qu.1.topic=Partial Differentiation and Applications@

qu.1.1.mode=Inline@
qu.1.1.name=MPK - CES@
qu.1.1.comment=<p>The answer is MPK=$anspretty.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=range(2,8);
$b=decimal(1,range(.1,.9,.1));
$c=range(2,8);
$d=1-$b;
$g=1-$b;
$h=$c*$g;
$F=$c*($b*L^$a+($g)*K^$a)^frac(1,$a);
$Fpretty=mathml($c*($b*L^$a+($g)*K^$a)^frac(1,$a));
$ans=$c*($b*L^$a+($g)*K^$a)^(-frac($a-1,$a))*($g)*K^($a-1);
$anspretty=mathml($h*($b*L^$a+($g)*K^$a)^(-frac($a-1,$a))*K^($a-1));@
qu.1.1.uid=03438ced-efd1-4902-9580-d915f9a0441c@
qu.1.1.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Mpk;
  Author=Asha Sadanand;
  Difficulty=Medium;
  Feature=Algorithmic;
@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer=$ans@
qu.1.1.part.1.mode=Formula@
qu.1.1.question=<p>If the production function is $Fpretty, what is the marginal product of capital (<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>K</mi></mrow></mstyle></math>)?</p><p>MPK=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Simple Partial Derivative@
qu.1.2.comment=<p>Remember to treat <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>&nbsp;as a constant.</p>@
qu.1.2.editing=useHTML@
qu.1.2.hint.1=Think of X as a constant.@
qu.1.2.solution=@
qu.1.2.algorithm=$a=range(1,20,1);
$b=range(1,20,1);
$c=range(2,50,1);
$d=range(2,50,1);
$func=frac($a,$b)*X+frac($c,$d)*X*Y+frac($b,$d)*Y;
$ans=frac($c,$d)*X+frac($b,$d);
$F=maple("
printf(MathML[ExportPresentation]($a/$b*X+$c/$d*X*Y+$b/$d*Y));
");@
qu.1.2.uid=42c6f681-97a4-4f4a-bf83-4e04a2cb03e6@
qu.1.2.info=  Course=Introductory Mathematical Economics;
  Topic=First Partial;
  Author=Asha Sadanand;
  Difficulty=Easy;
@
qu.1.2.weighting=1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.maple_answer=$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 $F 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>Y</mi></mrow></mstyle></math>.<span>&nbsp;</span></p><p><1><span>&nbsp;</span></p>@

qu.1.3.mode=Inline@
qu.1.3.name=Second Order Partials - Two Variables@
qu.1.3.comment=<p>Remember to treat the other variable in each case as a constant.</p>
<p>Second partial with respect to X 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span>$XXpretty</span></p>
<p>&nbsp;</p>
<p>Second partial with respect to Y is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>Y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=$YYpretty</p>
<p>&nbsp;</p>
<p>Cross-partial with respect to X,Y 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>XY</mi></mrow></mfrac></mrow></mstyle></math>=<span>$XYpretty</span></p>
<p>&nbsp;</p>
<p>Cross-partial with respect to Y,X is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>XY</mi></mrow></mfrac></mrow></mstyle></math>=$XYpretty<span> </span></p>@
qu.1.3.editing=useHTML@
qu.1.3.hint.1=Young's theorem can save some work here.@
qu.1.3.solution=@
qu.1.3.algorithm=$a=range(2,10);
$b=range(2,10);
$c=range(2,10);
$d=range(2,5);
$e=range(2,5);
$g=range(2,5);
$h=range(2,5);
$v=maple("
F:=$a*X^$d*Y^$h+$b*Y^$e-$c*X^$g:
Fpretty:=MathML[ExportPresentation](F):
v1:=diff(F,X):
v2:=diff(F,Y):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
v5:=diff(v1,X):
v6:=diff(v1,Y):
v7:=diff(v2,Y):
v8:=MathML[ExportPresentation](v5):
v9:=MathML[ExportPresentation](v6):
v10:=MathML[ExportPresentation](v7):
Fpretty,convert(v1,string),convert(v2,string),v3,v4,convert(v5,string),convert(v6,string),convert(v7,string),v8,v9,v10
");
$Fpretty=switch(0,$v);
$Xans=switch(1,$v);
$Yans=switch(2,$v);
$Xpretty=switch(3,$v);
$Ypretty=switch(4,$v);
$XXans=switch(5,$v);
$XYans=switch(6,$v);
$YYans=switch(7,$v);
$XXpretty=switch(8,$v);
$XYpretty=switch(9,$v);
$YYpretty=switch(10,$v);@
qu.1.3.uid=d47150e3-bede-42e8-bd3d-f62a7c004231@
qu.1.3.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Differentiation;
  Sub-Topic=Second Order Derivatives;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.1.3.weighting=1,1,1,1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.maple_answer=$XXans@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.libname=@
qu.1.3.part.1.mode=Maple@
qu.1.3.part.1.allow2d=1@
qu.1.3.part.1.plot=@
qu.1.3.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.1.type=formula@
qu.1.3.part.2.name=sro_id_2@
qu.1.3.part.2.maple_answer=$YYans@
qu.1.3.part.2.editing=useHTML@
qu.1.3.part.2.question=(Unset)@
qu.1.3.part.2.libname=@
qu.1.3.part.2.mode=Maple@
qu.1.3.part.2.allow2d=1@
qu.1.3.part.2.plot=@
qu.1.3.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.2.type=formula@
qu.1.3.part.3.name=sro_id_3@
qu.1.3.part.3.maple_answer=$XYans@
qu.1.3.part.3.editing=useHTML@
qu.1.3.part.3.question=(Unset)@
qu.1.3.part.3.libname=@
qu.1.3.part.3.mode=Maple@
qu.1.3.part.3.allow2d=1@
qu.1.3.part.3.plot=@
qu.1.3.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.3.type=formula@
qu.1.3.part.4.name=sro_id_4@
qu.1.3.part.4.maple_answer=$XYans@
qu.1.3.part.4.editing=useHTML@
qu.1.3.part.4.question=(Unset)@
qu.1.3.part.4.libname=@
qu.1.3.part.4.mode=Maple@
qu.1.3.part.4.allow2d=1@
qu.1.3.part.4.plot=@
qu.1.3.part.4.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.4.type=formula@
