qu.1.topic=fvsfpGraphs@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=from f, find fp@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(-2,2);
condition:ne($a,0);
$n=rint(2,5);
$f=switch(rint(7),"sin(x)","($a)*sin(x)","cos(x)","($a)*cos(x)",
"1/(x-($a))", "1/(x-($a))^2", "x^($n)-($a)");
$fp=maple("diff($f,x)");
$plot=plotmaple("plot($f,x=-5..5,y=-8..8,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$plota=plotmaple("plot($fp,x=-5..5,y=-8..8,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$plotb=plotmaple("plot(-($fp),x=-5..5,y=-8..8,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$plotc=plotmaple("plot($f,x=-5..5,y=-8..8,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$plotd=plotmaple("plot(-($f),x=-5..5,y=-8..8,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$plote=plotmaple("plot(($fp)^2-5,x=-5..5,y=-8..8,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.1.uid=951aa805-fab9-493a-b39b-5003ebf5f85d@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Derivative;
  Sub-Topic=Graphing derivatives;
  Difficulty=Medium;
@
qu.1.1.question=<p align="center">$plot</p>
<p>Above is the graph of a function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Which of the following is the graph of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msup><mi>f</mi><mi>&prime;</mi></msup></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>?</p>@
qu.1.1.answer=1@
qu.1.1.choice.1=$plota@
qu.1.1.choice.2=$plotb@
qu.1.1.choice.3=$plotc@
qu.1.1.choice.4=$plotd@
qu.1.1.choice.5=$plote@
qu.1.1.fixed=@

qu.2.topic=DerRules@

qu.2.1.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>if <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo></mrow></math>$G).</p>
<p>Remember that in TA, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>sin</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></math>means <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>.</p>
<p>Be very careful entering your answer and don't worry about simplifying. Make sure you use *, brackets, and ^ properly. Use the preview button to check your syntax.</p>@
qu.2.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=diff(($f)*($g),x)@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=ProductRule@
qu.2.1.comment=<p>Using the product rule, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><msup><mi>f</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&prime;</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>g</mi><msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>, take <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>$F and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$G.</p>
<p>&nbsp;</p>
<p>So,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>($F)<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow></math>($GP) + ($G)<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow></math>($F).</p>@
qu.2.1.editing=useHTML@
qu.2.1.hint.1=Use the product rule.@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(1,5);
$d=rint(1,5);
$e=rint(-5,5);
$h=rint(-5,5);
condition:ne($a,$d);
$f=switch(rint(9),"($a)*x+($b)","($a)*x^2+($b)*x+($c)","sin(($a)*x+($b))",
"cos(($a)*x+($b))","tan(($a)*x)","sec(($a)*x)","csc(($a)*x)",
"cot(($a)*x)","($a)*x^3+($b)*x+($c)");
$g=switch(rint(9),"($d)*x+($e)","($d)*x^2+($e)*x+($h)","sin(($d)*x+($e))",
"cos(($d)*x+($e))","tan(($d)*x)","sec(($d)*x)","csc(($d)*x)",
"cot(($d)*x)","($d)*x^3+($e)*x+($h)");
$M=maple("
f:=MathML[ExportPresentation]($f):
g:=MathML[ExportPresentation]($g):
fp:=MathML[ExportPresentation](diff($f,x)):
gp:=MathML[ExportPresentation](diff($g,x)):
f,g,fp,gp
");
$F=switch(0,$M);
$G=switch(1,$M);
$FP=switch(2,$M);
$GP=switch(3,$M);@
qu.2.1.uid=7742f070-0da1-47b4-b2d8-38295c5fdb19@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Derivative Rules;
  Sub-Topic=Product Rule;
@

