qu.1.topic=Differential@

qu.1.1.mode=Inline@
qu.1.1.name=1/sqrt(x)@
qu.1.1.comment=<p>$plot</p>
<p>The curve is black and the tangent line at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math> is red. Notice how closely the tangent line follows the curve near <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math>.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$f="x^(-1/2)";
$a=switch(rint(5),4,9,16,25,36);
$dx=switch(rint(4),.1,.2,-.1,-.2);
$A=($a)+($dx);
$fp="(-1/2)*x^(-3/2)";
$M=maple("convert(simplify(subs(x=$a,$f)),string),convert(simplify(subs(x=$a,$fp)),string)");
$y1=switch(0,$M);
$m=switch(1,$M);
$plot=plotmaple("plot([$f,($m)*(x-($a))+($y1)],x=$a-5..$a+5,y=$y1-1..$y1+1,thickness=2,
color=[black,red]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.1.uid=d3e97077-19fe-4c7a-ad7e-841926dc301a@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Differentials;
  Sub-Topic=Set-up estimate;
@
qu.1.1.weighting=1,1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer=x^(-1/2)@
qu.1.1.part.1.mode=Formula@
qu.1.1.part.2.name=sro_id_2@
qu.1.1.part.2.answer.units=@
qu.1.1.part.2.numStyle=   @
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.showUnits=false@
qu.1.1.part.2.question=(Unset)@
qu.1.1.part.2.mode=Numeric@
qu.1.1.part.2.grading=exact_value@
qu.1.1.part.2.negStyle=both@
qu.1.1.part.2.answer.num=$a@
qu.1.1.part.3.name=sro_id_3@
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.question=(Unset)@
qu.1.1.part.3.mode=Numeric@
qu.1.1.part.3.grading=exact_value@
qu.1.1.part.3.negStyle=both@
qu.1.1.part.3.answer.num=$dx@
qu.1.1.question=<p>In order to estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$A</mi></mrow></mfrac></mrow></msqrt></mrow></math>, choose an appropriate function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>, a value <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>at which to evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math>&nbsp;and an appropriate value for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>.</p><p>&nbsp;</p><p><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><span>&nbsp;</span><1><span>&nbsp;</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</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><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</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><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=sqrt(x)@
qu.1.2.comment=<p>$plot</p>
<p>The curve is black and the tangent line at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> is red. Notice how closely the tangent line follows the curve near <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'>$a</mi></mrow></math>.</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$f="sqrt(x)";
$a=switch(rint(5),4,9,16,25,36);
$dx=switch(rint(4),.1,.2,-.1,-.2);
$A=($a)+($dx);
$fp="(1/2)*x^(-1/2)";
$M=maple("convert(simplify(subs(x=$a,$f)),string),convert(simplify(subs(x=$a,$fp)),string)");
$y1=switch(0,$M);
$m=switch(1,$M);
$plot=plotmaple("plot([$f,($m)*(x-($a))+($y1)],x=$a-10..$a+10,thickness=2,
color=[black,red]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.2.uid=f8668ea2-a544-485d-84dd-37811b369e73@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Differentials;
  Sub-Topic=Set-up estimate;
@
qu.1.2.weighting=1,1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.answer=sqrt(x)@
qu.1.2.part.1.mode=Formula@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.answer.units=@
qu.1.2.part.2.numStyle=   @
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.showUnits=false@
qu.1.2.part.2.question=(Unset)@
qu.1.2.part.2.mode=Numeric@
qu.1.2.part.2.grading=exact_value@
qu.1.2.part.2.negStyle=both@
qu.1.2.part.2.answer.num=$a@
qu.1.2.part.3.name=sro_id_3@
qu.1.2.part.3.answer.units=@
qu.1.2.part.3.numStyle=   @
qu.1.2.part.3.editing=useHTML@
qu.1.2.part.3.showUnits=false@
qu.1.2.part.3.question=(Unset)@
qu.1.2.part.3.mode=Numeric@
qu.1.2.part.3.grading=exact_value@
qu.1.2.part.3.negStyle=both@
qu.1.2.part.3.answer.num=$dx@
qu.1.2.question=<p>In order to estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$A</mi></mrow></msqrt></mrow></math>, choose an appropriate function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>, a value <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>at which to evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math>&nbsp;and an appropriate value for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>.</p><p>&nbsp;</p><p><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><span>&nbsp;</span><1><span>&nbsp;</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</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><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</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><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.3.mode=Inline@
qu.1.3.name=x^(-1/3)@