qu.1.3.question=<p>Find the second order partial derivatives for 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><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p>Second partial 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><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>Second partial 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>Y</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><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>Y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>Cross-partial 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><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</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><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>YX</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p><p>Cross-partial 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>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><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><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>YX</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><4><span> <br /></span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=MPL - quasilinear@
qu.1.4.comment=<p>The marginal product of labour is MPL=$anspretty.</p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=range(2,8);
$c=range(2,8);
$d=1-$a;
$F=$c*(L^frac(1,$a))+K;
$Fpretty=mathml($c*(L^frac(1,$a))+K);
$ans=frac($c,$a)*(L^frac($d,$a));
$anspretty=mathml(frac($c,$a)*(L^frac($d,$a)));@
qu.1.4.uid=67c9e871-0487-4b75-9d01-6369b60f2291@
qu.1.4.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Mpl - Quasilinear;
  Author=Asha Sadanand;
  Difficulty=Easy;
  Feature=Algorithmic;
@
qu.1.4.weighting=1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.answer=$ans@
qu.1.4.part.1.mode=Formula@
qu.1.4.question=<p>If the production function is $Fpretty, what is the marginal product of labour (<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>L</mi></mrow></mstyle></math>)?</p><p><span>MPL= </span><1><span>&nbsp;</span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Partial Derivatives - polynomial 2 variables@
qu.1.5.comment=<p>Remember to treat <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>Y</mi></mrow></mstyle></math>&nbsp;as a constant.</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>=$anspretty</p>@
qu.1.5.editing=useHTML@
qu.1.5.hint.1=Treat Y as a constant.@
qu.1.5.solution=@
qu.1.5.algorithm=$a=range(1,20,1);
$b=range(1,20,1);
$c=range(1,20,1);
$d=range(2,9,1);
$e=range(2,9,1);
$f=range(2,9,1);
$g=range(2,9,1);
$v=maple("
randomize():
v1:=($a*X^($d)+$b*X^($e)*Y^($f)+$c*Y^($g)):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string)
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.5.uid=8896c8b7-eddd-4a6f-b207-64efb8b8e837@
qu.1.5.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Derivatives;
  Sub-Topic=Polynomial Two Variables;
  Author=Asha Sadanand;
  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(($ANSWER)=(resp));@
qu.1.5.part.1.type=formula@
qu.1.5.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$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=Partial Derivatives - polynomial 3 variables Y@
qu.1.6.comment=<p>Remember to treat the other variables as constants.</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>=$anspretty</p>@
qu.1.6.editing=useHTML@
qu.1.6.hint.1=Treat X and Z as a constants.@
qu.1.6.solution=@
qu.1.6.algorithm=$a=range(1,20,1);
$b=range(1,20,1);
$c=range(1,20,1);
$d=range(2,9,1);
$e=range(2,9,1);
$f=range(2,9,1);
$g=range(2,9,1);
$h=range(2,9,1);
$i=range(2,9,1);
$j=range(2,9,1);
$k=range(2,9,1);
$v=maple("
randomize():
v1:=($a*X^($d)*Z^($h)+$b*X^($e)*Y^($f)*Z^($k)+$c*Y^($g)*Z^($i)+Z^($j)*X):
v2:=diff(v1,Y):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
v1,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=bb1ea4d7-4017-4c5e-a496-f88c6ebc1e5e@
qu.1.6.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Derivatives;
  Sub-Topic=Polynomial Two Variables;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
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(($ANSWER)=(resp));@
qu.1.6.part.1.type=formula@
qu.1.6.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Z</mi></mrow></mfenced></mrow></mstyle></math>=$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>Y</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=MRTS@
qu.1.7.comment=<p>The MRTS(L,K)=<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>MPK</mi><mrow><mi>MPL</mi></mrow></mfrac></mrow></mstyle></math>=$MRTSpretty.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$a=range(2,8);
$b=range(2,8);
$c=range(2,8);
$d=1-$a;
$e=1-$b;
$F=$c*(L^frac(1,$a))*K^frac(1,$b);
$Fpretty=mathml($c*(L^frac(1,$a))*K^frac(1,$b));
$MPL=frac($c,$a)*(L^frac($d,$a))*K^frac(1,$b);
$MPLpretty=mathml(frac($c,$a)*(L^frac($d,$a))*K^frac(1,$b));
$MPK=frac($c,$b)*(L^frac(1,$a))*K^frac($e,$b);
$MPKpretty=mathml(frac($c,$b)*(L^frac(1,$a))*K^frac($e,$b));
$MRTS=frac($a,$b)*L/K;
$MRTSpretty=mathml(frac($a,$b)*L/K);@
qu.1.7.uid=912a84e8-6a7a-4bfb-9005-2bb8368f7ae2@
qu.1.7.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Differentiation;
  Sub-Topic=Mrts;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Algorithmic;
@
qu.1.7.weighting=1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.answer=$MRTS@
qu.1.7.part.1.mode=Formula@
qu.1.7.question=<p>If the production function is $Fpretty, what is the marginal rate of technical substitution for labour (<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>L</mi></mrow></mstyle></math>) with respect to capital (<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>K</mi></mrow></mstyle></math>)?</p><p>MRTS<sub>LK</sub>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Partial Derivatives - polynomial 2 variables@
qu.1.8.comment=<p>Remember to treat the other variables as constants.</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>=$anspretty</p>@
qu.1.8.editing=useHTML@
qu.1.8.hint.1=Treat X as a constant.@
qu.1.8.solution=@
qu.1.8.algorithm=$a=range(1,20,1);
$b=range(1,20,1);
$c=range(1,20,1);
$d=range(2,9,1);
$e=range(2,9,1);
