qu.1.topic=Rules of Differentiation@

qu.1.1.mode=Inline@
qu.1.1.name=Derivatives - e@
qu.1.1.comment=<p>In general, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>d</mi><mrow><mi>dx</mi></mrow></mfrac><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>d</mi><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.1.editing=useHTML@
qu.1.1.hint.1=Use the chain rule.@
qu.1.1.hint.2=Treat the question as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mi>U</mi></mrow></msup></mrow></mstyle></math>, where U is a function.@
qu.1.1.hint.3=The derivative of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mi>U</mi></mrow></msup></mrow></mstyle></math> with respect to U is just <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mi>U</mi></mrow></msup></mrow></mstyle></math>.@
qu.1.1.solution=@
qu.1.1.algorithm=$v=maple("
randomize():
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
v1[2]:=0:
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=exp(v1):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4,F,v5
");
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.1.uid=37f91b2b-a382-4710-b940-550d4327e6c3@
qu.1.1.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=E;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.maple_answer=$ans@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.libname=@
qu.1.1.part.1.mode=Maple@
qu.1.1.part.1.allow2d=1@
qu.1.1.part.1.plot=@
qu.1.1.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.1.part.1.type=formula@
qu.1.1.question=<p>Differentiate the following function with respect to X:</p><p>F(X)=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Derivatives - exponential, product rule@
qu.1.2.comment=<p>In general, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>d</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>df</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.2.editing=useHTML@
qu.1.2.hint.1=Treat the question as expU), where U is a function.@
qu.1.2.hint.2=Use the chain rule.@
qu.1.2.hint.3=Use product rule.@
qu.1.2.solution=@
qu.1.2.algorithm=$v=maple("
randomize():
tmp:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
tmp[2]:=0:
tmp:=PolynomialTools[FromCoefficientVector](tmp,X):
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
v1[2]:=0:
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=v1*exp(tmp):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.2.uid=ce24ae85-1dfb-4490-be25-9592ccdea355@
qu.1.2.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Exponential, Product Rule;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.2.weighting=1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.maple_answer=$ans@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.libname=@
qu.1.2.part.1.mode=Maple@
qu.1.2.part.1.allow2d=1@
qu.1.2.part.1.plot=@
qu.1.2.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.2.part.1.type=formula@
qu.1.2.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Fpretty.</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.3.mode=Inline@
qu.1.3.name=Derivatives - 4th degree polynomial@
qu.1.3.comment=<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.3.editing=useHTML@
qu.1.3.hint.1=Use the power rule.@
qu.1.3.solution=@
qu.1.3.algorithm=$v=maple("
randomize():
v1:=sort(RandomTools[Generate](polynom(nonzero(range=-9..9,denominator=1),X,degree=4))):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.3.uid=72f8f21a-0bc2-4810-b91d-ab006f114b11@
qu.1.3.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Polynomial;
  Author=Asha Sadanand;
  Difficulty=Easy;
@
qu.1.3.weighting=1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.maple_answer=$ans@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.libname=@
qu.1.3.part.1.mode=Maple@
qu.1.3.part.1.allow2d=1@
qu.1.3.part.1.plot=@
qu.1.3.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.1.type=formula@
qu.1.3.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Derivatives - ln  and exponential@
qu.1.4.comment=<p>In general, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>d</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>df</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>d</mi><mrow><mi>dx</mi></mrow></mfrac><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>df</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.4.editing=useHTML@
qu.1.4.hint.1=Treat the natural log term as ln(U), where U is a function.@
qu.1.4.hint.2=Treat the exponential term as exp(V), where V is a function.@
qu.1.4.hint.3=The derivative of a sum is the sum of the derivatives.@
qu.1.4.solution=@
qu.1.4.algorithm=$v=maple("
randomize():
tmp:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
tmp[2]:=0:
tmp:=PolynomialTools[FromCoefficientVector](tmp,X):
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
v1[2]:=0:
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=exp(v1)+ln(tmp):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.4.uid=b5f8d291-d30e-4b55-aa94-a38f81e021e3@
qu.1.4.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Ln And Exponential;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.1.4.weighting=1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.maple_answer=$ans@
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.libname=@
qu.1.4.part.1.mode=Maple@
qu.1.4.part.1.allow2d=1@
qu.1.4.part.1.plot=@
qu.1.4.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.4.part.1.type=formula@
qu.1.4.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Derivatives - ln@
qu.1.5.comment=<p>In general, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>d</mi><mrow><mi>dx</mi></mrow></mfrac><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac><mfrac><mrow><mi>d</mi><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.5.editing=useHTML@
qu.1.5.hint.1=Use the chain rule.@
qu.1.5.hint.2=Treat the question as ln(U), where U is a function.@
qu.1.5.solution=@
qu.1.5.algorithm=$v=maple("
randomize():
v1:=ln(sort(RandomTools[Generate](polynom(nonzero(range=-9..9,denominator=1),X,degree=2)))):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.5.uid=f01777ba-3267-492b-bf91-9eee03aa60f4@
qu.1.5.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Ln;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.1.5.weighting=1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.maple_answer=$ans@
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.libname=@
qu.1.5.part.1.mode=Maple@
qu.1.5.part.1.allow2d=1@
qu.1.5.part.1.plot=@
qu.1.5.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.5.part.1.type=formula@
qu.1.5.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Derivatives - quotient rule@
qu.1.6.comment=<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.6.editing=useHTML@
qu.1.6.hint.1=Use the quotient rule.@
qu.1.6.solution=@
qu.1.6.algorithm=$v=maple("
randomize():
tmp:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),2)),Vector):
tmp:=PolynomialTools[FromCoefficientVector](tmp,X):
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
v1[2]:=0:
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=tmp/v1:

v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.6.uid=eabeca6f-511a-437f-ad6a-d0420a2ad689@
qu.1.6.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Quotient Rule;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.1.6.weighting=1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.maple_answer=$ans@
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.libname=@
qu.1.6.part.1.mode=Maple@
qu.1.6.part.1.allow2d=1@
qu.1.6.part.1.plot=@
qu.1.6.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.6.part.1.type=formula@
qu.1.6.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Derivatives - exponential, quotient rule@
qu.1.7.comment=<p>In general, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>d</mi><mrow><mi>dx</mi></mrow></mfrac><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>d</mi><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.7.editing=useHTML@
qu.1.7.hint.1=Use quotient rule.@
qu.1.7.hint.2=Use the chain rule.@
qu.1.7.hint.3=Treat the question as expU), where U is a function.@
qu.1.7.solution=@
qu.1.7.algorithm=$v=maple("
randomize():
tmp:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
tmp[2]:=0:
tmp:=PolynomialTools[FromCoefficientVector](tmp,X):
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=exp(tmp)/v1:

v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.7.uid=47f87b50-1f31-4042-8f97-759340baef7c@
qu.1.7.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Exponential, Quotient Rule;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.7.weighting=1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.maple_answer=$ans@
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.libname=@
qu.1.7.part.1.mode=Maple@
qu.1.7.part.1.allow2d=1@
qu.1.7.part.1.plot=@
qu.1.7.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.7.part.1.type=formula@
qu.1.7.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Derivatives - quotient rule; chain rule@
qu.1.8.comment=<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.8.editing=useHTML@
qu.1.8.hint.1=Use the quotient rule.@
qu.1.8.hint.2=Use the chain rule.@
qu.1.8.hint.3=Treat the question as ln(U), where U is a function.@
qu.1.8.solution=@
qu.1.8.algorithm=$v=maple("
randomize():
tmp:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
tmp[2]:=0:
tmp:=PolynomialTools[FromCoefficientVector](tmp,X):
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
v1[2]:=0:
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=ln(tmp)/v1:

v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.8.uid=d41380ae-9893-4302-bca2-ecda7f8219d1@
qu.1.8.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Quotient Rule, Chain Rule;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.8.weighting=1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.maple_answer=$ans@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.question=(Unset)@
qu.1.8.part.1.libname=@
qu.1.8.part.1.mode=Maple@
qu.1.8.part.1.allow2d=1@
qu.1.8.part.1.plot=@
qu.1.8.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.8.part.1.type=formula@
qu.1.8.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.9.mode=Inline@
qu.1.9.name=Derivatives - exponential@
qu.1.9.comment=<p>In general, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>d</mi><mrow><mi>dx</mi></mrow></mfrac><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>d</mi><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.9.editing=useHTML@
qu.1.9.hint.1=Use the chain rule.@
qu.1.9.hint.2=Treat the question as expU), where U is a function.@
qu.1.9.solution=@
qu.1.9.algorithm=$v=maple("
randomize():
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
v1[2]:=0:
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=exp(v1):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.9.uid=35bc3686-616c-49a4-b303-5dea7185da14@
qu.1.9.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Exponential;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.1.9.weighting=1@
qu.1.9.numbering=alpha@
qu.1.9.part.1.name=sro_id_1@
qu.1.9.part.1.maple_answer=$ans@
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.question=(Unset)@
qu.1.9.part.1.libname=@
qu.1.9.part.1.mode=Maple@
qu.1.9.part.1.allow2d=1@
qu.1.9.part.1.plot=@
qu.1.9.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.9.part.1.type=formula@
qu.1.9.question=<p>Differentiate the following function with respect to X:</p><p>F(X)=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.10.mode=Inline@
qu.1.10.name=Derivatives - 3rd degree polynomial@
qu.1.10.comment=<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.10.editing=useHTML@
qu.1.10.hint.1=Use the power rule.@
qu.1.10.solution=@
qu.1.10.algorithm=$v=maple("
randomize():
v1:=RandomTools[Generate](polynom(nonzero(range=-9..9,denominator=1),X,degree=3)):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.10.uid=1b737869-b5da-42f5-a390-38ec0873c3f4@
qu.1.10.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=3rd Degree Polynomial;
  Author=Asha Sadanand;
  Difficulty=Easy;
@
qu.1.10.weighting=1@
qu.1.10.numbering=alpha@
qu.1.10.part.1.name=sro_id_1@
qu.1.10.part.1.maple_answer=$ans@
qu.1.10.part.1.editing=useHTML@
qu.1.10.part.1.question=(Unset)@
qu.1.10.part.1.libname=@
qu.1.10.part.1.mode=Maple@
qu.1.10.part.1.allow2d=1@
qu.1.10.part.1.plot=@
qu.1.10.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.10.part.1.type=formula@
qu.1.10.question=<p>Differentiate the following function with respect to X:</p><p>F(X)=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.11.mode=Inline@
qu.1.11.name=Derivatives - 3rd degree polynomial, chain rule@
qu.1.11.comment=<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.11.editing=useHTML@
qu.1.11.hint.1=Use the power rule.@
qu.1.11.hint.2=Use the chain rule.@
qu.1.11.solution=@
qu.1.11.algorithm=$v=maple("
randomize():
v1:=RandomTools[Generate](polynom(nonzero(range=-9..9,denominator=1),X,degree=3))^3:
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.11.uid=92fa2b99-195e-4047-8892-64e2fddf783d@
qu.1.11.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=3rd Degree Polynomial, Chain Rule;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.1.11.weighting=1@
qu.1.11.numbering=alpha@
qu.1.11.part.1.name=sro_id_1@
qu.1.11.part.1.maple_answer=$ans@
qu.1.11.part.1.editing=useHTML@
qu.1.11.part.1.question=(Unset)@
qu.1.11.part.1.libname=@
qu.1.11.part.1.mode=Maple@
qu.1.11.part.1.allow2d=1@
qu.1.11.part.1.plot=@
qu.1.11.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.11.part.1.type=formula@
qu.1.11.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.12.mode=Inline@
qu.1.12.name=Derivatives - simple quotient rule@
qu.1.12.comment=<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.12.editing=useHTML@
qu.1.12.hint.1=Use the quotient rule.@
qu.1.12.solution=@
qu.1.12.algorithm=$v=maple("
randomize():
tmp:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
tmp[2]:=0:
tmp:=PolynomialTools[FromCoefficientVector](tmp,X):
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),2)),Vector):
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=v1/(tmp):