qu.1.3.comment=<p>$plot</p>
<p>The curve is black and the tangent line at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> is red. Notice how closely the tangent line follows the curve near <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'>$a</mi></mrow></math>.</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$f="x^(-1/3)";
$a=switch(rint(5),1,8,27,64,125,216);
$dx=switch(rint(4),.1,.2,-.1,-.2);
$A=($a)+($dx);
$fp="(-1/3)*x^(-4/3)";
$M=maple("convert(simplify(subs(x=$a,$f)),string),convert(simplify(subs(x=$a,$fp)),string)");
$y1=switch(0,$M);
$m=switch(1,$M);
$plot=plotmaple("plot([$f,($m)*(x-($a))+($y1)],x=$a-5..$a+5,y=$y1-1..$y1+1, thickness=2,
color=[black,red]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.3.uid=e9da2fc0-a556-49f4-a6cc-a7973b4f34e3@
qu.1.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Differentials;
  Sub-Topic=Set-up estimate;
@
qu.1.3.weighting=1,1,1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.answer=x^(-1/3)@
qu.1.3.part.1.mode=Formula@
qu.1.3.part.2.name=sro_id_2@
qu.1.3.part.2.answer.units=@
qu.1.3.part.2.numStyle=   @
qu.1.3.part.2.editing=useHTML@
qu.1.3.part.2.showUnits=false@
qu.1.3.part.2.question=(Unset)@
qu.1.3.part.2.mode=Numeric@
qu.1.3.part.2.grading=exact_value@
qu.1.3.part.2.negStyle=both@
qu.1.3.part.2.answer.num=$a@
qu.1.3.part.3.name=sro_id_3@
qu.1.3.part.3.answer.units=@
qu.1.3.part.3.numStyle=   @
qu.1.3.part.3.editing=useHTML@
qu.1.3.part.3.showUnits=false@
qu.1.3.part.3.question=(Unset)@
qu.1.3.part.3.mode=Numeric@
qu.1.3.part.3.grading=exact_value@
qu.1.3.part.3.negStyle=both@
qu.1.3.part.3.answer.num=$dx@
qu.1.3.question=<p>In order to estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$A</mi></mrow></mfrac></mrow></mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math>, choose an appropriate function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>, a value <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>at which to evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math>&nbsp;and an appropriate value for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>.</p><p>&nbsp;</p><p><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><span>&nbsp;</span><1><span>&nbsp;</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</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><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</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><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=x^1/3@
qu.1.4.comment=<p>$plot</p>
<p>The curve is black and the tangent line at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> is red. Notice how closely the tangent line follows the curve near <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'>$a</mi></mrow></math>.</p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$f="x^(1/3)";
$a=switch(rint(5),1,8,27,64,125,216);
$dx=switch(rint(4),.1,.2,-.1,-.2);
$A=($a)+($dx);
$fp="(1/3)*x^(-2/3)";
$M=maple("convert(simplify(subs(x=$a,$f)),string),convert(simplify(subs(x=$a,$fp)),string)");
$y1=switch(0,$M);
$m=switch(1,$M);
$plot=plotmaple("plot([$f,($m)*(x-($a))+($y1)],x=$a-10..$a+10,thickness=2,
color=[black,red]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.4.uid=de348317-adaf-48a9-a019-52874cbb535f@
qu.1.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Differentials;
  Sub-Topic=Set-up estimate;
@
qu.1.4.weighting=1,1,1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.answer=x^(1/3)@
qu.1.4.part.1.mode=Formula@
qu.1.4.part.2.name=sro_id_2@
qu.1.4.part.2.answer.units=@
qu.1.4.part.2.numStyle=   @
qu.1.4.part.2.editing=useHTML@
qu.1.4.part.2.showUnits=false@
qu.1.4.part.2.question=(Unset)@
qu.1.4.part.2.mode=Numeric@
qu.1.4.part.2.grading=exact_value@
qu.1.4.part.2.negStyle=both@
qu.1.4.part.2.answer.num=$a@
qu.1.4.part.3.name=sro_id_3@
qu.1.4.part.3.answer.units=@
qu.1.4.part.3.numStyle=   @
qu.1.4.part.3.editing=useHTML@
qu.1.4.part.3.showUnits=false@
qu.1.4.part.3.question=(Unset)@
qu.1.4.part.3.mode=Numeric@
qu.1.4.part.3.grading=exact_value@
qu.1.4.part.3.negStyle=both@
qu.1.4.part.3.answer.num=$dx@
qu.1.4.question=<p>In order to estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math>, choose an appropriate function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>, a value <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>at which to evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math>&nbsp;and an appropriate value for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>.</p><p>&nbsp;</p><p><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><span>&nbsp;</span><1><span>&nbsp;</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</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><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</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><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.2.topic=find dy@