qu.2.2.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>if <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Q.</p>
<p>&nbsp;</p>
<p>Remember that in TA, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>sin</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></math>means <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>.</p>
<p>Be very careful entering your answer and don't worry about simplifying. Make sure you use *, brackets, ^, and / properly. Use the preview button to check your syntax.</p>@
qu.2.2.maple=ANS:=diff(($f)/($g),x);
ANS2:=(($g)*diff($f,x)-($f)*diff($g,x))/($g)^2;
evalb(simplify((ANS)-($RESPONSE))=0 or
simplify((ANS2)-($RESPONSE))=0);@
qu.2.2.allow2d=1@
qu.2.2.maple_answer=ANS:=diff(($f)/($g),x);
ANS2:=(($g)*diff($f,x)-($f)*diff($g,x))/($g)^2;
ANS or ANS2@
qu.2.2.type=formula@
qu.2.2.mode=Maple@
qu.2.2.name=QuotientRule@
qu.2.2.comment=<p>Using the quotient rule, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>f</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>g</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mi></mi><mi></mi></msup></mrow><mrow><mi></mi></mrow></math>.</p>
<p>&nbsp;</p>
<p>Taking <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>g</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>$G,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></math>($FP)<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow></math>($G) - ($GP)<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math>($F)&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></math>&nbsp;/&nbsp; ($G)<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi></mi><mrow><mn>2</mn></mrow></msup></mrow></math>.</p>
<p>&nbsp;</p>@
qu.2.2.editing=useHTML@
qu.2.2.hint.1=Use the quotient rule.@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(1,5);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint(1,5);
condition:ne($a,$d);
$e=rint(-5,5);
$h=rint(-5,5);
$f=switch(rint(9),"($a)*x+($b)","($a)*x^2+($b)*x+($c)","sin(($a)*x)",
"cos(($a)*x)","tan(($a)*x)","sec(($a)*x)","csc(($a)*x)",
"cot(($a)*x)","($a)*x^3+($b)*x+($c)");
$g=switch(rint(6),"($d)*x+($e)","($d)*x^2+($e)*x+($h)","sin(($d)*x)",
"cos(($d)*x)","tan(($d)*x)","($d)*x^3+($e)*x+($h)");
$M=maple("q:=MathML[ExportPresentation](($f)/($g)):
f:= MathML[ExportPresentation](($f)):
g:= MathML[ExportPresentation](($g)):
fp:= MathML[ExportPresentation](diff($f,x)):
gp:= MathML[ExportPresentation](diff($g,x)):


q,f,g,fp,gp");
$Q=switch(0,$M);
$F=switch(1,$M);
$G=switch(2,$M);
$FP=switch(3,$M);
$GP=switch(4,$M);@
qu.2.2.uid=0fa72e60-9c1a-4f84-9d9c-5672186fbe86@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Derivative Rules;
  Sub-Topic=Quotient Rule;
@

qu.2.3.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>if <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Q.</p>
<p>&nbsp;</p>
<p>Remember that in TA, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>sin</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></math>means <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>.</p>
<p>Be very careful entering your answer and don't worry about simplifying. Make sure you use *, brackets, ^, and / properly. Use the preview button to check your syntax.<strong><br />
</strong></p>@
qu.2.3.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.3.allow2d=1@
qu.2.3.maple_answer=diff($f,x)@
qu.2.3.type=formula@
qu.2.3.mode=Maple@
qu.2.3.name=ChainRule@
qu.2.3.comment=@
qu.2.3.editing=useHTML@
qu.2.3.hint.1=Use the chain rule.@
qu.2.3.solution=@
qu.2.3.algorithm=$a=rint(1,5);
$b=rint(-5,5);
$c=rint(-5,5);
$n=rint(2,10);
$m=rint(-10,10);
condition:eq(gcd($m,$n),1);
$q=switch(rint(2),$m,"$m/$n");
$f=switch(rint(6),"(($a)*x+($b))^($q)","(($a)*x^2+($b)*x+($c))^($q)",
"sin(($a)*x+($b))^($q)","cos(($a)*x+($b))^($q)","tan(($a)*x+($b))^($q)",
"sec(($a)*x+($b))^($q)");
$Q=maple("printf(MathML[ExportPresentation]($f))");@
qu.2.3.uid=a7f9a12a-5436-441b-9286-e4515183b416@
qu.2.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Derivative Rules;
  Sub-Topic=Chain Rule;
@

