b))/3):
convert(c1,string),
convert(c2,string), 
MathML[ExportPresentation](c1),
MathML[ExportPresentation](c2),
MathML[ExportPresentation](expand($f)),
convert(($A)*(($b)^2+($a)^2+($a)*($b)+($d)),string)
");
$c1=switch(0,$M);
$c2=switch(1,$M);
$C1=switch(2,$M);
$C2=switch(3,$M);
$F=switch(4,$M);
$m=switch(5,$M);
$y1="subs(x=$c1,$f)";
$y2="subs(x=$c2,$f)";
$ay="(subs(x=$a,$f))";
$plot=plotmaple("plot([$f,($m)*(x-($a))+($ay),($m)*(x-($c1))+($y1),
($m)*(x-($c2))+($y2)],x=$a-1..$b+1,thickness=2,
color=[black,red,blue,green]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.1.uid=9a1644e1-9644-40ea-a085-36b6a064f353@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Topic=Mean Value Theorem;
  Course=Introduction to Calculus I;
@
qu.2.1.weighting=1,1,1@
qu.2.1.numbering=alpha@
qu.2.1.part.1.editing=useHTML@
qu.2.1.part.1.question=(Unset)@
qu.2.1.part.1.name=sro_id_1@
qu.2.1.part.1.answer=$m@
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.name=sro_id_2@
qu.2.1.part.2.answer=$c1@
qu.2.1.part.2.mode=Formula@
qu.2.1.part.3.editing=useHTML@
qu.2.1.part.3.question=(Unset)@
qu.2.1.part.3.name=sro_id_3@
qu.2.1.part.3.answer=$c2@
qu.2.1.part.3.mode=Formula@
qu.2.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F satisfies the conditions for the Mean Value Theorem on the interval</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>. Find the slope <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfrac></mrow></math>and the guaranteed <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>value(s)<font size="5"><strong> </strong></font>between&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></math> that satisfies <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfrac></mrow></math></p><p>&nbsp;</p><p><font size="3"><font size="2">If there are two <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math> values enter the smaller one in the space for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>c</mi><mrow><mn>1</mn></mrow></msub></mrow></math> and the larger one in the space for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>c</mi><mrow><mn>2</mn></mrow></msub></mrow></math>.</font></font></p><p><font size="3"><font size="2">If there is just one <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>value, enter this value in the space for </font></font><font size="3"><font size="2"><font size="3"><font size="2"><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>c</mi><mrow><mn>1</mn></mrow></msub></mrow></math> </font></font>and leave </font></font><font size="3"><font size="2"><font size="3"><font size="2"><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>c</mi><mrow><mn>2</mn></mrow></msub></mrow></math></font></font> empty.</font></font></p><p><font size="3"><font size="2">(There can't be more than two <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfrac></mrow></math>values in this example. Be prepared to explain why not on the next midterm.)</font></font></p><p>&nbsp;</p><p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp; <span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>&nbsp;</span><1><span>&nbsp;</span></p><p><span><span>(smaller <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>)&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><msub><mi>c</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>&nbsp;&nbsp;</span><2></span></p><p><span><span>(larger <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>c</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.2.2.mode=Inline@
qu.2.2.name=MVT Quadratic@
qu.2.2.comment=<p>$plot</p>
<p>The curve <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F is black and the line joining&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfenced></mrow></math>&nbsp;is red (where <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>).</p>
<p>The guaranteed tangent line with slope&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfrac></mrow></math> (only one in this example) is in blue. The point of tangency is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></math>$C,$Y1<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math>.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$A=rint(-2,2);
condition:ne($A,0);
$a=rint(-5,0);
$b=rint(0,6);
$d=rint(-3,6);
condition:ne($d,0);
$D=rint(-3,6);
$f="($A)*(x^2+($d)*x+($D))";
$M=maple("
c:=(($a)+($b))/2:
convert(c,string), 
MathML[ExportPresentation](c),
MathML[ExportPresentation](expand($f)),
MathML[ExportPresentation](subs(x=$c,$f)),
convert(($A)*(($b)+($a)+($d)),string)
");
$c=switch(0,$M);
$C=switch(1,$M);
$F=switch(2,$M);
$Y1=switch(3,$M);
$m=switch(4,$M);
$y1="subs(x=$c,$f)";
$ay="subs(x=$a,$f)";
$plot=plotmaple("plot([$f,($m)*(x-($a))+($ay),($m)*(x-($c))+($y1)],
x=$a-2..$b+2,thickness=2,
color=[black,red,blue]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.2.uid=65a9fa35-ee8b-4e1b-9942-123a7c0eab83@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Topic=Mean Value Theorem;
  Course=Introduction to Calculus I;
@
qu.2.2.weighting=1,1@
qu.2.2.numbering=alpha@
qu.2.2.part.1.editing=useHTML@
qu.2.2.part.1.question=(Unset)@
qu.2.2.part.1.name=sro_id_1@
qu.2.2.part.1.answer=$m@
qu.2.2.part.1.mode=Formula@
qu.2.2.part.2.editing=useHTML@
qu.2.2.part.2.question=(Unset)@
qu.2.2.part.2.name=sro_id_2@
qu.2.2.part.2.answer=$c@
qu.2.2.part.2.mode=Formula@
qu.2.2.question=<p>&nbsp;</p><p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F satisfies the conditions for the Mean Value Theorem on the interval <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>. Find the slope <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfrac></mrow></math>and the guaranteed value <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>between&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></math> that satisfies <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>f</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfrac></mrow></math></p><p>(There may be more than one choice for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>f</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfrac></mrow></math> in some examples, but not here.)</p><p>&nbsp;</p><p>&nbsp;<span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>&nbsp;</span><1><span>&nbsp;</span></p><p><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>c</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>&nbsp;&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.3.topic=Rolle's Theorem@