$f=range(2,9,1);
$g=range(2,9,1);
$v=maple("
randomize():
v1:=($a*X^($d)+$b*X^($e)*Y^($f)+$c*Y^($g)):
v2:=diff(v1,Y):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string)
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.8.uid=f16b0a90-6056-4d16-951e-782e4255b8a9@
qu.1.8.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Derivatives;
  Sub-Topic=Polynomial Two Variables;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
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(($ANSWER)=(resp));@
qu.1.8.part.1.type=formula@
qu.1.8.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>$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>Y</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.9.mode=Inline@
qu.1.9.name=Three Variable Partial Differentiation@
qu.1.9.comment=<p>Remember to treat the other two variables in each case as constants.</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$a=range(2,10);
$b=range(2,10);
$c=range(2,10);
$d=range(2,5);
$e=range(2,5);
$g=range(2,5);
$v=maple("
F:=$a*X^$d*Y+$b*Y^$e-$c*X*Y*Z+Z^$g:
Fpretty:=MathML[ExportPresentation](F):
v1:=diff(F,X):
v2:=diff(F,Y):
v3:=diff(F,Z):
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
Fpretty,convert(v1,string),convert(v2,string),convert(v3,string),v4,v5,v6
");
$Xans=switch(1,$v);
$Yans=switch(2,$v);
$Zans=switch(3,$v);
$Xpretty=switch(4,$v);
$Ypretty=switch(5,$v);
$Zpretty=switch(6,$v);
$Fpretty=switch(0,$v);@
qu.1.9.uid=215cdcf9-ec0a-48df-84d5-f5207ad34588@
qu.1.9.weighting=1,1,1@
qu.1.9.numbering=alpha@
qu.1.9.part.1.name=sro_id_1@
qu.1.9.part.1.maple_answer=$Xans@
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.question=(Unset)@
qu.1.9.part.1.libname=@
qu.1.9.part.1.mode=Maple@
qu.1.9.part.1.allow2d=1@
qu.1.9.part.1.plot=@
qu.1.9.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.9.part.1.type=formula@
qu.1.9.part.2.name=sro_id_2@
qu.1.9.part.2.maple_answer=$Yans@
qu.1.9.part.2.editing=useHTML@
qu.1.9.part.2.question=(Unset)@
qu.1.9.part.2.libname=@
qu.1.9.part.2.mode=Maple@
qu.1.9.part.2.allow2d=1@
qu.1.9.part.2.plot=@
qu.1.9.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.9.part.2.type=formula@
qu.1.9.part.3.name=sro_id_3@
qu.1.9.part.3.maple_answer=$Zans@
qu.1.9.part.3.editing=useHTML@
qu.1.9.part.3.question=(Unset)@
qu.1.9.part.3.libname=@
qu.1.9.part.3.mode=Maple@
qu.1.9.part.3.allow2d=1@
qu.1.9.part.3.plot=@
qu.1.9.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.9.part.3.type=formula@
qu.1.9.question=<p>Partially differentiate the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Z</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p>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><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><p>&nbsp;</p><p>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>Y</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><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>Y</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>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>Z</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><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>Z</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.1.10.mode=Inline@
qu.1.10.name=Three Variable Partial Differentiation@
qu.1.10.comment=<p>Remember to treat the other two&nbsp;variables in each case as constants.</p>@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$a=range(2,10);
$b=range(2,10);
$c=range(2,10);
$d=range(2,5);
$e=range(2,5);
$g=range(2,5);
$v=maple("
F:=$a*X^$d*Y+$b*Y^$e-$c*X*Y*Z+Z^$g:
Fpretty:=MathML[ExportPresentation](F):
v1:=diff(F,X):
v2:=diff(F,Y):
v3:=diff(F,Z):
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
Fpretty,convert(v1,string),convert(v2,string),convert(v3,string),v4,v5,v6
");
$Xans=switch(1,$v);
$Yans=switch(2,$v);
$Zans=switch(3,$v);
$Xpretty=switch(4,$v);
$Ypretty=switch(5,$v);
$Zpretty=switch(6,$v);
$Fpretty=switch(0,$v);@
qu.1.10.uid=b1080acd-995b-4a13-9485-1c8fbb0c2733@
qu.1.10.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.1.10.weighting=1,1,1@
qu.1.10.numbering=alpha@
qu.1.10.part.1.name=sro_id_1@
qu.1.10.part.1.maple_answer=$Xans@
qu.1.10.part.1.editing=useHTML@
qu.1.10.part.1.question=(Unset)@
qu.1.10.part.1.libname=@
qu.1.10.part.1.mode=Maple@
qu.1.10.part.1.allow2d=1@
qu.1.10.part.1.plot=@
qu.1.10.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.10.part.1.type=formula@
qu.1.10.part.2.name=sro_id_2@
qu.1.10.part.2.maple_answer=$Yans@
qu.1.10.part.2.editing=useHTML@
qu.1.10.part.2.question=(Unset)@
qu.1.10.part.2.libname=@
qu.1.10.part.2.mode=Maple@
qu.1.10.part.2.allow2d=1@
qu.1.10.part.2.plot=@
qu.1.10.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.10.part.2.type=formula@
qu.1.10.part.3.name=sro_id_3@
qu.1.10.part.3.maple_answer=$Zans@
qu.1.10.part.3.editing=useHTML@
qu.1.10.part.3.question=(Unset)@
qu.1.10.part.3.libname=@
qu.1.10.part.3.mode=Maple@
qu.1.10.part.3.allow2d=1@
qu.1.10.part.3.plot=@
qu.1.10.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.10.part.3.type=formula@
qu.1.10.question=<p>Partially differentiate the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Z</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p>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><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><p>&nbsp;</p><p>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>Y</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><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>Y</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>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>Z</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><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>Z</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.1.11.mode=Inline@