qu.2.1.question=<p>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>$F, then the differential of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math></p>@
qu.2.1.maple=evalb(simplify(($ANSWER/(dx))-($RESPONSE/(d*x)))=0);@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=$ANS@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=Find dy@
qu.2.1.comment=<p><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>$F.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi></mi></mrow></math>$FP, therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</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><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></math>$FP<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>dx</mi></mrow><mrow><mi></mi></mrow></math>.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(1,5);
$poly=maple("randomize(): randpoly(x,degree=3,terms=3,coeffs=rand(-5..5))");
$n=switch(rint(4),"1/2","1/3","-1/2","-1/3");
$f=switch(rint(10),"$poly","sin($a*x)","tan($a*x)","cos($a*x)","csc($a*x)",
"sec($a*x)","cot($a*x)","x^($n)");
$M=maple("fp:=diff($f,x):
convert(fp,string), MathML[ExportPresentation]($f), MathML[ExportPresentation](fp)");
$F=switch(1,$M);
$fp=switch(0,$M);
$FP=switch(2,$M);
$ANS="($fp)*dx";@
qu.2.1.uid=f79d12aa-7737-4d1e-a136-e5d76b1146ed@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Differentials;
  Sub-Topic=Find dy;
  Difficulty=Easy;
@

qu.3.topic=Estimate@

qu.3.1.question=<p>Estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$A</mi></mrow></mfrac></mrow></mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math>, using differentials. Please do not defeat the purpose of the question by using a calculator. Enter your answer as a rational<strong> </strong>number. (For example, enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2.1</mn></mrow></math>as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>21</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math>.)</p>@
qu.3.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=$ANS@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=Estimate (a+dx)^(-1/3)@
qu.3.1.comment=<p>Choose <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><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>x</mi></mrow></mfrac></mrow></mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>x</mi><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow><mrow></mrow></math>, <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'>$a</mi></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$dx</mi></mrow></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow><mrow></mrow></math>, therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>dx</mi></mrow></math>.</p>
<p>This means for our chosen values of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$dx</mi></mrow></mfenced></mrow><mrow><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><mi mathvariant='normal'>$dy</mi></mrow></math>.</p>
<p>Therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$A</mi></mrow></mfrac></mrow></mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>&Delta;y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>dy</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><msup><mi mathvariant='normal'>$a</mi><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$dy</mi></mrow><mrow><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><mi mathvariant='normal'>$ANS</mi></mrow><mrow></mrow><mrow></mrow></math>.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$f="x^(-1/3)";
$a=switch(rint(3),1,8,27);
$dx=switch(rint(4),.1,.2,-.1,-.2);
$A=($a)+($dx);
$fp="(-1/3)*x^(-4/3)";
$M=maple("
m:=simplify(subs(x=$a,$fp)):
y:=simplify(subs(x=$a,$f)):
ans:=convert((m)*($dx)+(y),rational):
dy:=ans-y:
convert(m,string),convert(y,string), convert(ans,string), convert(dy,string)
");
$m=switch(0,$M);
$y1=switch(1,$M);
$ANS=switch(2,$M);
$dy=switch(3,$M);@
qu.3.1.uid=c5262788-497a-4770-b19d-366a3855fd35@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Differentials;
  Sub-Topic=Estimate;
  Difficulty=Medium;
@

qu.3.2.question=<p>Estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$A</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'></mo></mrow></msqrt></mrow></math>, using differentials. Please do not defeat the purpose of the question by using a calculator. Enter your answer as a rational<strong> </strong>number. (For example, enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2.1</mn></mrow></math>as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>21</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math>.)</p>@
qu.3.2.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.2.allow2d=1@
qu.3.2.maple_answer=$ANS@
qu.3.2.type=formula@
qu.3.2.mode=Maple@
qu.3.2.name=Estimate 1/sqrt(a+dx)@
qu.3.2.comment=<p>Choose <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><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>x</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></msup></mrow></math>, <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'>$a</mi></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$dx</mi></mrow></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math>, therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>dx</mi></mrow></math>.</p>
<p>This means for our chosen values of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$dx</mi></mrow></mfenced></mrow><mrow><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><mi mathvariant='normal'>$dy</mi></mrow></math>.</p>
<p>Therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$A</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>&Delta;y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>dy</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><mfrac><mn>1</mn><mrow><msqrt><mrow><mi mathvariant='normal'>$a</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$dy</mi></mrow><mrow><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><mi mathvariant='normal'>$ANS</mi></mrow></math>.</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$f="1/sqrt(x)";
$a=switch(rint(3),1,4,9);
$dx=switch(rint(4),.1,.2,-.1,-.2);
$A=($a)+($dx);
$fp="(-1/2)*x^(-3/2)";
$M=maple("
m:=simplify(subs(x=$a,$fp)):
y:=simplify(subs(x=$a,$f)):
ans:=convert((m)*($dx)+(y),rational):
dy:=ans-y:
convert(m,string),convert(y,string), convert(ans,string), convert(dy,string)
");
$m=switch(0,$M);
$y1=switch(1,$M);
$ANS=switch(2,$M);
$dy=switch(3,$M);@
qu.3.2.uid=bd4be2c2-18e9-41de-bd74-5c266b885e7d@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Differentials;
  Sub-Topic=Estimate;
  Difficulty=Medium;
@