qu.3.1.mode=Formula@
qu.3.1.name=Rolle's Theorem(quad)@
qu.3.1.comment=<p>$plot</p>
<p>The curve is black and the line joining <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$r1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$C</mi></mrow></mfenced></mrow></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$r2</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$C</mi></mrow></mfenced></mrow></math> is red. The guaranteed tangent line with slope <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn></mrow></math> is blue.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(-3,3);
condition:ne($a,0);
$C=rint(-3,3);
condition:ne($C,0);
$r1=rint(-5,1);
$r2=rint($r1+1,5);
$M=maple("
c:=(($r1)+($r2))/2:
f:=($a)*(x-($r1))*(x-($r2))+($C):
y1:=simplify(subs(x=$c,$f)):
convert(c,string),
convert(f,string),
convert(y1,string),
MathML[ExportPresentation](expand($f))
");
$c=switch(0,$M);
$f=switch(1,$M);
$y1=switch(2,$M);
$F=switch(3,$M);
$plot=plotmaple("plot([$f,$C,$y1],x=$r1-2..$r2+2,thickness=2,
color=[black,red,blue]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.1.uid=7fac4fb0-6d68-425a-a252-4a607874ee7d@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Rolle's Theorem;
@
qu.3.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F satisfies the conditions for Rolle's Theorem on the interval <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$r1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$r2</mi></mrow></mfenced></mrow></math>. Find the guaranteed <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>value between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$r1</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$r2</mi></mrow></math> that satisfies <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>f</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></math></p>
<p>(There may be more than one choice for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math> in some examples, but not here.)</p>
<p>&nbsp;</p>@
qu.3.1.answer=$c@

qu.3.2.mode=Formula@
qu.3.2.name=Rolle's Theorem(cubic)@
qu.3.2.comment=<p>$plot</p>
<p>The curve is black and the line joining <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$C</mi></mrow></mfenced></mrow></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$r</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$C</mi></mrow></mfenced></mrow></math> is red. The guaranteed tangent line with slope <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn></mrow></math> is in blue.</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(-3,3);
condition:ne($a,0);
$C=rint(-3,3);
condition:ne($C,0);
$r=rint(3,12,3);
$M=maple("
c:=2*($r)/3:
f:=($a)*x^2*(x-($r))+($C):
y1:=simplify(subs(x=$c,$f)):
convert(c,string),
convert(f,string),
convert(y1,string),
MathML[ExportPresentation](expand($f))

");
$c=switch(0,$M);
$f=switch(1,$M);
$F=switch(3,$M);
$y1=switch(2,$M);
$plot=plotmaple("plot([$f,$C,$y1],x=-2..$r+2,thickness=2,
color=[black,red,blue]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.2.uid=39dc1088-c259-42af-a4f3-ad3bd6a071af@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Rolle's Theorem;
@
qu.3.2.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F satisfies the conditions for Rolle's Theorem on the interval <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$r</mi></mrow></mfenced></mrow></math>. Find the guaranteed <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>value between&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mn>0</mn></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$r</mi></mrow></math> that satisfies <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>f</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.</mn></mrow></math></p>
<p>(There may be more than one choice for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math> in some examples, but not here.)</p>
<p>&nbsp;</p>@
qu.3.2.answer=$c@