qu.3.topic=ImplicitDerivatives@

qu.3.1.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math>if $F.</p>@
qu.3.1.maple=ANS2:=-x^(($n)-2)*(($n)-1)*(($a))/y^(2*($n)-1):

evalb(simplify(($ANSWER)-($RESPONSE)=0)or simplify((ANS2)-($RESPONSE)=0));@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=implicitdiff($f,y,x,x)@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=Implicit2ndDeriv@
qu.3.1.comment=@
qu.3.1.editing=useHTML@
qu.3.1.hint.1=Take the derivative implicitly.@
qu.3.1.hint.2=Implicit differentiation is an application of the chain rule.@
qu.3.1.solution=@
qu.3.1.algorithm=$n=rint(2,5);
$a=rint(1,10);
$f="x^($n)+y^($n)=($a)";
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.3.1.uid=9a33a9a4-afac-4d38-b16e-95b5ef9fa23d@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
  Topic=Implicit differentiation;
  Sub-Topic=Implicit second derivative;
@

qu.3.2.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>if $F.</p>@
qu.3.2.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.3.2.allow2d=1@
qu.3.2.maple_answer=implicitdiff($f,y,x)@
qu.3.2.type=formula@
qu.3.2.mode=Maple@
qu.3.2.name=ImplicitDeriv@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.hint.1=Take the derivative implicitly.@
qu.3.2.hint.2=Implicit differentiation is an application of the chain rule.@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(1,5);
$f=switch(rint(5),"($a)*x^2+x*y+($b)*y^3=($c)+$y","sin(($a)*x*y)=x+y",
"cos(($a)*x*y)=x+y","tan(($a)*x*y)=x+y","x^($a)*y^($b)=x-y");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.3.2.uid=95b42a8e-2b38-43ad-8875-67dd878108a6@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Implicit differentiation;
  Sub-Topic=Implicit first derivative;
  Difficulty=Medium;
@

qu.4.topic=HigherDer@

qu.4.1.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math>if <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow></mrow></math>$F.</p>@
qu.4.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.4.1.allow2d=1@
qu.4.1.maple_answer=diff($f,x,x)@
qu.4.1.type=formula@
qu.4.1.mode=Maple@
qu.4.1.name=HigherDer@
qu.4.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$F</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$FP</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$FPP</p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(1,5);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint(1,5);
$n=rint(4,6);
$p=maple("randomize(): randpoly(x,coeffs = rand(-5..5),degree=$n,terms=5)");
$f=switch(rint(8),"(($a)*x+($b))","($a)*x^2+($b)*x+($c)","sin(($a)*x+($b))",
"cos(($a)*x+($b))","($d)*x^3+($a)*x^2+($b)*x+($c)","$p","$p","$p");
$M=maple("f:=MathML[ExportPresentation]($f):
fp:=MathML[ExportPresentation](diff($f,x)):
fpp:=MathML[ExportPresentation](diff($f,x,x)):
f,fp,fpp");
$F=switch(0,$M);
$FP=switch(1,$M);
$FPP=switch(2,$M);@
qu.4.1.uid=f5eb38bb-d3c3-4aec-8239-e43b6093cd69@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Derivatives;
  Sub-Topic=Higher order derivatives;
@