v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.12.uid=1f2d9ef4-a269-468c-95df-0ea1168fc4a8@
qu.1.12.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Quotient Rule;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.1.12.weighting=1@
qu.1.12.numbering=alpha@
qu.1.12.part.1.name=sro_id_1@
qu.1.12.part.1.maple_answer=$ans@
qu.1.12.part.1.editing=useHTML@
qu.1.12.part.1.question=(Unset)@
qu.1.12.part.1.libname=@
qu.1.12.part.1.mode=Maple@
qu.1.12.part.1.allow2d=1@
qu.1.12.part.1.plot=@
qu.1.12.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.12.part.1.type=formula@
qu.1.12.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.13.mode=Inline@
qu.1.13.name=Derivatives - product rule; chain rule@
qu.1.13.comment=<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.13.editing=useHTML@
qu.1.13.hint.1=Use the product rule.@
qu.1.13.hint.2=Use the chain rule.@
qu.1.13.hint.3=Treat the question as ln(U), where U is a function.@
qu.1.13.solution=@
qu.1.13.algorithm=$v=maple("
randomize():
tmp:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),2)),Vector):
tmp:=PolynomialTools[FromCoefficientVector](tmp,X):
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),2)),Vector):
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=v1*ln(tmp):

v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.13.uid=e9add840-173b-4848-bb11-e12157eb4c90@
qu.1.13.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Product Rule, Chain Rule;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.13.weighting=1@
qu.1.13.numbering=alpha@
qu.1.13.part.1.name=sro_id_1@
qu.1.13.part.1.maple_answer=$ans@
qu.1.13.part.1.editing=useHTML@
qu.1.13.part.1.question=(Unset)@
qu.1.13.part.1.libname=@
qu.1.13.part.1.mode=Maple@
qu.1.13.part.1.allow2d=1@
qu.1.13.part.1.plot=@
qu.1.13.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.13.part.1.type=formula@
qu.1.13.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.14.mode=Inline@
qu.1.14.name=Derivatives - Polynomial and ln@
qu.1.14.comment=<p>In general, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>d</mi><mrow><mi>dx</mi></mrow></mfrac><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac><mfrac><mrow><mi>d</mi><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.1.14.editing=useHTML@
qu.1.14.hint.1=Remember: the derivative of a sum is simply the sum of the derivatives.@
qu.1.14.hint.2=Treat the question as ln(U), where U is a function.@
qu.1.14.solution=@
qu.1.14.algorithm=$v=maple("
randomize():
tmp:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
tmp[2]:=0:
tmp:=PolynomialTools[FromCoefficientVector](tmp,X):
v1:=convert(RandomTools[Generate](list(nonzeroint(range=-9..9),3)),Vector):
v1[2]:=0:
v1:=sort(PolynomialTools[FromCoefficientVector](v1,X)):
v1:=v1-ln(tmp):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.14.uid=f227839e-3723-4019-ba72-b65237c847ef@
qu.1.14.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Polynomial And Ln;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.1.14.weighting=1@
qu.1.14.numbering=alpha@
qu.1.14.part.1.name=sro_id_1@
qu.1.14.part.1.maple_answer=$ans@
qu.1.14.part.1.editing=useHTML@
qu.1.14.part.1.question=(Unset)@
qu.1.14.part.1.libname=@
qu.1.14.part.1.mode=Maple@
qu.1.14.part.1.allow2d=1@
qu.1.14.part.1.plot=@
qu.1.14.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.14.part.1.type=formula@
qu.1.14.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.topic=Higher Order Derivatives: Concavity & Convexity@

qu.2.1.mode=Inline@
qu.2.1.name=Optimization - One Variable with S.O.C. and steps@
qu.2.1.comment=<p>Please provide a single numerical response per answer field. To account for a small amount of rounding, there is a range of values around the correct answer that will be awarded full marks.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$v=maple("
randomize():
v1:=randpoly(X,degree=2):
v2:=diff(v1,X):
v3:=diff(v2,X):
if v3>0 then k:=0
elif v3<0 then k:=1
elif v3=0 then k:=2
end if:
v5:=solve(v2=0,X):
v6:=MathML[ExportPresentation](v1):
convert(v1,string),convert(v2,string),convert(v3,string),k,convert(v5,string),convert(v6,string)
");
$F=switch(5,$v);
$fx=switch(1,$v);
$fxx=switch(2,$v);
$ans1=switch(4,$v);
$k=switch(3,$v);
$ans2=switch($k,'min','max','unknown');
$wrong1=switch($k,'max','unknown','min');
$wrong2=switch($k,'unknown','min','max');@
qu.2.1.uid=3bbb0d74-5306-4047-ad21-833947a8bb05@
qu.2.1.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=One Variable;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Walks Students Through Steps;
@
qu.2.1.weighting=1,1,1,1,1@
qu.2.1.numbering=alpha@
qu.2.1.part.1.name=sro_id_1@
qu.2.1.part.1.maple_answer=$fx@
qu.2.1.part.1.editing=useHTML@
qu.2.1.part.1.question=(Unset)@
qu.2.1.part.1.libname=@
qu.2.1.part.1.mode=Maple@
qu.2.1.part.1.allow2d=1@
qu.2.1.part.1.plot=@
qu.2.1.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.1.type=formula@
qu.2.1.part.2.name=sro_id_2@
qu.2.1.part.2.answer.units=@
qu.2.1.part.2.numStyle=   @
qu.2.1.part.2.editing=useHTML@
qu.2.1.part.2.showUnits=false@
qu.2.1.part.2.question=(Unset)@
qu.2.1.part.2.mode=Numeric@
qu.2.1.part.2.grading=exact_value@
qu.2.1.part.2.negStyle=both@
qu.2.1.part.2.answer.num=0@
qu.2.1.part.3.name=sro_id_3@
qu.2.1.part.3.answer.units=@
qu.2.1.part.3.numStyle=   arithmetic@
qu.2.1.part.3.editing=useHTML@
qu.2.1.part.3.showUnits=false@
qu.2.1.part.3.err=0.05@
qu.2.1.part.3.question=(Unset)@
qu.2.1.part.3.mode=Numeric@
qu.2.1.part.3.grading=toler_abs@
qu.2.1.part.3.negStyle=minus@
qu.2.1.part.3.answer.num=$ans1@
qu.2.1.part.4.name=sro_id_4@
qu.2.1.part.4.maple_answer=$fxx@
qu.2.1.part.4.editing=useHTML@
qu.2.1.part.4.question=(Unset)@
qu.2.1.part.4.libname=@
qu.2.1.part.4.mode=Maple@
qu.2.1.part.4.allow2d=1@
qu.2.1.part.4.plot=@
qu.2.1.part.4.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.4.type=formula@
qu.2.1.part.5.grader=exact@
qu.2.1.part.5.name=sro_id_5@
qu.2.1.part.5.editing=useHTML@
qu.2.1.part.5.display.permute=true@
qu.2.1.part.5.answer.3=$wrong2@
qu.2.1.part.5.question=(Unset)@
qu.2.1.part.5.answer.2=$wrong1@
qu.2.1.part.5.answer.1=$ans2@
qu.2.1.part.5.mode=List@
qu.2.1.part.5.display=menu@
qu.2.1.part.5.credit.3=0.0@
qu.2.1.part.5.credit.2=0.0@
qu.2.1.part.5.credit.1=1.0@
qu.2.1.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">Find the first order condition for optimizing this function:</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>F</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></mstyle></math><span>&nbsp;</span><1><span> </span>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above first order condition, what is the value of the critical point (stationary point)?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">What is the second derivative of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math> with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</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></mstyle></math><span>&nbsp;</span><4><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above second derivative, is the critical point a maximum, minimum or is it indeterminate?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi></mrow></mstyle></math> is a <span>&nbsp;</span><5><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.2.2.mode=Inline@
qu.2.2.name=One variable -- SOC@
qu.2.2.comment=@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=range(0,2);
$A=switch($a, "maximum", "minimum", "point of inflection");
$Ans1=switch($a,"F''(X)<0","F''(X)>0","F''(X)=0");
$W11=switch($a,"F''(X)>0","F''(X)=0","F''(X)<0");
$W12=switch($a,"F''(X)=0","F''(X)<0","F''(X)>0");
$Ans2=switch($a,"sufficient","sufficient","necessary");
$W21=switch($a,"necessary","necessary","sufficient");@
qu.2.2.uid=c65493e4-1996-495a-bd5e-e82ed2c0320f@
qu.2.2.info=  Course=Introductory Mathematical Economics;
  Topic=Second Order Conditions -One Variable;
  Sub-Topic=Fill Blanks;
  Difficulty=Easy;
  Author=Asha Sadanand;
@
qu.2.2.weighting=1,1@
qu.2.2.numbering=alpha@
qu.2.2.part.1.grader=exact@
qu.2.2.part.1.name=sro_id_1@
qu.2.2.part.1.editing=useHTML@
qu.2.2.part.1.display.permute=true@
qu.2.2.part.1.answer.3=$W12@
qu.2.2.part.1.question=(Unset)@
qu.2.2.part.1.answer.2=$W11@
qu.2.2.part.1.answer.1=$Ans1@
qu.2.2.part.1.mode=List@
qu.2.2.part.1.display=menu@
qu.2.2.part.1.credit.3=0.0@
qu.2.2.part.1.credit.2=0.0@
qu.2.2.part.1.credit.1=1.0@
qu.2.2.part.2.grader=exact@
qu.2.2.part.2.name=sro_id_2@
qu.2.2.part.2.editing=useHTML@
qu.2.2.part.2.display.permute=true@
qu.2.2.part.2.question=(Unset)@
qu.2.2.part.2.answer.2=$W21@
qu.2.2.part.2.answer.1=$Ans2@
qu.2.2.part.2.mode=List@
qu.2.2.part.2.display=menu@
qu.2.2.part.2.credit.2=0.0@
qu.2.2.part.2.credit.1=1.0@
qu.2.2.question=<p>To find the extrema of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math> we differentiate and set the derivative to zero, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>. Then we check the second order condition which for a $A is that&nbsp; <1><span> </span>is a <span>&nbsp;</span><2><span>&nbsp; </span>condition.</p>@