qu.1.11.name=Partial Derivatives - polynomial 3 variables X@
qu.1.11.comment=<p>Remember to treat the other two variables as constants.</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>=$anspretty</p>@
qu.1.11.editing=useHTML@
qu.1.11.hint.1=Treat Y and Z as a constants.@
qu.1.11.solution=@
qu.1.11.algorithm=$a=range(1,20,1);
$b=range(1,20,1);
$c=range(1,20,1);
$d=range(2,9,1);
$e=range(2,9,1);
$f=range(2,9,1);
$g=range(2,9,1);
$h=range(2,9,1);
$i=range(2,9,1);
$j=range(2,9,1);
$k=range(2,9,1);
$v=maple("
randomize():
v1:=($a*X^($d)*Z^($h)+$b*X^($e)*Y^($f)*Z^($k)+$c*Y^($g)*Z^($i)+Z^($j)*X):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string)
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.11.uid=935eabad-7b5e-4a3b-9716-e7907c9e69bb@
qu.1.11.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Derivatives;
  Sub-Topic=Polynomial Two Variables;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
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(($ANSWER)=(resp));@
qu.1.11.part.1.type=formula@
qu.1.11.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Z</mi></mrow></mfenced></mrow></mstyle></math>=$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=MPK -CD@
qu.1.12.comment=<p>The answer is MPK=$anspretty.</p>@
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$a=range(2,8);
$b=range(2,8);
$c=range(2,8);
$d=1-$b;
$F=$c*(L^frac(1,$a))*K^frac(1,$b);
$Fpretty=mathml($c*(L^frac(1,$a))*K^frac(1,$b));
$ans=frac($c,$b)*(L^frac(1,$a))*K^frac($d,$b);
$anspretty=mathml(frac($c,$b)*(L^frac(1,$a))*K^frac($d,$b));@
qu.1.12.uid=21355f9c-9596-4b0d-9e2c-0cea505a55eb@
qu.1.12.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Mpk;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Algorithmic;
@
qu.1.12.weighting=1@
qu.1.12.numbering=alpha@
qu.1.12.part.1.editing=useHTML@
qu.1.12.part.1.question=(Unset)@
qu.1.12.part.1.name=sro_id_1@
qu.1.12.part.1.answer=$ans@
qu.1.12.part.1.mode=Formula@
qu.1.12.question=<p>If the production function is $Fpretty, what is the marginal product of capital (<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>K</mi></mrow></mstyle></math>)?</p><p>MPK=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.13.mode=Inline@
qu.1.13.name=MPL-CD@
qu.1.13.comment=<p>The marginal product of labour is MPL=$anspretty.</p>@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$a=range(2,8);
$b=range(2,8);
$c=range(2,8);
$d=1-$a;
$F=$c*(L^frac(1,$a))*K^frac(1,$b);
$Fpretty=mathml($c*(L^frac(1,$a))*K^frac(1,$b));
$ans=frac($c,$a)*(L^frac($d,$a))*K^frac(1,$b);
$anspretty=mathml(frac($c,$a)*(L^frac($d,$a))*K^frac(1,$b));@
qu.1.13.uid=44864c9b-a382-4fd3-aa6c-dff3a20a0df4@
qu.1.13.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Mpl - Cd;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Algorithmic;
@
qu.1.13.weighting=1@
qu.1.13.numbering=alpha@
qu.1.13.part.1.editing=useHTML@
qu.1.13.part.1.question=(Unset)@
qu.1.13.part.1.name=sro_id_1@
qu.1.13.part.1.answer=$ans@
qu.1.13.part.1.mode=Formula@
qu.1.13.question=<p>If the production function is $Fpretty, what is the marginal product of labour (<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>L</mi></mrow></mstyle></math>)?</p><p><span>MPL= </span><1><span>&nbsp;</span></p>@

qu.1.14.mode=Inline@
qu.1.14.name=Simple Partial Derivative ii@
qu.1.14.comment=<p>Remember to treat <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>&nbsp;as a constant.</p>@
qu.1.14.editing=useHTML@
qu.1.14.hint.1=Think of X as a constant.@
qu.1.14.solution=@
qu.1.14.algorithm=$a=range(1,20,1);
$b=range(1,20,1);
$c=range(2,50,1);
$d=range(2,50,1);
$e=range(1,20,1);
$func=frac($a,$b)*X^2+frac($c,$d)*X*Y^3+frac($b,$e)*Y^2;
$ans="3*($c/$d)*X*Y^(2)+2*($b/$e)*Y";
$F=maple("
printf(MathML[ExportPresentation]($a/$b*X^2+$c/$d*X*Y^3+$b/$e*Y^2));
");@
qu.1.14.uid=635ce5d8-686f-4df2-a6fa-5e4410156f3e@
qu.1.14.info=  Course=Introductory Mathematical Economics;
  Topic=First Partial;
  Author=Asha Sadanand;
  Difficulty=Easy;
@
qu.1.14.weighting=1@
qu.1.14.numbering=alpha@
qu.1.14.part.1.name=sro_id_1@
qu.1.14.part.1.maple_answer=$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 $F 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>Y</mi></mrow></mstyle></math>.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.15.mode=Inline@
qu.1.15.name=CD: MPL, MPK, MRTS@
qu.1.15.comment=<p>The MPL is $MPLpretty.</p>
<p>The MPK is $MPKpretty.</p>
<p>The MRTS<sub>LK</sub> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>MPK</mi></mrow><mrow><mi>MPL</mi></mrow></mfrac></mrow></mstyle></math>=$MRTSpretty.</p>@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$a=range(2,8);
$b=range(2,8);
$c=range(2,8);
$d=1-$a;
$e=1-$b;
$F=$c*(L^frac(1,$a))*K^frac(1,$b);
$Fpretty=mathml($c*(L^frac(1,$a))*K^frac(1,$b));
$MPL=frac($c,$a)*(L^frac($d,$a))*K^frac(1,$b);
$MPLpretty=mathml(frac($c,$a)*(L^frac($d,$a))*K^frac(1,$b));
$MPK=frac($c,$b)*(L^frac(1,$a))*K^frac($e,$b);
$MPKpretty=mathml(frac($c,$b)*(L^frac(1,$a))*K^frac($e,$b));
$MRTS=frac($a,$b)*L/K;
$MRTSpretty=mathml(frac($a,$b)*L/K);@
qu.1.15.uid=647c5862-8c70-4030-8d42-13355ddc645c@
qu.1.15.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Differentiation;
  Sub-Topic=Mpk, Mpl, Mrts;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Algorithmic;
@
qu.1.15.weighting=1,1,1@
qu.1.15.numbering=alpha@
qu.1.15.part.1.editing=useHTML@
qu.1.15.part.1.question=(Unset)@
qu.1.15.part.1.name=sro_id_1@
qu.1.15.part.1.answer=$MPL@
qu.1.15.part.1.mode=Formula@
qu.1.15.part.2.editing=useHTML@