qu.3.3.question=<p>Estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math>, using differentials. Please do not defeat the purpose of the question by using a calculator. Enter your answer as a rational<strong> </strong>number. (For example, enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2.1</mn></mrow></math>as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>21</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math>.)</p>@
qu.3.3.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.3.allow2d=1@
qu.3.3.maple_answer=$ANS@
qu.3.3.type=formula@
qu.3.3.mode=Maple@
qu.3.3.name=Estimate (a+dx)^1/3@
qu.3.3.comment=<p>Choose <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><msup><mi>x</mi><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow><mrow></mrow></math>, <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'>$a</mi></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$dx</mi></mrow></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow><mrow><mi></mi></mrow></math>, therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>dx</mi></mrow></math>.</p>
<p>This means for our chosen values of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$dx</mi></mrow></mfenced></mrow><mrow><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><mi mathvariant='normal'>$dy</mi></mrow></math>.</p>
<p>Therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>$A</mi><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>&Delta;y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>dy</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi mathvariant='normal'>$a</mi><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$dy</mi></mrow><mrow><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><mi mathvariant='normal'>$ANS</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></math>.</p>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$f="x^(1/3)";
$a=switch(rint(3),1,8,27);
$dx=switch(rint(4),.1,.2,-.1,-.2);
$A=($a)+($dx);
$fp="(1/3)*x^(-2/3)";
$M=maple("
m:=simplify(subs(x=$a,$fp)):
y:=simplify(subs(x=$a,$f)):
ans:=convert((m)*($dx)+(y),rational):
dy:=ans-y:
convert(m,string),convert(y,string), convert(ans,string), convert(dy,string)
");
$m=switch(0,$M);
$y1=switch(1,$M);
$ANS=switch(2,$M);
$dy=switch(3,$M);@
qu.3.3.uid=97be2584-05d7-46d2-b172-1283c7c6dc31@
qu.3.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Differentials;
  Sub-Topic=Estimate;
  Difficulty=Medium;
@

qu.3.4.question=<p>Estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mrow><mi mathvariant='normal'>$A</mi></mrow></msqrt></mrow></math>, using differentials. Please do not defeat the purpose of the question by using a calculator. Enter your answer as a rational number. (For example, enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2.1</mn></mrow></math>as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>21</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math>.)</p>@
qu.3.4.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.4.allow2d=1@
qu.3.4.maple_answer=$ANS@
qu.3.4.type=formula@
qu.3.4.mode=Maple@
qu.3.4.name=Estimate sqrt(a+dx)@
qu.3.4.comment=<p>Choose <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><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow><mrow><mi></mi></mrow></math>, <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'>$a</mi></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$dx</mi></mrow></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math>, therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>dx</mi></mrow></math>.</p>
<p>This means for our chosen values of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dy</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$dx</mi></mrow></mfenced></mrow><mrow><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><mi mathvariant='normal'>$dy</mi></mrow></math>.</p>
<p>Therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msqrt><mrow><mi mathvariant='normal'>$A</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>&Delta;y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>dy</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$a</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$dy</mi></mrow><mrow><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><mi mathvariant='normal'>$ANS</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></math>.</p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$f="sqrt(x)";
$a=switch(rint(3),1,4,9);
$dx=switch(rint(4),.1,.2,-.1,-.2);
$A=($a)+($dx);
$fp="(1/2)*x^(-1/2)";
$M=maple("
m:=simplify(subs(x=$a,$fp)):
y:=simplify(subs(x=$a,$f)):
ans:=convert((m)*($dx)+(y),rational):
dy:=ans-y:
convert(m,string),convert(y,string), convert(ans,string), convert(dy,string)
");
$m=switch(0,$M);
$y1=switch(1,$M);
$ANS=switch(2,$M);
$dy=switch(3,$M);@
qu.3.4.uid=9e5e8706-454b-4026-9b04-d1180d33975c@
qu.3.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Differentials;
  Sub-Topic=Estimate;
  Difficulty=Medium;
@