qu.2.3.mode=Inline@
qu.2.3.name=Optimization - One Variable with S.O.C. - no steps@
qu.2.3.comment=<p>Please provide a single numerical response per answer field.&nbsp; To account for a&nbsp;small amount of rounding, there is a range of values around the correct answer that will be awarded full marks.</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$v=maple("
randomize():
v1:=randpoly(X,degree=2):
v2:=diff(v1,X):
v3:=diff(v2,X):
if v3>0 then k:=0
elif v3<0 then k:=1
elif v3=0 then k:=2
end if:
v5:=solve(v2=0,X):
v6:=MathML[ExportPresentation](v1):
convert(v1,string),convert(v2,string),convert(v3,string),k,convert(v5,string),convert(v6,string)
");
$F=switch(5,$v);
$fx=switch(1,$v);
$fxx=switch(2,$v);
$ans1=switch(4,$v);
$k=switch(3,$v);
$ans2=switch($k,'min','max','unknown');
$wrong1=switch($k,'max','unknown','min');
$wrong2=switch($k,'unknown','min','max');@
qu.2.3.uid=82024123-d76e-43c7-8d0d-a075422f52fb@
qu.2.3.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=One Variable;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Walks Students Through Steps;
@
qu.2.3.weighting=1,1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.name=sro_id_1@
qu.2.3.part.1.answer.units=@
qu.2.3.part.1.numStyle=   arithmetic@
qu.2.3.part.1.editing=useHTML@
qu.2.3.part.1.showUnits=false@
qu.2.3.part.1.err=0.05@
qu.2.3.part.1.question=(Unset)@
qu.2.3.part.1.mode=Numeric@
qu.2.3.part.1.grading=toler_abs@
qu.2.3.part.1.negStyle=minus@
qu.2.3.part.1.answer.num=$ans1@
qu.2.3.part.2.grader=exact@
qu.2.3.part.2.name=sro_id_2@
qu.2.3.part.2.editing=useHTML@
qu.2.3.part.2.display.permute=true@
qu.2.3.part.2.answer.3=$wrong2@
qu.2.3.part.2.question=(Unset)@
qu.2.3.part.2.answer.2=$wrong1@
qu.2.3.part.2.answer.1=$ans2@
qu.2.3.part.2.mode=List@
qu.2.3.part.2.display=menu@
qu.2.3.part.2.credit.3=0.0@
qu.2.3.part.2.credit.2=0.0@
qu.2.3.part.2.credit.1=1.0@
qu.2.3.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">&nbsp;</p><p align="left">What is the value of the critical point (stationary point) for this function?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">Is this critical point a maximum, minimum or is it indeterminate?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi></mrow></mstyle></math> is a <span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.2.4.mode=Inline@
qu.2.4.name=Optimization - Quadratic One Variable, with S.O.C. and steps@
qu.2.4.comment=<pre><span class="bold"><strong><strong>Here is what the function looks like:</strong></strong></span></pre>
<pre><span class="bold"><strong><strong>$plot<br /></strong></strong></span></pre>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$neg=range(-1,1,2);
$neg2=range(-1,1,2);
$a=$neg*range(-4,-2);
$b=$neg*range(2,4);
$d=$neg2*range(1,4);
$v=maple("
v1:=($d)*(((X^4)/4)-(($a)+($b)+1)*((X^3)/3)+(($a)*($b)+($b)+($a))*((X^2)/2)-($a)*($b)*X):
v2:=MathML[ExportPresentation](v1):
v3:=diff(v1,X):
v4:=diff(v3,X):

if $neg=1 then
v5:=eval(v4,X=$a)
else v5:=eval(v4,X=$b)
end if:

v6:=eval(v4,X=1):

if $neg=1 then
v7:=eval(v4,X=$b)
else v7:=eval(v4,X=$a)
end if:

if $neg=1 then
crit1:=$a
else crit1:=$b
end if:

if $neg=1 then
crit3:=$b
else crit3:=$a
end if:

if $neg2=1 then
k:=1
else k:=0
end if:
v1,v2,convert(v3,string),convert(v4,string),v5,v6,v7,crit1,crit3,k
");
$Fmath=switch(0,$v);
$F=switch(1,$v);
$crit1=switch(7,$v);
$crit2=1;
$crit3=switch(8,$v);
$eval1=switch(4,$v);
$eval2=switch(5,$v);
$eval3=switch(6,$v);
$k=switch(9,$v);
$ans1=switch($k,"a local maximum","a local minimum");
$wrong1=switch($k,"a local minimum","a local maximum");
$ans2=switch($k,"a local minimum","a local maximum");
$wrong2=switch($k,"a local maximum","a local minimum");
$ans3=switch($k,"a local maximum","a local minimum");
$wrong3=switch($k,"a local minimum","a local maximum");
$fx=switch(2,$v);
$fxx=switch(3,$v);
$plot=plotmaple("plot($Fmath, X=-5..5), plotdevice='gif', plotoptions='height=250, width=250'");@
qu.2.4.uid=e75bc9ab-552c-4975-9745-efba2a2a8f8e@
qu.2.4.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=One Variable;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Walks Students Through Steps;
@
qu.2.4.weighting=1,1,1,1,1,1,1,1,1,1,1,1@
qu.2.4.numbering=alpha@
qu.2.4.part.1.name=sro_id_1@
qu.2.4.part.1.maple_answer=$fx@
qu.2.4.part.1.editing=useHTML@
qu.2.4.part.1.question=(Unset)@
qu.2.4.part.1.libname=@
qu.2.4.part.1.mode=Maple@
qu.2.4.part.1.allow2d=1@
qu.2.4.part.1.plot=@
qu.2.4.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.4.part.1.type=formula@
qu.2.4.part.2.name=sro_id_2@
qu.2.4.part.2.answer.units=@
qu.2.4.part.2.numStyle=   @
qu.2.4.part.2.editing=useHTML@
qu.2.4.part.2.showUnits=false@
qu.2.4.part.2.question=(Unset)@
qu.2.4.part.2.mode=Numeric@
qu.2.4.part.2.grading=exact_value@
qu.2.4.part.2.negStyle=both@
qu.2.4.part.2.answer.num=0@
qu.2.4.part.3.name=sro_id_3@
qu.2.4.part.3.answer.units=@
qu.2.4.part.3.numStyle=   @
qu.2.4.part.3.editing=useHTML@
qu.2.4.part.3.showUnits=false@
qu.2.4.part.3.question=(Unset)@
qu.2.4.part.3.mode=Numeric@
qu.2.4.part.3.grading=exact_value@
qu.2.4.part.3.negStyle=both@
qu.2.4.part.3.answer.num=$crit1@
qu.2.4.part.4.name=sro_id_4@
qu.2.4.part.4.answer.units=@
qu.2.4.part.4.numStyle=   @
qu.2.4.part.4.editing=useHTML@
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qu.2.4.part.4.question=(Unset)@
qu.2.4.part.4.mode=Numeric@
qu.2.4.part.4.grading=exact_value@
qu.2.4.part.4.negStyle=both@
qu.2.4.part.4.answer.num=$crit2@
qu.2.4.part.5.name=sro_id_5@
qu.2.4.part.5.answer.units=@
qu.2.4.part.5.numStyle=   @
qu.2.4.part.5.editing=useHTML@
qu.2.4.part.5.showUnits=false@
qu.2.4.part.5.question=(Unset)@
qu.2.4.part.5.mode=Numeric@
qu.2.4.part.5.grading=exact_value@
qu.2.4.part.5.negStyle=both@
qu.2.4.part.5.answer.num=$crit3@
qu.2.4.part.6.name=sro_id_6@
qu.2.4.part.6.maple_answer=$fxx@
qu.2.4.part.6.editing=useHTML@
qu.2.4.part.6.question=(Unset)@
qu.2.4.part.6.libname=@
qu.2.4.part.6.mode=Maple@
qu.2.4.part.6.allow2d=1@
qu.2.4.part.6.plot=@
qu.2.4.part.6.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.4.part.6.type=formula@
qu.2.4.part.7.name=sro_id_7@
qu.2.4.part.7.answer.units=@
qu.2.4.part.7.numStyle=   arithmetic@
qu.2.4.part.7.editing=useHTML@
qu.2.4.part.7.showUnits=false@
qu.2.4.part.7.err=0.01@
qu.2.4.part.7.question=(Unset)@
qu.2.4.part.7.mode=Numeric@
qu.2.4.part.7.grading=toler_abs@
qu.2.4.part.7.negStyle=minus@
qu.2.4.part.7.answer.num=$eval1@
qu.2.4.part.8.name=sro_id_8@
qu.2.4.part.8.answer.units=@
qu.2.4.part.8.numStyle=   arithmetic@
qu.2.4.part.8.editing=useHTML@
qu.2.4.part.8.showUnits=false@
qu.2.4.part.8.err=0.01@
qu.2.4.part.8.question=(Unset)@
qu.2.4.part.8.mode=Numeric@
qu.2.4.part.8.grading=toler_abs@
qu.2.4.part.8.negStyle=minus@
qu.2.4.part.8.answer.num=$eval2@
qu.2.4.part.9.name=sro_id_9@
qu.2.4.part.9.answer.units=@
qu.2.4.part.9.numStyle=   arithmetic@
qu.2.4.part.9.editing=useHTML@
qu.2.4.part.9.showUnits=false@
qu.2.4.part.9.err=0.01@
qu.2.4.part.9.question=(Unset)@
qu.2.4.part.9.mode=Numeric@
qu.2.4.part.9.grading=toler_abs@
qu.2.4.part.9.negStyle=minus@
qu.2.4.part.9.answer.num=$eval3@
qu.2.4.part.10.grader=exact@
qu.2.4.part.10.name=sro_id_10@
qu.2.4.part.10.editing=useHTML@
qu.2.4.part.10.display.permute=true@
qu.2.4.part.10.answer.3=indeterminate@
qu.2.4.part.10.question=(Unset)@
qu.2.4.part.10.answer.2=$wrong1@
qu.2.4.part.10.answer.1=$ans1@
qu.2.4.part.10.mode=List@
qu.2.4.part.10.display=menu@
qu.2.4.part.10.credit.3=0.0@
qu.2.4.part.10.credit.2=0.0@
qu.2.4.part.10.credit.1=1.0@
qu.2.4.part.11.grader=exact@
qu.2.4.part.11.name=sro_id_11@
qu.2.4.part.11.editing=useHTML@
qu.2.4.part.11.display.permute=true@
qu.2.4.part.11.answer.3=indeterminate@
qu.2.4.part.11.question=(Unset)@
qu.2.4.part.11.answer.2=$wrong2@
qu.2.4.part.11.answer.1=$ans2@
qu.2.4.part.11.mode=List@
qu.2.4.part.11.display=menu@
qu.2.4.part.11.credit.3=0.0@
qu.2.4.part.11.credit.2=0.0@
qu.2.4.part.11.credit.1=1.0@
qu.2.4.part.12.grader=exact@
qu.2.4.part.12.name=sro_id_12@
qu.2.4.part.12.editing=useHTML@
qu.2.4.part.12.display.permute=true@
qu.2.4.part.12.answer.3=indeterminate@
qu.2.4.part.12.question=(Unset)@
qu.2.4.part.12.answer.2=$wrong3@
qu.2.4.part.12.answer.1=$ans3@
qu.2.4.part.12.mode=List@
qu.2.4.part.12.display=menu@
qu.2.4.part.12.credit.3=0.0@
qu.2.4.part.12.credit.2=0.0@
qu.2.4.part.12.credit.1=1.0@
qu.2.4.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">Find the first order condition for optimizing this function:</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>F</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></mstyle></math><span>&nbsp;</span><1><span> </span>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above first order condition, what are the values of the critical points (stationary points)?</p><p align="left">(Enter one critical point in each answer box, starting with the lowest and ending with the highest. Eg. -8, then 0 then 6)</p><p align="left">&nbsp;</p><p align="left">critical point #1=<span>&nbsp;</span><3><span>&nbsp;</span></p><p align="left">critical point #2=<span>&nbsp;</span><4><span>&nbsp;</span></p><p align="left">critical point #3=<span>&nbsp;</span><5><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">What is the second derivative of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math> with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>?</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</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></mstyle></math><span>&nbsp;</span><6><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">Evaluate the second derivative at each critical point.</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</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='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>critical</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>point</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&num;</mo><mi>1</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><7><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi mathvariant='normal'>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>critical</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>point</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&num;</mo><mi>2</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi></mi></mrow></msup><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>critical</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>point</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&num;</mo><mi>3</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><9><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left"><span>Given what you evaluated above, state whether each point is a local maximum, a local minimum, or it is indeterminate.</span></p><p align="left"><span>critical point #1 is </span><span>&nbsp;</span><10><span>&nbsp;</span></p><p align="left">critical point #2 is <span>&nbsp;</span><11><span>&nbsp;</span></p><p align="left">critical point #3 is <span>&nbsp;</span><12><span>&nbsp;</span></p>@