qu.1.15.part.2.question=(Unset)@
qu.1.15.part.2.name=sro_id_2@
qu.1.15.part.2.answer=$MPK@
qu.1.15.part.2.mode=Formula@
qu.1.15.part.3.editing=useHTML@
qu.1.15.part.3.question=(Unset)@
qu.1.15.part.3.name=sro_id_3@
qu.1.15.part.3.answer=$MRTS@
qu.1.15.part.3.mode=Formula@
qu.1.15.question=<p>If the production function is $Fpretty,</p><p>&nbsp;</p><p>What is the marginal product of labour (<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>L</mi></mrow></mstyle></math>)?</p><p>MPL=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the marginal product of capital (<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>K</mi></mrow></mstyle></math>)?</p><p>MPK=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the marginal rate of technical substitution for labour with respect to capital (MRTS<sub>LK</sub>)?</p><p>MRTS<sub>LK</sub>=<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.16.mode=Inline@
qu.1.16.name=Partial Derivatives - ln@
qu.1.16.comment=<p>Remember to treat <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>Y</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>&nbsp;as a constant.</p>
<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><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>Ax</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>Ax</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mi>x</mi></mrow></mfrac></mrow></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>=$anspretty</p>@
qu.1.16.editing=useHTML@
qu.1.16.hint.1=Treat Y as a constant.@
qu.1.16.hint.2=Use the chain rule.@
qu.1.16.hint.3=Use the Product rule.@
qu.1.16.hint.4=Treat the question as ln(U), where U is a function.@
qu.1.16.solution=@
qu.1.16.algorithm=$a=range(1,9,1);
$b=range(1,9,1);
$c=range(1,9,1);
$d=range(2,9,1);
$e=range(2,9,1);
$f=range(2,9,1);
$g=range(2,9,1);
$v=maple("
randomize():
v1:=$a*X^($d)+$b*X^($e)*Y^($f)*ln(sort(randpoly(X, degree=2, coeffs=rand(0..9)))+$c*X^($g)*Y):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string)
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.16.uid=0823c1c6-cccf-4b32-b4d4-d94b36833852@
qu.1.16.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Derivatives;
  Sub-Topic=Ln;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.16.weighting=1@
qu.1.16.numbering=alpha@
qu.1.16.part.1.name=sro_id_1@
qu.1.16.part.1.maple_answer=$ans@
qu.1.16.part.1.editing=useHTML@
qu.1.16.part.1.question=(Unset)@
qu.1.16.part.1.libname=@
qu.1.16.part.1.mode=Maple@
qu.1.16.part.1.allow2d=1@
qu.1.16.part.1.plot=@
qu.1.16.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.16.part.1.type=formula@
qu.1.16.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced></mrow></mstyle></math>=$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.1.17.mode=Inline@
qu.1.17.name=Partial Derivatives - exponential 2 variables@
qu.1.17.comment=<p>Remember to treat <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>Y</mi></mrow></mstyle></math>&nbsp;as a constant.</p>
<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>Ax</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>Ae</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></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>=$anspretty</p>@
qu.1.17.editing=useHTML@
qu.1.17.hint.1=Treat X as a constant.@
qu.1.17.hint.2=Treat the exponential term as exp(V), where V is a function.@
qu.1.17.hint.3=Use the chain rule.@
qu.1.17.hint.4=The derivative of a sum is the sum of@
qu.1.17.solution=@
qu.1.17.algorithm=$a=range(1,20,1);
$b=range(1,20,1);
$c=range(1,20,1);
$d=range(2,9,1);
$e=range(2,9,1);
$f=range(2,9,1);
$g=range(2,9,1);
$v=maple("
randomize():
v1:=($a*X^($d)*exp((sort(randpoly(Y, degree=2, coeffs=rand(-9..9)))))+$b*X^($e)*Y^($f)+$c*Y^($g)):
v2:=diff(v1,Y):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string)
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.17.uid=8925c0e0-573a-484e-a02c-8b53c73f9aa0@
qu.1.17.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Derivatives;
  Sub-Topic=Polynomial Two Variables;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.17.weighting=1@
qu.1.17.numbering=alpha@
qu.1.17.part.1.name=sro_id_1@
qu.1.17.part.1.maple_answer=$ans@
qu.1.17.part.1.editing=useHTML@
qu.1.17.part.1.question=(Unset)@
qu.1.17.part.1.libname=@
qu.1.17.part.1.mode=Maple@
qu.1.17.part.1.allow2d=1@
qu.1.17.part.1.plot=@
qu.1.17.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.17.part.1.type=formula@
qu.1.17.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced></mrow></mstyle></math>=$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>Y</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.18.mode=Inline@
qu.1.18.name=Partial Derivatives - polynomial 3 variables Z@
qu.1.18.comment=<p>Remember to treat the other two variables as constants.</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>=$anspretty</p>@
qu.1.18.editing=useHTML@
qu.1.18.hint.1=Treat X and Y as a constants.@
qu.1.18.solution=@
qu.1.18.algorithm=$a=range(1,20,1);
$b=range(1,20,1);
$c=range(1,20,1);
$d=range(2,9,1);
$e=range(2,9,1);
$f=range(2,9,1);
$g=range(2,9,1);
$h=range(2,9,1);
$i=range(2,9,1);
$j=range(2,9,1);
$k=range(2,9,1);
$v=maple("
randomize():
v1:=($a*X^($d)*Z^($h)+$b*X^($e)*Y^($f)*Z^($k)+$c*Y^($g)*Z^($i)+Z^($j)*X):
v2:=diff(v1,Z):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string)
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.18.uid=f79b8411-94e0-42e7-b21b-692eda759fc1@
qu.1.18.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Derivatives;
  Sub-Topic=Polynomial Two Variables;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.18.weighting=1@
qu.1.18.numbering=alpha@