qu.5.topic=RelatedRatesCircle@

qu.5.1.question=<p>A point moves along the circle <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$as</mi></mrow></math> so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math> cm/min. Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></math>at the point <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$x</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$I</mi></mrow></msqrt></mrow></mrow></mfenced></mrow></math>.</p>@
qu.5.1.answer=$ANS cm/min@
qu.5.1.mode=Dimensioned Formula@
qu.5.1.name=dy/dt, y POS@
qu.5.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$as</mi></mrow></math></p>
<p>Therefore, differentiating both sides with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>t</mi></mrow></math>yields:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><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><mn>0</mn></mrow></math>.</p>
<p>Plugging in the information we know, this gives:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi mathvariant='normal'>$I</mi></mrow></msqrt></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></math>.</p>
<p>Solving for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dt</mi></mrow></mfrac></mrow></math>, we find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dt</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$ANS</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>cm/min.</p>@
qu.5.1.editing=useHTML@
qu.5.1.hint.1=Take the derivative with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>t</mi></mrow></math>implicitly on&nbsp;both sides, then plug in the information you know.@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(5,13);
$as=$a^2;
$x=rint(-$a+1,$a);
$I=($a)^2-($x)^2;
$b=rint(-13,13,3);
condition: ne($b,0);
$m=maple("y:=sqrt($I):
ans:=-($x)*($b)/(y):
convert(y,string),convert(ans,string)");
$y=switch(0,$m);
$ANS=switch(1,$m);@
qu.5.1.uid=037dabe5-d251-4693-b259-35c37cd3e345@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Implicit Derivatives;
  Sub-Topic=Related Rates;
@

qu.5.2.question=<p>A point moves along the circle <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$as</mi></mrow></math> so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math> cm/min. Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></math>at the point <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$x</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msqrt><mrow><mi mathvariant='normal'>$I</mi></mrow></msqrt></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math>.</p>@
qu.5.2.answer=$ANS cm/min@
qu.5.2.mode=Dimensioned Formula@
qu.5.2.name=dx/dt, y NEG@
qu.5.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$as</mi></mrow></math>&nbsp;therefore, differentiating both sides with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>t</mi></mrow></math>yields:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>y</mi><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></math></p>
<p>plugging in the information we know, this gives:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></math>.</p>
<p>Solving for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></math>, we find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$ANS</mi></mrow></math>cm/min.</p>@
qu.5.2.editing=useHTML@
qu.5.2.hint.1=Take the derivative with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>t</mi></mrow></math>implicitly on both sides, then plug in the information you know.@
qu.5.2.solution=@
qu.5.2.algorithm=$a=rint(5,13);
$as=$a^2;
$x=rint(-$a+1,$a);
condition: ne($x,0);
$I=($a)^2-($x)^2;
$b=rint(-13,13,3);
condition: ne($b,0);
$m=maple("
y:=-sqrt($I):
ans:=-(y)*($b)/($x):
convert(y,string), convert(ans,string)");
$y=switch(0,$m);
$ANS=switch(1,$m);@
qu.5.2.uid=c226af00-acfc-4cbe-ac14-e634ade4f904@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
  Topic=Implicit derivatives;
  Sub-Topic=Related Rates;
@