qu.3.topic=Applications/Optimization@

qu.3.1.mode=Inline@
qu.3.1.name=Optimization - One Variable with S.O.C. and steps@
qu.3.1.comment=<p>Please provide a single numerical response per answer field. To account for a small amount of rounding, there is a range of values around the correct answer that will be awarded full marks.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$v=maple("
randomize():
v1:=randpoly(X,degree=2):
v2:=diff(v1,X):
v3:=diff(v2,X):
if v3>0 then k:=0
elif v3<0 then k:=1
elif v3=0 then k:=2
end if:
v5:=solve(v2=0,X):
v6:=MathML[ExportPresentation](v1):
convert(v1,string),convert(v2,string),convert(v3,string),k,convert(v5,string),convert(v6,string)
");
$F=switch(5,$v);
$fx=switch(1,$v);
$fxx=switch(2,$v);
$ans1=switch(4,$v);
$k=switch(3,$v);
$ans2=switch($k,'min','max','unknown');
$wrong1=switch($k,'max','unknown','min');
$wrong2=switch($k,'unknown','min','max');@
qu.3.1.uid=3bbb0d74-5306-4047-ad21-833947a8bb05@
qu.3.1.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=One Variable;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Walks Students Through Steps;
@
qu.3.1.weighting=1,1,1,1,1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.maple_answer=$fx@
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.libname=@
qu.3.1.part.1.mode=Maple@
qu.3.1.part.1.allow2d=1@
qu.3.1.part.1.plot=@
qu.3.1.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.1.part.1.type=formula@
qu.3.1.part.2.name=sro_id_2@
qu.3.1.part.2.answer.units=@
qu.3.1.part.2.numStyle=   @
qu.3.1.part.2.editing=useHTML@
qu.3.1.part.2.showUnits=false@
qu.3.1.part.2.question=(Unset)@
qu.3.1.part.2.mode=Numeric@
qu.3.1.part.2.grading=exact_value@
qu.3.1.part.2.negStyle=both@
qu.3.1.part.2.answer.num=0@
qu.3.1.part.3.name=sro_id_3@
qu.3.1.part.3.answer.units=@
qu.3.1.part.3.numStyle=   arithmetic@
qu.3.1.part.3.editing=useHTML@
qu.3.1.part.3.showUnits=false@
qu.3.1.part.3.err=0.05@
qu.3.1.part.3.question=(Unset)@
qu.3.1.part.3.mode=Numeric@
qu.3.1.part.3.grading=toler_abs@
qu.3.1.part.3.negStyle=minus@
qu.3.1.part.3.answer.num=$ans1@
qu.3.1.part.4.name=sro_id_4@
qu.3.1.part.4.maple_answer=$fxx@
qu.3.1.part.4.editing=useHTML@
qu.3.1.part.4.question=(Unset)@
qu.3.1.part.4.libname=@
qu.3.1.part.4.mode=Maple@
qu.3.1.part.4.allow2d=1@
qu.3.1.part.4.plot=@
qu.3.1.part.4.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.1.part.4.type=formula@
qu.3.1.part.5.grader=exact@
qu.3.1.part.5.name=sro_id_5@
qu.3.1.part.5.editing=useHTML@
qu.3.1.part.5.display.permute=true@
qu.3.1.part.5.answer.3=$wrong2@
qu.3.1.part.5.question=(Unset)@
qu.3.1.part.5.answer.2=$wrong1@
qu.3.1.part.5.answer.1=$ans2@
qu.3.1.part.5.mode=List@
qu.3.1.part.5.display=menu@
qu.3.1.part.5.credit.3=0.0@
qu.3.1.part.5.credit.2=0.0@
qu.3.1.part.5.credit.1=1.0@
qu.3.1.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">Find the first order condition for optimizing this function:</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>F</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></mstyle></math><span>&nbsp;</span><1><span> </span>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above first order condition, what is the value of the critical point (stationary point)?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">What is the second derivative of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math> with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</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></mstyle></math><span>&nbsp;</span><4><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above second derivative, is the critical point a maximum, minimum or is it indeterminate?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi></mrow></mstyle></math> is a <span>&nbsp;</span><5><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.3.2.mode=Inline@
qu.3.2.name=Perfect Competition - two corner solutions with fixed cost@
qu.3.2.comment=<p>In the short run, it may still be profitable for the firm to produce some quantity even though it is making negative profit. This is because its only alternative is to make -$FC.</p>
<p>However, if the price is so low that the firm cannot even equate MC=P, it is optimal for the firm to produce zero in both the short run and the long run.</p>@
qu.3.2.editing=useHTML@
qu.3.2.hint.1=quantities cannot be negative@
qu.3.2.hint.2=check for corner solutions@
qu.3.2.solution=@
qu.3.2.algorithm=$a=range(1,5);
$b=range(1,10);
$FC=range(10,20);
$TC=$a*q^2+$b*q+$FC;
$v=maple("
v1:=MathML[ExportPresentation]($a*q^2+$b*q+$FC):
v1
");
$cpretty=switch(0,$v);
$P=range(5,15);
$q=decimal(2,($P-$b)/(2*$a));
$profit=$q*$P-($a*$q^2+$b*$q+$FC);
$ansLR=abs(if(lt($profit,0),0,$q));
$ansSR=abs(if(lt($P,$b),0,$q));@
qu.3.2.uid=46716b0f-3000-4ea3-8ab5-3867a81e3032@
qu.3.2.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Corner Solutions;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.3.2.weighting=1,1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.answer.units=@
qu.3.2.part.1.numStyle=   @
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.showUnits=false@
qu.3.2.part.1.question=(Unset)@
qu.3.2.part.1.mode=Numeric@
qu.3.2.part.1.grading=exact_value@
qu.3.2.part.1.negStyle=both@
qu.3.2.part.1.answer.num=$ansSR@
qu.3.2.part.2.name=sro_id_2@
qu.3.2.part.2.answer.units=@
qu.3.2.part.2.numStyle=   @
qu.3.2.part.2.editing=useHTML@
qu.3.2.part.2.showUnits=false@
qu.3.2.part.2.question=(Unset)@
qu.3.2.part.2.mode=Numeric@
qu.3.2.part.2.grading=exact_value@
qu.3.2.part.2.negStyle=both@
qu.3.2.part.2.answer.num=$ansLR@
qu.3.2.question=<p>A perfectly competitive firm faces a price of P=$P and has a total cost function of C=$cpretty.</p><p>&nbsp;</p><p>What quantity should the firm produce in the short run? Remember that this means the firm still needs to pay the fixed part of its cost function.</p><p>&nbsp;</p><p>(Round answer to two decimal places if necessary. For example, 1.6666 becomes 1.67.)</p><p>q SR=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What quantity should the firm produce in the long run? (If now the firm has the option of shutting down and paying no cost.)</p><p>&nbsp;</p><p>(Round to two decimal places.)</p><p>q LR=<span>&nbsp;</span><2><span>&nbsp;</span></p>@

qu.3.3.mode=Inline@
qu.3.3.name=Perfect Competition - corner solution with fixed cost@
qu.3.3.comment=<p>Please provide a single numerical response per answer field. To account for a small amount of rounding, there is a range of values around the correct answer that will be awarded full marks.</p>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$a=range(1,5);
$b=range(1,10);
$FC=range(10,20);
$TC=$a*q^2+$b*q+$FC;
$v=maple("
v1:=MathML[ExportPresentation]($a*q^2+$b*q+$FC):
v1
");
$cpretty=switch(0,$v);
$P=range(11,20);
$q=decimal(2,($P-$b)/(2*$a));
$profit=$q*$P-($a*$q^2+$b*$q+$FC);
$ansLR=if(lt($profit,0),0,$q);@
qu.3.3.uid=b05fe16d-e8d2-453c-b104-9009a8468cc1@
qu.3.3.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Corner Solutions;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.3.3.weighting=1,1@
qu.3.3.numbering=alpha@
qu.3.3.part.1.name=sro_id_1@
qu.3.3.part.1.answer.units=@
qu.3.3.part.1.numStyle=   @
qu.3.3.part.1.editing=useHTML@
qu.3.3.part.1.showUnits=false@
qu.3.3.part.1.err=0.05@
qu.3.3.part.1.question=(Unset)@
qu.3.3.part.1.mode=Numeric@
qu.3.3.part.1.grading=toler_abs@
qu.3.3.part.1.negStyle=both@
qu.3.3.part.1.answer.num=$q@
qu.3.3.part.2.name=sro_id_2@
qu.3.3.part.2.answer.units=@
qu.3.3.part.2.numStyle=   @
qu.3.3.part.2.editing=useHTML@
qu.3.3.part.2.showUnits=false@
qu.3.3.part.2.err=0.05@
qu.3.3.part.2.question=(Unset)@
qu.3.3.part.2.mode=Numeric@
qu.3.3.part.2.grading=toler_abs@
qu.3.3.part.2.negStyle=both@
qu.3.3.part.2.answer.num=$ansLR@
qu.3.3.question=<p>A perfectly competitive firm faces a price of P=$P and has a total cost function of C=$cpretty.</p><p>&nbsp;</p><p>What quantity should the firm produce in the short run? Remember that this means the firm still needs to pay the fixed part of its cost function.</p><p>&nbsp;</p><p>(Round answer to two decimal places if necessary. For example, 1.6666 becomes 1.67.)</p><p>q SR=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What quantity should the firm produce in the long run?</p><p>(Round to two decimal places.)</p><p>q LR=<span>&nbsp;</span><2><span>&nbsp;</span></p>@