qu.1.18.part.1.name=sro_id_1@
qu.1.18.part.1.maple_answer=$ans@
qu.1.18.part.1.editing=useHTML@
qu.1.18.part.1.question=(Unset)@
qu.1.18.part.1.libname=@
qu.1.18.part.1.mode=Maple@
qu.1.18.part.1.allow2d=1@
qu.1.18.part.1.plot=@
qu.1.18.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.18.part.1.type=formula@
qu.1.18.question=<p>Differentiate the following function with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Z</mi></mrow></mfenced></mrow></mstyle></math>=$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>Z</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.19.mode=Inline@
qu.1.19.name=Partial Derivatives -Poly in X multiplied by poly in Y@
qu.1.19.comment=<p>Remember to treat the other variables as constants.</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>=$anspretty</p>@
qu.1.19.editing=useHTML@
qu.1.19.hint.1=Treat Y as a constant.@
qu.1.19.solution=@
qu.1.19.algorithm=$v=maple("
randomize():
v1:=(sort(randpoly(X, degree=2, coeffs=rand(-9..9))))*(sort(randpoly(Y, degree=2, coeffs=rand(-9..9)))):
v2:=diff(v1,X):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
v1,convert(v2,string),v3,v4
");
$F=switch(0,$v);
$ans=switch(1,$v);
$Fpretty=switch(2,$v);
$anspretty=switch(3,$v);@
qu.1.19.uid=e0fd81fa-0667-4e8a-af23-68047a79d965@
qu.1.19.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - One Variable;
  Sub-Topic=Ln;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.1.19.weighting=1@
qu.1.19.numbering=alpha@
qu.1.19.part.1.name=sro_id_1@
qu.1.19.part.1.maple_answer=$ans@
qu.1.19.part.1.editing=useHTML@
qu.1.19.part.1.question=(Unset)@
qu.1.19.part.1.libname=@
qu.1.19.part.1.mode=Maple@
qu.1.19.part.1.allow2d=1@
qu.1.19.part.1.plot=@
qu.1.19.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.19.part.1.type=formula@
qu.1.19.question=<p>Differentiate the following function with respect to X:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$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.1.20.mode=Inline@
qu.1.20.name=CES: MPL, MPK, MRTS@
qu.1.20.comment=<p>The MPL is $MPLpretty.</p>
<p>The MPK is $MPKpretty.</p>
<p>The MRTS<sub>LK</sub> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>MPK</mi></mrow><mrow><mi>MPL</mi></mrow></mfrac></mrow></mstyle></math>=$MRTSpretty.</p>@
qu.1.20.editing=useHTML@
qu.1.20.solution=@
qu.1.20.algorithm=$a=range(2,8);
$b=decimal(1,range(.1,.9,.1));
$c=range(2,8);
$d=1-$a;
$e=1-$b;
$g=$c*$e;
$h=$c*$b;
$r=decimal(1,$e/$b);
$F=$c*($b*L^$a+($e)*K^$a)^frac(1,$a);
$Fpretty=mathml($c*($b*L^$a+($e)*K^$a)^frac(1,$a));
$MPL=$h*($b*L^$a+($e)*K^$a)^frac(1-$a,$a)*L^($a-1);
$MPLpretty=mathml($h*($b*L^$a+($e)*K^$a)^frac(1-$a,$a)*L^($a-1));
$MPK=$g*($b*L^$a+($e)*K^$a)^frac(1-$a,$a)*K^($a-1);
$MPKpretty=mathml($g*($b*L^$a+($e)*K^$a)^frac(1-$a,$a)*K^($a-1));
$MRTS=$r*K^($a-1)/L^($a-1);
$MRTSpretty=mathml($r*K^($a-1)/L^($a-1));@
qu.1.20.uid=245136ff-cf51-4113-9a1d-0ef6df053ae1@
qu.1.20.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Differentiation;
  Sub-Topic=Ces  Mpk, Mpl, Mrts;
  Author=Asha Sadanand;
  Difficulty=Medium;
  Feature=Algorithmic;
@
qu.1.20.weighting=1,1,1@
qu.1.20.numbering=alpha@
qu.1.20.part.1.editing=useHTML@
qu.1.20.part.1.question=(Unset)@
qu.1.20.part.1.name=sro_id_1@
qu.1.20.part.1.answer=$MPL@
qu.1.20.part.1.mode=Formula@
qu.1.20.part.2.editing=useHTML@
qu.1.20.part.2.question=(Unset)@
qu.1.20.part.2.name=sro_id_2@
qu.1.20.part.2.answer=$MPK@
qu.1.20.part.2.mode=Formula@
qu.1.20.part.3.name=sro_id_3@
qu.1.20.part.3.maple_answer=$MRTS@
qu.1.20.part.3.editing=useHTML@
qu.1.20.part.3.question=(Unset)@
qu.1.20.part.3.libname=@
qu.1.20.part.3.mode=Maple@
qu.1.20.part.3.allow2d=1@
qu.1.20.part.3.plot=@
qu.1.20.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.20.part.3.type=formula@
qu.1.20.question=<p>If the production function is $Fpretty,</p><p>&nbsp;</p><p>What is the marginal product of labour (<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>L</mi></mrow></mstyle></math>)?</p><p>MPL=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the marginal product of capital (<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>K</mi></mrow></mstyle></math>)?</p><p>MPK=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the marginal rate of technical substitution for labour with respect to capital (MRTS<sub>LK</sub>)?</p><p>MRTS<sub>LK</sub>=<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.2.topic=Second Order Partials@

qu.2.1.mode=Inline@
qu.2.1.name=Second Order Partials - Two Variables@
qu.2.1.comment=<p>Remember to treat the other variable in each case as a constant.</p>
<p>Second partial with respect to X 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span>$XXpretty</span></p>
<p>&nbsp;</p>
<p>Second partial with respect to Y is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>Y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=$YYpretty</p>
<p>&nbsp;</p>
<p>Cross-partial with respect to X,Y 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>XY</mi></mrow></mfrac></mrow></mstyle></math>=<span>$XYpretty</span></p>
<p>&nbsp;</p>
<p>Cross-partial with respect to Y,X is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>XY</mi></mrow></mfrac></mrow></mstyle></math>=$XYpretty<span> </span></p>@
qu.2.1.editing=useHTML@
qu.2.1.hint.1=Young's theorem can save some work here.@
qu.2.1.solution=@
qu.2.1.algorithm=$a=range(2,10);
$b=range(2,10);
$c=range(2,10);
$d=range(2,5);
$e=range(2,5);
$g=range(2,5);
$h=range(2,5);
$v=maple("
F:=$a*X^$d*Y^$h+$b*Y^$e-$c*X^$g:
Fpretty:=MathML[ExportPresentation](F):
v1:=diff(F,X):
v2:=diff(F,Y):
v3:=MathML[ExportPresentation](v1):
v4:=MathML[ExportPresentation](v2):