qu.6.topic=RelatedRateSnowperson@

qu.6.1.question=<p align="center"><img alt="" src=" __BASE_URI__Images_Intervals/snowballb.jpg" /></p>
<p>&nbsp;</p>
<p>The radius of a spherical snowperson's head is melting under the mild sun at the rate of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$DR</mi></mrow></math> cm/h (centimetres per hour.) Find the rate at which the volume is changing when the volume is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$V</mi></mrow></math>. <strong>Use the abbreviation cc/h for cubic centimetres per hour. </strong>Don't forget to use * for multiplicaton and brackets where necessary.</p>
<p>(The volume of a sphere is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>4</mn><mrow><mn>3</mn></mrow></mfrac><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>r</mi><mrow><mn>3</mn></mrow></msup></mrow></math>.)</p>@
qu.6.1.answer=$ANS cc/h@
qu.6.1.mode=Dimensioned Formula@
qu.6.1.name=VofSphere, Find dV/dt@
qu.6.1.comment=<p>Volume is represented by the formula<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>4</mn><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>r</mi><mrow><mn>3</mn></mrow></msup></mrow></math>.</p>
<p>Taking the derivative with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>t</mi></mrow></math>on both sides, we find</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>V</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></mrow></math> (*).</p>
<p>We know <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$DR</mi></mrow></math>, we need to&nbsp;find what <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>r</mi></mrow></math>is when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$V</mi></mrow></math>. To&nbsp;do&nbsp;this we solve</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>4</mn><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>r</mi><mrow><mn>3</mn></mrow></msup></mrow></math>, giving <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$r</mi></mrow></math> cm.&nbsp;Substituting this values into&nbsp;&nbsp;(*) we find,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>V</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='normal'>4</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$r</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$DR</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$ANS</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math> cc/hr.</p>
<p>&nbsp;</p>@
qu.6.1.editing=useHTML@
qu.6.1.hint.1=Take the derivative with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>t</mi></mrow></math>implicitly, then plug in the information you know.@
qu.6.1.solution=@
qu.6.1.algorithm=$dr=rint(-8,-2,2);
$a=switch(rint(4),27,64,125,216);
$m=maple("
DR:=($dr)/10:
V:=4*Pi*($a)/3:
r:=surd($a,3):
ans:=4*Pi*($r)^2*(DR):
convert(DR, string),convert(V, string),convert(r, string),convert(ans, string)
");
$DR=switch(0,$m);
$V=switch(1,$m);
$r=switch(2,$m);
$ANS=switch(3,$m);@
qu.6.1.uid=63f256fa-8e70-466a-97b7-3190c6925576@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Implicit derivatives;
  Sub-Topic=Related Rates;
@

qu.6.2.question=<p><tt><a href="javascript: window.open('__BASE_URI__Images_Intervals/snowballb.jpg', 'website').focus();"><img alt="" width="442" height="255" src="__BASE_URI__Images_Intervals/snowballb.jpg" /></a></tt></p>
<p>The spherical head of a snowperson is melting under the HOT sun at the rate of $M cc/h (cubic centimetres per hour.) Find the rate at which the radius is changing when the radius <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$r. <strong>Use cm/h for the units. </strong>Don't forget to use * for multiplication. Also, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>1</mn><mrow><mn>8</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow></mrow></mfrac></mrow></math>would be entered as 1/(8*Pi). Don't forget those brackets!</p>
<p>(The volume of a sphere is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>4</mn><mrow><mn>3</mn></mrow></mfrac><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>r</mi><mrow><mn>3</mn></mrow></msup></mrow></math>.)</p>@
qu.6.2.answer=$ANS cm/h@
qu.6.2.mode=Dimensioned Formula@
qu.6.2.name=VofSphere, Find dr/dt@
qu.6.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>4</mn><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>r</mi><mrow><mn>3</mn></mrow></msup></mrow></math>, taking the derivative with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>t</mi></mrow></math>on both sides gives us:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dV</mi><mrow><mi>dt</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></math>, substituting the information we know we find:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>$M</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><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$r</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></mrow></math>, solving for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dr</mi><mrow><mi>dt</mi></mrow></mfrac></mrow></math>&nbsp;we find:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dr</mi><mrow><mi>dt</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$ans cm/h</p>@
qu.6.2.editing=useHTML@
qu.6.2.hint.1=Take the derivative with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>t</mi></mrow></math>implicitly on both sides, then plug in the information you know.@
qu.6.2.solution=@
qu.6.2.algorithm=$M=rint(-200,-50,10);
$r=-$M/10;
$ANS=maple("$M/(4*Pi*($r)^2)");
$ans=maple("MathML[ExportPresentation]($ANS)");@
qu.6.2.uid=65485a52-0976-4caf-a815-66e93cdc5939@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Implicit derivatives;
  Sub-Topic=Related Rates;
@