qu.3.4.mode=Inline@
qu.3.4.name=Perfect Competition - P=MC - (q whole number, messy cost)@
qu.3.4.comment=<p>The marginal cost is the derivative of $Cpretty, which is $MCpretty.</p>
<p>Setting P equal to MC, and solving for q gets q=$q.</p>@
qu.3.4.editing=useHTML@
qu.3.4.hint.1=Competitive firms pick the quantity that makes P=MC.@
qu.3.4.solution=@
qu.3.4.algorithm=$P=range(100,200);
$q=range(10,20);
$b=range(1,5);
$e=range(10,50);
$m=maple("
randomize():
m1:=solve(a*$q+$b=$P,a):
m2:=int(m1*q+$b,q)+$e:
m3:=MathML[ExportPresentation](m2):
m4:=MathML[ExportPresentation](m1*q+$b):
convert(m1,string),convert(m2,string),convert(m3,string),m4
");
$a=switch(0,$m);
$C=switch(1,$m);
$Cpretty=switch(2,$m);
$MCpretty=switch(3,$m);@
qu.3.4.uid=041799f3-cc2f-44b2-8f48-ec4680ec2810@
qu.3.4.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Perfect Competition, Solving For Q;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Algorithmic;
@
qu.3.4.weighting=1,1@
qu.3.4.numbering=alpha@
qu.3.4.part.1.name=sro_id_1@
qu.3.4.part.1.maple_answer=printf("$MCpretty")@
qu.3.4.part.1.editing=useHTML@
qu.3.4.part.1.question=(Unset)@
qu.3.4.part.1.libname=@
qu.3.4.part.1.mode=Maple@
qu.3.4.part.1.allow2d=1@
qu.3.4.part.1.plot=@
qu.3.4.part.1.maple=evalb(($a*q+$b)-($RESPONSE) = 0);@
qu.3.4.part.1.type=formula@
qu.3.4.part.2.name=sro_id_2@
qu.3.4.part.2.answer.units=@
qu.3.4.part.2.numStyle=   @
qu.3.4.part.2.editing=useHTML@
qu.3.4.part.2.showUnits=false@
qu.3.4.part.2.question=(Unset)@
qu.3.4.part.2.mode=Numeric@
qu.3.4.part.2.grading=exact_value@
qu.3.4.part.2.negStyle=both@
qu.3.4.part.2.answer.num=$q@
qu.3.4.question=<p>A competitive firm faces a price of $P and a total cost function of TC=$Cpretty.</p><p>What is this firm's marginal cost function? Leave your answer in fraction form and remember to put brackets around any fractions.</p><p><span>MC(q)= </span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What quantity should this firm produce?</p><p>q=<span>&nbsp;</span><2><span>&nbsp;</span></p>@

qu.3.5.mode=Inline@
qu.3.5.name=Optimization - One Variable with S.O.C. - no steps@
qu.3.5.comment=<p>Please provide a single numerical response per answer field.&nbsp; To account for a&nbsp;small amount of rounding, there is a range of values around the correct answer that will be awarded full marks.</p>@
qu.3.5.editing=useHTML@
qu.3.5.solution=@
qu.3.5.algorithm=$v=maple("
randomize():
v1:=randpoly(X,degree=2):
v2:=diff(v1,X):
v3:=diff(v2,X):
if v3>0 then k:=0
elif v3<0 then k:=1
elif v3=0 then k:=2
end if:
v5:=solve(v2=0,X):
v6:=MathML[ExportPresentation](v1):
convert(v1,string),convert(v2,string),convert(v3,string),k,convert(v5,string),convert(v6,string)
");
$F=switch(5,$v);
$fx=switch(1,$v);
$fxx=switch(2,$v);
$ans1=switch(4,$v);
$k=switch(3,$v);
$ans2=switch($k,'min','max','unknown');
$wrong1=switch($k,'max','unknown','min');
$wrong2=switch($k,'unknown','min','max');@
qu.3.5.uid=82024123-d76e-43c7-8d0d-a075422f52fb@
qu.3.5.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=One Variable;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Walks Students Through Steps;
@
qu.3.5.weighting=1,1@
qu.3.5.numbering=alpha@
qu.3.5.part.1.name=sro_id_1@
qu.3.5.part.1.answer.units=@
qu.3.5.part.1.numStyle=   arithmetic@
qu.3.5.part.1.editing=useHTML@
qu.3.5.part.1.showUnits=false@
qu.3.5.part.1.err=0.05@
qu.3.5.part.1.question=(Unset)@
qu.3.5.part.1.mode=Numeric@
qu.3.5.part.1.grading=toler_abs@
qu.3.5.part.1.negStyle=minus@
qu.3.5.part.1.answer.num=$ans1@
qu.3.5.part.2.grader=exact@
qu.3.5.part.2.name=sro_id_2@
qu.3.5.part.2.editing=useHTML@
qu.3.5.part.2.display.permute=true@
qu.3.5.part.2.answer.3=$wrong2@
qu.3.5.part.2.question=(Unset)@
qu.3.5.part.2.answer.2=$wrong1@
qu.3.5.part.2.answer.1=$ans2@
qu.3.5.part.2.mode=List@
qu.3.5.part.2.display=menu@
qu.3.5.part.2.credit.3=0.0@
qu.3.5.part.2.credit.2=0.0@
qu.3.5.part.2.credit.1=1.0@
qu.3.5.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">&nbsp;</p><p align="left">What is the value of the critical point (stationary point) for this function?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">Is this critical point a maximum, minimum or is it indeterminate?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi></mrow></mstyle></math> is a <span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.3.6.mode=Inline@
qu.3.6.name=Elasticity - price elasticity, inverse linear demand@
qu.3.6.comment=<p>Remember, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>P</mi><mrow><mi>Q</mi></mrow></mfrac><mrow><mfrac><mi>dQ</mi><mrow><mi>dP</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.3.6.editing=useHTML@
qu.3.6.solution=@
qu.3.6.algorithm=$e=decimal(2,range(1,7)/4);
$ne=-$e;
$b=range(2,9);
$a=range(10,20);
$q=$a-$b*P;
$c=decimal(2,$a/$b);
$d=decimal(2,1/$b);
$f=decimal(2,1/$d);
$g=decimal(2,$c/$d);
$invD=$c-$d*Q;
$invDpretty=mathml($invD);
$v=maple("
v1:=(($ne)*$a)/((-1+($ne))*$b):
v2:=diff($q,P):
v3:=((-($f))*P)/($g-($f)*P):
v4:=MathML[ExportPresentation](v3):
if $e < 1 then k := 0 
elif $e = 1 then k := 1
elif $e > 1 then k := 2 
end if:
v1,convert(v2,string),convert(v3,string),convert(v4,string),k
");
$P=decimal(2,switch(0,$v));
$elast=switch(2,$v);
$elastpretty=switch(3,$v);
$k=switch(4,$v);
$Answers = switch($k,"Inelastic", "Unit Elastic", "Elastic");
$Distractors = switch($k,"Unit Elastic", "Elastic", "Unit Elastic");
$Distractors2 = switch($k,"Elastic", "Inelastic", "Inelastic");@
qu.3.6.uid=77d493e9-edab-44b8-8754-b4df61635c85@
qu.3.6.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Elasticity;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Algorithmic;
@
qu.3.6.weighting=1,1,1@
qu.3.6.numbering=alpha@
qu.3.6.part.1.name=sro_id_1@
qu.3.6.part.1.maple_answer=$elast@
qu.3.6.part.1.editing=useHTML@
qu.3.6.part.1.question=(Unset)@
qu.3.6.part.1.libname=@
qu.3.6.part.1.mode=Maple@
qu.3.6.part.1.allow2d=1@
qu.3.6.part.1.plot=@
qu.3.6.part.1.maple=is(abs($ANSWER)-abs($RESPONSE) = 0);@
qu.3.6.part.1.type=formula@
qu.3.6.part.2.name=sro_id_2@
qu.3.6.part.2.maple_answer=$e@
qu.3.6.part.2.editing=useHTML@
qu.3.6.part.2.question=(Unset)@
qu.3.6.part.2.libname=@
qu.3.6.part.2.mode=Maple@
qu.3.6.part.2.allow2d=1@
qu.3.6.part.2.plot=@
qu.3.6.part.2.maple=evalb($ANSWER-abs($RESPONSE) = 0);@
qu.3.6.part.2.type=formula@
qu.3.6.part.3.grader=exact@
qu.3.6.part.3.name=sro_id_3@
qu.3.6.part.3.editing=useHTML@
qu.3.6.part.3.display.permute=true@
qu.3.6.part.3.answer.4=Not Enough Information@
qu.3.6.part.3.answer.3=$Distractors2@
qu.3.6.part.3.question=(Unset)@
qu.3.6.part.3.answer.2=$Distractors@
qu.3.6.part.3.answer.1=$Answers@
qu.3.6.part.3.mode=List@
qu.3.6.part.3.display=menu@
qu.3.6.part.3.credit.4=0.0@
qu.3.6.part.3.credit.3=0.0@
qu.3.6.part.3.credit.2=0.0@
qu.3.6.part.3.credit.1=1.0@
qu.3.6.question=<p>If Inverse Demand is P=$invDpretty, what is the formula for the price-elasticity of demand? (Round to two decimals.)</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='1.0' scriptminsize='8.0pt'><mrow><mi mathsize='15'>&varepsilon;</mi><mo mathsize='15' mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>P</mi></mrow></mfenced></mrow></mstyle></math>= </span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>If the price is $P, what is the elasticity? (Round to the nearest quarter.)</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='1.0' scriptminsize='8.0pt'><mrow><mi mathsize='15'>&varepsilon;</mi><mo mathsize='15' mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$P</mi></mrow></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>Therefore when the price is $P, demand is <span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.3.7.mode=Inline@
qu.3.7.name=one variable optimization@
qu.3.7.comment=<p>To find how much Gizmo Inc would produce we first have to maximize their profit. To find their profit we take the price set by the government,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$P, multiplied by the number of grams of flubber they produce and sell, denoted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math> which gives us revenue, and then subtract the cost as given in the question, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$C. Taking these steps, we find profit to be $R1.</p>
<p>To maximize we need to differentiate and set the derivative equal to zero. $F1. The solutions to this equation are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$a, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$b. To see that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>is a solution, we substitute <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>in the first order condition, as we see that this amounts to just summing up the coefficients. Since the coefficients sum to zero, indeed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>is a solution. Now we divide the first order condition by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>since if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math> is a solution then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math> divides it. The result is the quadratic equation $N2, which can be solved using the quadratic formula to find the other two roots to be $a and $b.</p>
<p>&nbsp;</p>
<p>We find that the profit is maximized at $ans since there, the profit is $val, which is the highest profit over all the solutions. The graphs below illustrate the situation:</p>
<p>&nbsp;</p>
<p>Cost</p>
<p>$plot1</p>
<p>&nbsp;</p>
<p>Profit</p>
<p>$plot2</p>
<p>&nbsp;</p>
<p>Next the government imposes a production limit of $L and Gizmo Inc. cannot avoid any of its costs by shutting down. $ANS</p>
<p>Now suppose it can avoid all of its costs by shutting down. $ANS2</p>@
qu.3.7.editing=useHTML@
qu.3.7.solution=@
qu.3.7.algorithm=$a=range(2,4,1);
$b=range((($a)+1),(($a)+3),1);
$P=range(1,9,1);
$L=range(2,5,1);
$Q=maple("
c:=3*F^4-4*(1+$a+$b)*F^3+6*(($a)*($b)+$a+$b)*F^2-(12*($a)*($b)-$P)*F:
d:=1+($b):
m:=minimize(c,F=0 .. d):
e:=floor(m):
c:=c-(e):
C1:=MathML[ExportPresentation](c):
R:=(($P)*F-(c)):
R1:=MathML[ExportPresentation](R):
f1:=diff(R,F):
F1:=MathML[ExportPresentation](f1=0):
N:=(f1)/(F-1):
N1:=simplify(N):
N2:=MathML[ExportPresentation](N1=0):
a1:=$a:
b1:=$b:
if $L<b1 then k:=1 else k:=0  end if:
if $L<b1 then b1:=$L  end if:
PB:=eval($R,F=$b):
P1:=eval($R,F=1):
if P1>PB  then i:=0 elif P1=PB then i:=1  else i:=2 end if:
PBL:=eval($R,F=b1):
if P1>PBL  then j:=0 elif P1=PBL then j:=1  else j:=2 end if:
if k=1 then k:=i+1 end if:
if P1<0 then h1:=1 else h1:=0 end if:
if PBL<0 then h2:=1 else h2:=0 end if:
h:=(h1)*(h2):
PB,P1,PBL,C1,R,F1,i,b1,j,R1,N2,c,d,k,h
");
$C=switch(3,$Q);
$F1=switch(5,$Q);
$i=switch(6,$Q);
$ans=switch(($i),1,"1 or $b",($b));
$PB=switch(0,$Q);
$P1=switch(1,$Q);
$val=switch(($i),($P1),($PB),($PB));
$b1=switch(7,$Q);
$j=switch(8,$Q);
$R1=switch(9,$Q);
$N2=switch(10,$Q);
$ans2=switch(($j),1,"1 or $b1",($b1));
$c=switch(11,$Q);
$d=switch(12,$Q);
$R=switch(4,$Q);
$plot1=plotmaple("plot($c,F=0..$d), plotoptions='width=250, height=250'");
$plot2=plotmaple("plot($R,F=0..$d), plotoptions='width=250, height=250'");
$k=switch(13,$Q);
$ANS=switch(($k),"The government limit is not restrictive and so Gizmo Inc. continues to produce as before.","The government limit is not restrictive and so Gizmo Inc. continues to produce as before.","Before the government limit Gizmo Inc. could maximize profit by either choosing 1 gram or $b grams, with the restriction it can only choose 1 gram." ,"Before the restriction Gizmo Inc. would have chosen $b grams but now it must choose F= $ans2.");
$h=switch(14,$Q);
$ANS2=switch(($h), "Although costs can be avoided by shutting down, since Gizmo Inc. is making a non-negative profit, it prefers to produce F= $ans2  of flubber.","Since Gizmo Inc. is making a loss by producing F=$ans2 of flubber, it will prefer to shut down.");
$ans3=switch(($h),$ans2,0);@
qu.3.7.uid=272eec21-e304-402f-883c-31af4cb0017c@
qu.3.7.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization One Variable --Multiple Solutions;
  Sub-Topic=Interior Max And Corner;
  Difficulty=Hard;
  Author=Asha Sadanand;
@
qu.3.7.weighting=1,1,1@
qu.3.7.numbering=alpha@
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qu.3.7.part.1.name=sro_id_1@
qu.3.7.part.1.answer=$ans@
qu.3.7.part.1.mode=Formula@
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qu.3.7.part.2.question=(Unset)@
qu.3.7.part.2.name=sro_id_2@
qu.3.7.part.2.answer=$ans2@
qu.3.7.part.2.mode=Formula@
qu.3.7.part.3.editing=useHTML@
qu.3.7.part.3.question=(Unset)@
qu.3.7.part.3.name=sro_id_3@
qu.3.7.part.3.answer=$ans3@
qu.3.7.part.3.mode=Formula@
qu.3.7.question=<p>A mad scientist has recently uncovered the process for making flubber. The cost<br />of producing F grams of flubber is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mfenced open='(' close=')' separators=','><mrow><mi>F</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$C. Gizmos Incorporated has<br />obtained the formula and wants to sell flubber to maximize its profit. Since flubber is a controlled<br />substance, the government has fixed the price per gram at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$P.</p><p>How many grams of flubber should Gizmos Inc. produce to maximize its profit? <span>&nbsp;</span><1><span>&nbsp;</span></p><p>If the government also limits how much can be produced to a maximum of $L grams, and Gizmo Inc. cannot avoid any of its costs by shutting down then<br />how much should Gizmos Inc. produce? <span>&nbsp;</span><2><span>&nbsp;</span></p><p>Now suppose that it can avoid all of its costs by shutting down, and choosing<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.</mn></mrow></mstyle></math>Now how many grams of flubber will it choose to produce?<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p>@