v5:=diff(v1,X):
v6:=diff(v1,Y):
v7:=diff(v2,Y):
v8:=MathML[ExportPresentation](v5):
v9:=MathML[ExportPresentation](v6):
v10:=MathML[ExportPresentation](v7):
Fpretty,convert(v1,string),convert(v2,string),v3,v4,convert(v5,string),convert(v6,string),convert(v7,string),v8,v9,v10
");
$Fpretty=switch(0,$v);
$Xans=switch(1,$v);
$Yans=switch(2,$v);
$Xpretty=switch(3,$v);
$Ypretty=switch(4,$v);
$XXans=switch(5,$v);
$XYans=switch(6,$v);
$YYans=switch(7,$v);
$XXpretty=switch(8,$v);
$XYpretty=switch(9,$v);
$YYpretty=switch(10,$v);@
qu.2.1.uid=d47150e3-bede-42e8-bd3d-f62a7c004231@
qu.2.1.info=  Course=Introductory Mathematical Economics;
  Topic=Partial Differentiation;
  Sub-Topic=Second Order Derivatives;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.2.1.weighting=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=$XXans@
qu.2.1.part.1.editing=useHTML@
qu.2.1.part.1.question=(Unset)@
qu.2.1.part.1.libname=@
qu.2.1.part.1.mode=Maple@
qu.2.1.part.1.allow2d=1@
qu.2.1.part.1.plot=@
qu.2.1.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.1.type=formula@
qu.2.1.part.2.name=sro_id_2@
qu.2.1.part.2.maple_answer=$YYans@
qu.2.1.part.2.editing=useHTML@
qu.2.1.part.2.question=(Unset)@
qu.2.1.part.2.libname=@
qu.2.1.part.2.mode=Maple@
qu.2.1.part.2.allow2d=1@
qu.2.1.part.2.plot=@
qu.2.1.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.2.type=formula@
qu.2.1.part.3.name=sro_id_3@
qu.2.1.part.3.maple_answer=$XYans@
qu.2.1.part.3.editing=useHTML@
qu.2.1.part.3.question=(Unset)@
qu.2.1.part.3.libname=@
qu.2.1.part.3.mode=Maple@
qu.2.1.part.3.allow2d=1@
qu.2.1.part.3.plot=@
qu.2.1.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.3.type=formula@
qu.2.1.part.4.name=sro_id_4@
qu.2.1.part.4.maple_answer=$XYans@
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.question=<p>Find the second order partial derivatives for 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><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced></mrow></mstyle></math>=$Fpretty</p><p>&nbsp;</p><p>Second partial 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><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>Second partial 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>Y</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><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>Y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>Cross-partial 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><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</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><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>YX</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p><p>Cross-partial 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>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><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><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>YX</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><4><span> <br /></span></p>@

qu.3.topic=Second Order Conditions - Concavity and Convexity@

qu.3.1.mode=Inline@
qu.3.1.name=Hessian matrix -- negative definite@
qu.3.1.comment=<p>The first order conditions are found by differentiating the 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> first, then 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>Y</mi></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math> and finally 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>Z</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>to find:</p>
<p>$Jx =0,</p>
<p>$Jy =0,</p>
<p>$Jz =0.</p>
<p>These equations can be solved using various methods such as Gauss-Jordan, Cramer's rule, or row operations to reach the row echelon form.</p>
<p>The solutions are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$a, <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>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$b, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$c.</p>
<p>&nbsp;</p>
<p>The Hessian is found by differentiating the first order conditions again, and arranging them in a matrix.</p>
<p>$Hpretty</p>
<p>To determine the nature of the Hessian we look at the principal minors, the determinants of the whole matrix, of the matrix formed by dropping the last row and column, and the top left entry.</p>
<p>The latter is $H1, the 2x2 determinant is det $H2</p>
<p>and finally the determinant of the whole matrix is det $Hpretty.</p>
<p>You can verify that these are alternating in sign starting with negative, which means the that Hessian is negative definite, and thus the solution we found is a maximum.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=range(-9,9,1);
$b=range(-9,9,1);
$c=range(-9,9,1);
$Z=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
V:=Matrix([[X-($a)],[Y-($b)],[Z-($c)]]):
V1:=m1.V:
V2:=(LinearAlgebra[Transpose](V1)).V1:
F:=(-1)*(V2[1,1]):
F1:=expand(F):
E:=MathML[ExportPresentation](F1):
H:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y),diff(diff($F,X),Z)],[diff(diff($F,Y),X),diff(diff($F,Y),Y),diff(diff($F,Y),Z)],[diff(diff($F,Z),X),diff(diff($F,Z),Y),diff(diff($F,Z),Z)]]):
H1:=diff(diff($F,X),X):
H2:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y)],[diff(diff($F,Y),X),diff(diff($F,Y),Y)]]):
G:=MathML[ExportPresentation](H):
G1:=MathML[ExportPresentation](H1):
G2:=MathML[ExportPresentation](H2):
JX:=diff($F,X):
JY:=diff($F,Y):
JZ:=diff($F,Z):
Jx:=MathML[ExportPresentation](JX):
Jy:=MathML[ExportPresentation](JY):
Jz:=MathML[ExportPresentation](JZ):
F,E,G,G1,G2,convert(H,string), convert(H2,string),convert(JX,string),convert(JY,string),convert(JZ,string),convert(V,string),convert(V1,string),convert(m1,string),convert(V2,string),Jx,Jy,Jz