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qu.3.8.name=Corner Solution - formal method@
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qu.3.8.solution=@
qu.3.8.algorithm=$b=range(1,8);
$v=maple("
if $b=2 then k:=1
elif $b=4 then k:=1
else k:=0
end if:

if $b=3 then i:=0
else i:=1
end if:

if $b<2 then j:=0
else j:=1
end if:

if $b<4 then h:=0
else h:=1
end if:

if $b=1 then m:=0
elif $b=2 then m:=1
elif $b=3 then m:=1
elif $b=4 then m:=2
elif $b=5 then m:=3
elif $b>5 then m:=4
end if:

k,i,j,h,m
");
$k=switch(0,$v);
$ans1=switch($k,'No','Yes');
$wrong1=switch($k,'Yes','No');
$i=switch(1,$v);
$ans2=switch($i,'No','Yes');
$wrong2=switch($i,'Yes','No');
$j=switch(2,$v);
$ans3=switch($j,'No','Yes');
$wrong3=switch($j,'Yes','No');
$h=switch(3,$v);
$ans4=switch($h,'No','Yes');
$wrong4=switch($h,'Yes','No');
$m=switch(4,$v);
$cand1=switch($m,1,2,2,2,2);
$cand2=switch($m,'','',4,4,4);
$cand3=switch($m,'','','',5,$b);
$cand4='';
$ans5=switch($m,1,2,2,2,$b);
$ans6=switch($m,'','','',5,'');@
qu.3.8.uid=c7426cf6-f874-49ea-ba0e-3109eeae22d7@
qu.3.8.info=  Course=Introductory Mathematical Economics;
  Topic=Constrained Optimization;
  Sub-Topic=Corner Solutions;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Walks Students Through Steps;
@
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qu.3.8.part.3.answer.3=Not Enough Info@
qu.3.8.part.3.question=(Unset)@
qu.3.8.part.3.answer.2=Yes@
qu.3.8.part.3.answer.1=No@
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qu.3.8.part.4.answer.1=No@
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qu.3.8.part.6.answer.3=Not Enough Info@
qu.3.8.part.6.question=(Unset)@
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qu.3.8.question=<p>Find the maximum of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><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><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>9</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>24</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>16</mn></mrow></mstyle></math> if x cannot be smaller than 0 or larger than $b. (<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$b</mi></mrow></mstyle></math>).</p><p>&nbsp;</p><p>What are the critical points (stationary points) of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><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><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>9</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>24</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>16</mn></mrow></mstyle></math>? (The points at which <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>.)</p><p>Enter the critical points from smallest to largest.</p><p><span>x1= </span><1><span>&nbsp; <br /></span></p><p>x2=<2><span>&nbsp;</span></p><p>&nbsp;</p><p>For an&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></mstyle></math> to be a candidate to maximize a function, one of the following conditions must hold:</p><p>(1) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>0</mn></mrow></mstyle></math> AND <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&ast;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfenced><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p><p>and/or</p><p>(2) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn></mrow></mstyle></math>AND <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&ast;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>b</mi></mrow></mfenced><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p><p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi></mrow></mstyle></math>is the lower boundary on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math> (in this case it is 0), and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>b</mi></mrow></mstyle></math>is the upper boundary on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math> (in this case it is $b).</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></mfenced></mrow></mstyle></math> is the first derivative of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>, evaluated at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></mstyle></math>.</p><p>&nbsp;</p><p>Does&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>satisfy condition (1)?</p><p><span>&nbsp;</span><3><span>&nbsp;</span></p><p>Does&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>satisfy condition (2)?&nbsp;</p><p><span>&nbsp;</span><4><span>&nbsp;</span></p><p>&nbsp;</p><p>Does <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$b satisfy condition (1)?</p><p><span>&nbsp;</span><5><span>&nbsp;</span></p><p>Does <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$b satisfy condition (2)?</p><p><span>&nbsp;</span><6><span>&nbsp;</span></p><p>&nbsp;</p><p>Does <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>x1 satisfy condition (1)?</p><p><span>&nbsp;</span><7><span>&nbsp;</span></p><p>Does <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>x1 satisfy condition (2)?</p><p><span>&nbsp;</span><8><span>&nbsp;</span></p><p>Is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>x1 within the boundaries on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>?</p><p>&nbsp;<9><span>&nbsp;</span></p><p>&nbsp;</p><p>Does <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>x2 satisfy condition (1)?</p><p>&nbsp;<10><span> </span></p><p>Does <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>x2 satisfy condition (2)?</p><p><span>&nbsp;</span><11><span> <br /></span></p><p>&nbsp;Is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>x2 within the boundaries on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>?</p><p>&nbsp;<span>&nbsp;</span><12><span>&nbsp;</span></p><p>&nbsp;</p><p><span>Given the above results, what are the candidates for a maximum? </span></p><p><span>Enter the candidates from the smallest to the largest and don't repeat any numbers. </span>If there are fewer candidates than boxes below, leave the last boxes empty.</p><p><span>&nbsp;</span><13><span>&nbsp;</span></p><p><span>&nbsp;</span><14><span>&nbsp;</span></p><p><span>&nbsp;</span><15><span>&nbsp;</span></p><p><span>&nbsp;</span><16><span>&nbsp;</span></p><p>&nbsp;</p><p><span>Out of the above candidates, which one(s) maximize the function?</span> If there are multiple values that maximize the function, enter them from smallest to largest.</p><p><span>&nbsp;</span><17><span>&nbsp;</span></p><p><span>&nbsp;</span><18><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.3.9.mode=Inline@
qu.3.9.name=Optimization - One Variable with steps@
qu.3.9.comment=@
qu.3.9.editing=useHTML@
qu.3.9.solution=@
qu.3.9.algorithm=$v=maple("
randomize():
v1:=randpoly(X,degree=2):
v2:=diff(v1,X):
v3:=diff(v2,X):
if v3>0 then k:=0
elif v3<0 then k:=1
elif v3=0 then k:=2
end if:
v5:=solve(v2=0,X):
v6:=MathML[ExportPresentation](v1):
convert(v1,string),convert(v2,string),convert(v3,string),k,convert(v5,string),convert(v6,string)
");
$F=switch(5,$v);
$fx=switch(1,$v);
$fxx=switch(2,$v);
$ans1=switch(4,$v);
$k=switch(3,$v);
$ans2=switch($k,'min','max','unknown');
$wrong1=switch($k,'max','unknown','min');
$wrong2=switch($k,'unknown','min','max');@
qu.3.9.uid=cb3ea860-7a90-4103-b699-61ad4c629a82@
qu.3.9.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=One Variable;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Walks Students Through Steps;
@
qu.3.9.weighting=1,1,1@
qu.3.9.numbering=alpha@
qu.3.9.part.1.editing=useHTML@
qu.3.9.part.1.question=(Unset)@
qu.3.9.part.1.name=sro_id_1@
qu.3.9.part.1.answer=$fx@
qu.3.9.part.1.mode=Formula@
qu.3.9.part.2.name=sro_id_2@
qu.3.9.part.2.answer.units=@
qu.3.9.part.2.numStyle=   @
qu.3.9.part.2.editing=useHTML@
qu.3.9.part.2.showUnits=false@
qu.3.9.part.2.question=(Unset)@
qu.3.9.part.2.mode=Numeric@
qu.3.9.part.2.grading=exact_value@
qu.3.9.part.2.negStyle=both@
qu.3.9.part.2.answer.num=0@
qu.3.9.part.3.name=sro_id_3@
qu.3.9.part.3.answer.units=@
qu.3.9.part.3.numStyle=   arithmetic@
qu.3.9.part.3.editing=useHTML@
qu.3.9.part.3.showUnits=false@
qu.3.9.part.3.err=0.05@
qu.3.9.part.3.question=(Unset)@
qu.3.9.part.3.mode=Numeric@
qu.3.9.part.3.grading=toler_abs@
qu.3.9.part.3.negStyle=minus@
qu.3.9.part.3.answer.num=$ans1@
qu.3.9.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">Find the first order condition for optimizing this function:</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>F</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></mstyle></math><span>&nbsp;</span><1><span> </span>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above first order condition, what is the value of the critical point (stationary point)?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span>&nbsp;</span></p><p align="left">&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p>@

qu.3.10.mode=Inline@
qu.3.10.name=Optimization - Quadratic One Variable, with S.O.C. and steps@
qu.3.10.comment=<pre><span class="bold"><strong><strong>Here is what the function looks like:</strong></strong></span></pre>
<pre><span class="bold"><strong><strong>$plot<br /></strong></strong></span></pre>@
qu.3.10.editing=useHTML@
qu.3.10.solution=@
qu.3.10.algorithm=$neg=range(-1,1,2);
$neg2=range(-1,1,2);
$a=$neg*range(-4,-2);
$b=$neg*range(2,4);
$d=$neg2*range(1,4);
$v=maple("
v1:=($d)*(((X^4)/4)-(($a)+($b)+1)*((X^3)/3)+(($a)*($b)+($b)+($a))*((X^2)/2)-($a)*($b)*X):
v2:=MathML[ExportPresentation](v1):
v3:=diff(v1,X):
v4:=diff(v3,X):

if $neg=1 then
v5:=eval(v4,X=$a)
else v5:=eval(v4,X=$b)
end if:

v6:=eval(v4,X=1):

if $neg=1 then
v7:=eval(v4,X=$b)
else v7:=eval(v4,X=$a)
end if:

if $neg=1 then
crit1:=$a
else crit1:=$b
end if:

if $neg=1 then
crit3:=$b
else crit3:=$a
end if:

if $neg2=1 then
k:=1
else k:=0
end if:
v1,v2,convert(v3,string),convert(v4,string),v5,v6,v7,crit1,crit3,k
");
$Fmath=switch(0,$v);
$F=switch(1,$v);
$crit1=switch(7,$v);
$crit2=1;
$crit3=switch(8,$v);
$eval1=switch(4,$v);
$eval2=switch(5,$v);
$eval3=switch(6,$v);
$k=switch(9,$v);
$ans1=switch($k,"a local maximum","a local minimum");
$wrong1=switch($k,"a local minimum","a local maximum");
$ans2=switch($k,"a local minimum","a local maximum");
$wrong2=switch($k,"a local maximum","a local minimum");
$ans3=switch($k,"a local maximum","a local minimum");
$wrong3=switch($k,"a local minimum","a local maximum");
$fx=switch(2,$v);
$fxx=switch(3,$v);
$plot=plotmaple("plot($Fmath, X=-5..5), plotdevice='gif', plotoptions='height=250, width=250'");@
qu.3.10.uid=e75bc9ab-552c-4975-9745-efba2a2a8f8e@
qu.3.10.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=One Variable;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Walks Students Through Steps;
@
qu.3.10.weighting=1,1,1,1,1,1,1,1,1,1,1,1@
qu.3.10.numbering=alpha@
qu.3.10.part.1.name=sro_id_1@
qu.3.10.part.1.maple_answer=$fx@
qu.3.10.part.1.editing=useHTML@
qu.3.10.part.1.question=(Unset)@
qu.3.10.part.1.libname=@
qu.3.10.part.1.mode=Maple@
qu.3.10.part.1.allow2d=1@
qu.3.10.part.1.plot=@
qu.3.10.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.10.part.1.type=formula@
qu.3.10.part.2.name=sro_id_2@
qu.3.10.part.2.answer.units=@
qu.3.10.part.2.numStyle=   @
qu.3.10.part.2.editing=useHTML@
qu.3.10.part.2.showUnits=false@
qu.3.10.part.2.question=(Unset)@
qu.3.10.part.2.mode=Numeric@
qu.3.10.part.2.grading=exact_value@
qu.3.10.part.2.negStyle=both@
qu.3.10.part.2.answer.num=0@
qu.3.10.part.3.name=sro_id_3@
qu.3.10.part.3.answer.units=@
qu.3.10.part.3.numStyle=   @
qu.3.10.part.3.editing=useHTML@
qu.3.10.part.3.showUnits=false@
qu.3.10.part.3.question=(Unset)@
qu.3.10.part.3.mode=Numeric@
qu.3.10.part.3.grading=exact_value@
qu.3.10.part.3.negStyle=both@
qu.3.10.part.3.answer.num=$crit1@
qu.3.10.part.4.name=sro_id_4@
qu.3.10.part.4.answer.units=@
qu.3.10.part.4.numStyle=   @
qu.3.10.part.4.editing=useHTML@
qu.3.10.part.4.showUnits=false@
qu.3.10.part.4.question=(Unset)@
qu.3.10.part.4.mode=Numeric@
qu.3.10.part.4.grading=exact_value@
qu.3.10.part.4.negStyle=both@
qu.3.10.part.4.answer.num=$crit2@
qu.3.10.part.5.name=sro_id_5@
qu.3.10.part.5.answer.units=@
qu.3.10.part.5.numStyle=   @
qu.3.10.part.5.editing=useHTML@
qu.3.10.part.5.showUnits=false@
qu.3.10.part.5.question=(Unset)@
qu.3.10.part.5.mode=Numeric@
qu.3.10.part.5.grading=exact_value@
qu.3.10.part.5.negStyle=both@
qu.3.10.part.5.answer.num=$crit3@
qu.3.10.part.6.name=sro_id_6@
qu.3.10.part.6.maple_answer=$fxx@
qu.3.10.part.6.editing=useHTML@
qu.3.10.part.6.question=(Unset)@
qu.3.10.part.6.libname=@
qu.3.10.part.6.mode=Maple@
qu.3.10.part.6.allow2d=1@
qu.3.10.part.6.plot=@
qu.3.10.part.6.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.10.part.6.type=formula@
qu.3.10.part.7.name=sro_id_7@
qu.3.10.part.7.answer.units=@
qu.3.10.part.7.numStyle=   arithmetic@
qu.3.10.part.7.editing=useHTML@
qu.3.10.part.7.showUnits=false@
qu.3.10.part.7.err=0.01@
qu.3.10.part.7.question=(Unset)@
qu.3.10.part.7.mode=Numeric@
qu.3.10.part.7.grading=toler_abs@
qu.3.10.part.7.negStyle=minus@
qu.3.10.part.7.answer.num=$eval1@
qu.3.10.part.8.name=sro_id_8@
qu.3.10.part.8.answer.units=@
qu.3.10.part.8.numStyle=   arithmetic@
qu.3.10.part.8.editing=useHTML@
qu.3.10.part.8.showUnits=false@
qu.3.10.part.8.err=0.01@
qu.3.10.part.8.question=(Unset)@
qu.3.10.part.8.mode=Numeric@
qu.3.10.part.8.grading=toler_abs@
qu.3.10.part.8.negStyle=minus@
qu.3.10.part.8.answer.num=$eval2@
qu.3.10.part.9.name=sro_id_9@
qu.3.10.part.9.answer.units=@
qu.3.10.part.9.numStyle=   arithmetic@
qu.3.10.part.9.editing=useHTML@
qu.3.10.part.9.showUnits=false@
qu.3.10.part.9.err=0.01@
qu.3.10.part.9.question=(Unset)@
qu.3.10.part.9.mode=Numeric@
qu.3.10.part.9.grading=toler_abs@
qu.3.10.part.9.negStyle=minus@
qu.3.10.part.9.answer.num=$eval3@
qu.3.10.part.10.grader=exact@
qu.3.10.part.10.name=sro_id_10@
qu.3.10.part.10.editing=useHTML@
qu.3.10.part.10.display.permute=true@
qu.3.10.part.10.answer.3=indeterminate@
qu.3.10.part.10.question=(Unset)@
qu.3.10.part.10.answer.2=$wrong1@
qu.3.10.part.10.answer.1=$ans1@
qu.3.10.part.10.mode=List@
qu.3.10.part.10.display=menu@
qu.3.10.part.10.credit.3=0.0@
qu.3.10.part.10.credit.2=0.0@
qu.3.10.part.10.credit.1=1.0@
qu.3.10.part.11.grader=exact@
qu.3.10.part.11.name=sro_id_11@
qu.3.10.part.11.editing=useHTML@
qu.3.10.part.11.display.permute=true@
qu.3.10.part.11.answer.3=indeterminate@
qu.3.10.part.11.question=(Unset)@
qu.3.10.part.11.answer.2=$wrong2@
qu.3.10.part.11.answer.1=$ans2@
qu.3.10.part.11.mode=List@
qu.3.10.part.11.display=menu@
qu.3.10.part.11.credit.3=0.0@
qu.3.10.part.11.credit.2=0.0@
qu.3.10.part.11.credit.1=1.0@
qu.3.10.part.12.grader=exact@
qu.3.10.part.12.name=sro_id_12@
qu.3.10.part.12.editing=useHTML@
qu.3.10.part.12.display.permute=true@
qu.3.10.part.12.answer.3=indeterminate@
qu.3.10.part.12.question=(Unset)@
qu.3.10.part.12.answer.2=$wrong3@
qu.3.10.part.12.answer.1=$ans3@
qu.3.10.part.12.mode=List@
qu.3.10.part.12.display=menu@
qu.3.10.part.12.credit.3=0.0@
qu.3.10.part.12.credit.2=0.0@
qu.3.10.part.12.credit.1=1.0@
qu.3.10.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">Find the first order condition for optimizing this function:</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>F</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></mstyle></math><span>&nbsp;</span><1><span> </span>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above first order condition, what are the values of the critical points (stationary points)?</p><p align="left">(Enter one critical point in each answer box, starting with the lowest and ending with the highest. Eg. -8, then 0 then 6)</p><p align="left">&nbsp;</p><p align="left">critical point #1=<span>&nbsp;</span><3><span>&nbsp;</span></p><p align="left">critical point #2=<span>&nbsp;</span><4><span>&nbsp;</span></p><p align="left">critical point #3=<span>&nbsp;</span><5><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">What is the second derivative of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math> with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>?</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</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></mstyle></math><span>&nbsp;</span><6><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">Evaluate the second derivative at each critical point.</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</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='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>critical</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>point</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&num;</mo><mi>1</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><7><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi mathvariant='normal'>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>critical</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>point</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&num;</mo><mi>2</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi></mi></mrow></msup><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>critical</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>point</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&num;</mo><mi>3</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><9><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left"><span>Given what you evaluated above, state whether each point is a local maximum, a local minimum, or it is indeterminate.</span></p><p align="left"><span>critical point #1 is </span><span>&nbsp;</span><10><span>&nbsp;</span></p><p align="left">critical point #2 is <span>&nbsp;</span><11><span>&nbsp;</span></p><p align="left">critical point #3 is <span>&nbsp;</span><12><span>&nbsp;</span></p>@

qu.3.11.mode=Inline@
qu.3.11.name=Elasticity - price elasticity, linear demand@
qu.3.11.comment=<p>Remember, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&varepsilon;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>P</mi><mrow><mi>Q</mi></mrow></mfrac><mrow><mfrac><mi>dQ</mi><mrow><mi>dP</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.3.11.editing=useHTML@
qu.3.11.solution=@
qu.3.11.algorithm=$e=decimal(2,range(1,7)/4);
$ne=-$e;
$b=range(2,9);
$a=range(10,20);
$q="$a-$b*P";
$qpretty=mathml($q);
$v=maple("
v1:=(($ne)*$a)/((-1+($ne))*$b):
v2:=diff($q,P):
v3:=-(v2*P)/($a-$b*P):
v4:=MathML[ExportPresentation](v3):
if $e < 1 then k := 0 
elif $e = 1 then k := 1
elif $e > 1 then k := 2 
end if:
v1,v2,convert(v3,string),v4,k
");
$P=decimal(2,switch(0,$v));
$elast=switch(2,$v);
$elastpretty=switch(3,$v);
$k=switch(4,$v);
$Answers = switch($k,"Inelastic", "Unit Elastic", "Elastic");
$Distractors = switch($k,"Unit Elastic", "Elastic", "Unit Elastic");
$Distractors2 = switch($k,"Elastic", "Inelastic", "Inelastic");@
qu.3.11.uid=9c3e0567-b4ea-4473-b88e-8b393b16c8a8@
qu.3.11.weighting=1,1,1@
qu.3.11.numbering=alpha@
qu.3.11.part.1.name=sro_id_1@
qu.3.11.part.1.maple_answer=$elast@
qu.3.11.part.1.editing=useHTML@
qu.3.11.part.1.question=(Unset)@
qu.3.11.part.1.libname=@
qu.3.11.part.1.mode=Maple@
qu.3.11.part.1.allow2d=1@
qu.3.11.part.1.plot=@
qu.3.11.part.1.maple=is(abs($ANSWER)-abs($RESPONSE) = 0);@
qu.3.11.part.1.type=formula@
qu.3.11.part.2.name=sro_id_2@
qu.3.11.part.2.maple_answer=$e@
qu.3.11.part.2.editing=useHTML@
qu.3.11.part.2.question=(Unset)@
qu.3.11.part.2.mode=Maple@
qu.3.11.part.2.allow2d=1@
qu.3.11.part.2.plot=@
qu.3.11.part.2.maple=evalb($ANSWER-abs($RESPONSE) = 0);@
qu.3.11.part.2.type=formula@
qu.3.11.part.3.grader=exact@
qu.3.11.part.3.name=sro_id_3@
qu.3.11.part.3.editing=useHTML@
qu.3.11.part.3.display.permute=true@
qu.3.11.part.3.answer.4=Not Enough Information@
qu.3.11.part.3.answer.3=$Distractors2@
qu.3.11.part.3.question=(Unset)@
qu.3.11.part.3.answer.2=$Distractors@
qu.3.11.part.3.answer.1=$Answers@
qu.3.11.part.3.mode=List@
qu.3.11.part.3.display=menu@
qu.3.11.part.3.credit.4=0.0@
qu.3.11.part.3.credit.3=0.0@
qu.3.11.part.3.credit.2=0.0@
qu.3.11.part.3.credit.1=1.0@
qu.3.11.question=<p>If Demand is Q=$qpretty, what is the formula for the price-elasticity of demand?</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='1.0' scriptminsize='8.0pt'><mrow><mi mathsize='15'>&varepsilon;</mi><mo mathsize='15' mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>P</mi></mrow></mfenced></mrow></mstyle></math>= </span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>If the price is $P, what is the elasticity? (Round to the nearest quarter.)</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='1.0' scriptminsize='8.0pt'><mrow><mi mathsize='15'>&varepsilon;</mi><mo mathsize='15' mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$P</mi></mrow></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>Therefore when the price is $P, demand is <span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.3.12.mode=Inline@
qu.3.12.name=Perfect Competition - corner solution@
qu.3.12.comment=<p>Please provide a single numerical response per answer field. To account for a small amount of rounding, there is a range of values around the correct answer that will be awarded full marks.</p>
<p>If the price is low enough, the firm can't find a positive quantity that equates marginal cost and price. Therefore when this happens, the firm is best to produce zero quantity.</p>@
qu.3.12.editing=useHTML@
qu.3.12.hint.1=quantities cannot be negative@
qu.3.12.hint.2=check for corner solutions@
qu.3.12.solution=@
qu.3.12.algorithm=$a=range(1,5);
$b=range(1,10);
$c=mathml($a*q^2+$b*q);
$P=range(5,15);
$q=decimal(2,($P-$b)/(2*$a));
$ans=if(lt($q,0),0,$q);@
qu.3.12.uid=bbcf9fab-eecd-4045-9d03-22d83f7f94ff@
qu.3.12.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Corner Solutions;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.12.weighting=1@
qu.3.12.numbering=alpha@
qu.3.12.part.1.name=sro_id_1@
qu.3.12.part.1.answer.units=@
qu.3.12.part.1.numStyle=   @
qu.3.12.part.1.editing=useHTML@
qu.3.12.part.1.showUnits=false@
qu.3.12.part.1.err=0.05@
qu.3.12.part.1.question=(Unset)@
qu.3.12.part.1.mode=Numeric@
qu.3.12.part.1.grading=toler_abs@
qu.3.12.part.1.negStyle=both@
qu.3.12.part.1.answer.num=$ans@
qu.3.12.question=<p>A perfectly competitive firm faces a price of P=$P and has a total cost function of C=$c. What quantity should the firm produce?</p><p>&nbsp;</p><p>(Round answer to two decimal places if necessary. For example, 1.6666 becomes 1.67.)</p><p>q=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.13.mode=Inline@
qu.3.13.name=Perfect Competition - P=MC - (q can be a fraction)@
qu.3.13.comment=<p>The marginal cost is the derivative of $Cpretty, which is $MCpretty.</p>
<p>Setting P=MC, and solving for q, gives q=$ans.</p>@
qu.3.13.editing=useHTML@
qu.3.13.hint.1=Competitive firms pick the quantity that makes P=MC.@
qu.3.13.solution=@
qu.3.13.algorithm=$P=range(100,200);
$m=maple("
randomize():
m1:=sort(randpoly(q, degree=2, coeffs=rand(2..9))):
m2:=diff(m1, q):
m3:=solve(m2 = $P):
m4:=MathML[ExportPresentation](m1):
m5:=MathML[ExportPresentation](m2):
convert(m1,string),convert(m2,string),convert(m3,string), m4,m5 
");
$C=switch(0,$m);
$MC=switch(1,$m);
$ans=switch(2,$m);
$Cpretty=switch(3,$m);
$MCpretty=switch(4,$m);@
qu.3.13.uid=3e64316e-e21f-4e23-be73-8809d5d5293a@
qu.3.13.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Perfect Competition, Solving For Q;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Algorithmic;
@
qu.3.13.weighting=1,1@
qu.3.13.numbering=alpha@
qu.3.13.part.1.name=sro_id_1@
qu.3.13.part.1.maple_answer=printf("$MCpretty")@
qu.3.13.part.1.editing=useHTML@
qu.3.13.part.1.question=(Unset)@
qu.3.13.part.1.libname=@
qu.3.13.part.1.mode=Maple@
qu.3.13.part.1.allow2d=1@
qu.3.13.part.1.plot=@
qu.3.13.part.1.maple=evalb(($MC)-($RESPONSE) = 0);@
qu.3.13.part.1.type=formula@
qu.3.13.part.2.name=sro_id_2@
qu.3.13.part.2.answer.units=@
qu.3.13.part.2.numStyle=   arithmetic@
qu.3.13.part.2.editing=useHTML@
qu.3.13.part.2.showUnits=false@
qu.3.13.part.2.question=(Unset)@
qu.3.13.part.2.mode=Numeric@
qu.3.13.part.2.grading=exact_value@
qu.3.13.part.2.negStyle=minus@
qu.3.13.part.2.answer.num=$ans@
qu.3.13.question=<p>A competitive firm faces a price of $P and a total cost function of TC=$Cpretty.</p><p>What is this firm's marginal cost function?</p><p><span>MC(q)= </span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What quantity should this firm produce?</p><p>Leave your answer in fraction form (if necessary).</p><p>q=<span>&nbsp;</span><2><span>&nbsp;</span></p>@

qu.3.14.mode=Inline@
qu.3.14.name=Optimization - One Variable - No Steps@
qu.3.14.comment=<p>Please provide a single numerical response per answer field. To account for a small amount of rounding, there is a range of values around the correct answer that will be awarded full marks.</p>@
qu.3.14.editing=useHTML@
qu.3.14.solution=@
qu.3.14.algorithm=$v=maple("
randomize():
v1:=randpoly(X,degree=2):
v2:=diff(v1,X):
v3:=diff(v2,X):
if v3>0 then k:=0
elif v3<0 then k:=1
elif v3=0 then k:=2
end if:
v5:=solve(v2=0,X):
v6:=MathML[ExportPresentation](v1):
convert(v1,string),convert(v2,string),convert(v3,string),k,convert(v5,string),convert(v6,string)
");
$F=switch(5,$v);
$fx=switch(1,$v);
$fxx=switch(2,$v);
$ans1=switch(4,$v);
$k=switch(3,$v);
$ans2=switch($k,'min','max','unknown');
$wrong1=switch($k,'max','unknown','min');
$wrong2=switch($k,'unknown','min','max');@
qu.3.14.uid=8fd6eda6-d505-4941-853a-467a33540e0d@
qu.3.14.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=One Variable;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.14.weighting=1@
qu.3.14.numbering=alpha@
qu.3.14.part.1.name=sro_id_1@
qu.3.14.part.1.answer.units=@
qu.3.14.part.1.numStyle=   arithmetic@
qu.3.14.part.1.editing=useHTML@
qu.3.14.part.1.showUnits=false@
qu.3.14.part.1.err=0.05@
qu.3.14.part.1.question=(Unset)@
qu.3.14.part.1.mode=Numeric@
qu.3.14.part.1.grading=toler_abs@
qu.3.14.part.1.negStyle=minus@
qu.3.14.part.1.answer.num=$ans1@
qu.3.14.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">&nbsp;</p><p align="left">Given the above first order condition, what is the value of the critical point (stationary point)?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p align="left">&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p>@

qu.4.topic=Rules of differentiation - no Maple call@

qu.4.1.mode=Inline@
qu.4.1.name=Derivatives - quotient rule; chain rule - no maple@
qu.4.1.comment=<p>The derivative is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=$anspretty</p>@
qu.4.1.editing=useHTML@
qu.4.1.hint.1=Use the quotient rule.@
qu.4.1.hint.2=Use the chain rule.@
qu.4.1.hint.3=Treat the question as ln(U), where U is a function.@
qu.4.1.solution=@
qu.4.1.algorithm=$a=range(2,4,1);
$b=range(1,9,1);
$c=range(1,9,1);
$d=range(2,4,1);
$e=range(-9,-1,1);
$f=range(-3,3,1);
$F=ln(($c)*x^($a)+($b))/(($e)*x^($d)+($f));
$Fpretty=mathml($F);
$ans=(($c)*($a)*x^(($a)-1))/((($c)*x^($a)+($b))*(($e)*x^($d)+($f)))-(ln(($c)*x^($a)+($b)))*(($e)*($d)*x^(($d)-1))/((($e)*x^($d)+($f))^2);
$anspretty=mathml($ans);@
qu.4.1.uid=c557de69-c6cf-429c-938e-35069f90d1ca@
qu.4.1.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Quotient Rule, Chain Rule;
  Author=Asha Sadanand;
  Difficulty=Hard;
  Feature=No Maple Call;
@
qu.4.1.weighting=1@
qu.4.1.numbering=alpha@
qu.4.1.part.1.editing=useHTML@
qu.4.1.part.1.question=(Unset)@
qu.4.1.part.1.name=sro_id_1@
qu.4.1.part.1.answer=$ans@
qu.4.1.part.1.mode=Formula@
qu.4.1.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