");
$E=switch(1,$Z);
$H=switch(5,$Z);
$JX=switch(7,$Z);
$JY=switch(8,$Z);
$JZ=switch(9,$Z);
$Hpretty=switch(2,$Z);
$H1=switch(3,$Z);
$H2=switch(4,$Z);
$Jx=switch(14,$Z);
$Jy=switch(15,$Z);
$Jz=switch(16,$Z);@
qu.3.1.uid=9f33b2a2-259c-4fd6-9d5d-93d48e0f8f6f@
qu.3.1.info=  Course=Introductory Mathematical Economics;
  Topic=Second Order Conditions;
  Sub-Topic=Unconstrained Optimization;
  Difficulty=Hard;
  Author=Asha Sadanand;
  Feature=Many Steps;
@
qu.3.1.weighting=1,1,1,1,1,1,1,1,1,1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.maple_answer=$JX@
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.libname=@
qu.3.1.part.1.mode=Maple@
qu.3.1.part.1.allow2d=1@
qu.3.1.part.1.plot=@
qu.3.1.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.1.part.1.type=formula@
qu.3.1.part.2.name=sro_id_2@
qu.3.1.part.2.maple_answer=$JY@
qu.3.1.part.2.editing=useHTML@
qu.3.1.part.2.question=(Unset)@
qu.3.1.part.2.libname=@
qu.3.1.part.2.mode=Maple@
qu.3.1.part.2.allow2d=1@
qu.3.1.part.2.plot=@
qu.3.1.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.1.part.2.type=formula@
qu.3.1.part.3.name=sro_id_3@
qu.3.1.part.3.maple_answer=$JZ@
qu.3.1.part.3.editing=useHTML@
qu.3.1.part.3.question=(Unset)@
qu.3.1.part.3.libname=@
qu.3.1.part.3.mode=Maple@
qu.3.1.part.3.allow2d=1@
qu.3.1.part.3.plot=@
qu.3.1.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.1.part.3.type=formula@
qu.3.1.part.4.editing=useHTML@
qu.3.1.part.4.question=(Unset)@
qu.3.1.part.4.name=sro_id_4@
qu.3.1.part.4.answer=$a@
qu.3.1.part.4.mode=Formula@
qu.3.1.part.5.editing=useHTML@
qu.3.1.part.5.question=(Unset)@
qu.3.1.part.5.name=sro_id_5@
qu.3.1.part.5.answer=$b@
qu.3.1.part.5.mode=Formula@
qu.3.1.part.6.editing=useHTML@
qu.3.1.part.6.question=(Unset)@
qu.3.1.part.6.name=sro_id_6@
qu.3.1.part.6.answer=$c@
qu.3.1.part.6.mode=Formula@
qu.3.1.part.7.name=sro_id_7@
qu.3.1.part.7.maple_answer=printf("$Hpretty");@
qu.3.1.part.7.editing=useHTML@
qu.3.1.part.7.question=(Unset)@
qu.3.1.part.7.libname=@
qu.3.1.part.7.mode=Maple@
qu.3.1.part.7.allow2d=2@
qu.3.1.part.7.plot=@
qu.3.1.part.7.maple=ans:=$H:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.3.1.part.7.type=maple@
qu.3.1.part.8.grader=exact@
qu.3.1.part.8.name=sro_id_8@
qu.3.1.part.8.editing=useHTML@
qu.3.1.part.8.answer.5=Negative semi-definite@
qu.3.1.part.8.display.permute=true@
qu.3.1.part.8.answer.4=Positive semi-definite@
qu.3.1.part.8.answer.3=Indefinite@
qu.3.1.part.8.question=(Unset)@
qu.3.1.part.8.answer.2=Negative definite@
qu.3.1.part.8.answer.1=Positive definite@
qu.3.1.part.8.mode=List@
qu.3.1.part.8.display=menu@
qu.3.1.part.8.credit.5=0.0@
qu.3.1.part.8.credit.4=0.0@
qu.3.1.part.8.credit.3=0.0@
qu.3.1.part.8.credit.2=1.0@
qu.3.1.part.8.credit.1=0.0@
qu.3.1.part.9.grader=exact@
qu.3.1.part.9.name=sro_id_9@
qu.3.1.part.9.editing=useHTML@
qu.3.1.part.9.display.permute=false@
qu.3.1.part.9.answer.3=neither necessary nor sufficient@
qu.3.1.part.9.question=(Unset)@
qu.3.1.part.9.answer.2=sufficient@
qu.3.1.part.9.answer.1=necessary@
qu.3.1.part.9.mode=List@
qu.3.1.part.9.display=menu@
qu.3.1.part.9.credit.3=0.0@
qu.3.1.part.9.credit.2=1.0@
qu.3.1.part.9.credit.1=0.0@
qu.3.1.part.10.grader=exact@
qu.3.1.part.10.name=sro_id_10@
qu.3.1.part.10.editing=useHTML@
qu.3.1.part.10.display.permute=false@
qu.3.1.part.10.answer.3=an inconclusive result@
qu.3.1.part.10.question=(Unset)@
qu.3.1.part.10.answer.2=a minimum@
qu.3.1.part.10.answer.1=a maximum@
qu.3.1.part.10.mode=List@
qu.3.1.part.10.display=menu@
qu.3.1.part.10.credit.3=0.0@
qu.3.1.part.10.credit.2=0.0@
qu.3.1.part.10.credit.1=1.0@
qu.3.1.question=<p>Consider the function $E. To optimize this function we find the first order conditions for <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>, <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>Y</mi></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><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>Please give the first order conditions in the order <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>,<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>Y</mi></mrow></mstyle></math> and lastly <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>Z</mi></mrow></mstyle></math>.</p><p><span>&nbsp;</span><1><span> </span>=0</p><p><span>&nbsp;</span><2><span> </span>=0</p><p><span>&nbsp;</span><3><span> </span>=0.</p><p>&nbsp;</p><p>Solving, we find <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='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span> </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='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><5><span>, and&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>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><6><span>.&nbsp;</span> </span></p><p>&nbsp;</p><p>Find the Hessian for this function. <span>&nbsp;</span><7><span> </span>.</p><p>&nbsp;</p><p>The Hessian is <span>&nbsp;</span><8><span> </span>, and this is<span> a </span><9><span>&nbsp;</span>condition for&nbsp;&nbsp; <10><span> </span>.</p>@

qu.3.2.mode=Inline@
qu.3.2.name=Hessian matrix -- positive definite@
qu.3.2.comment=<p>The first order conditions are found by differentiating the 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> first, then 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>Y</mi></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math> and finally 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>Z</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>to find:</p>
<p>$Jx =0,</p>
<p>$Jy =0,</p>
<p>$Jz =0.</p>
<p>These equations can be solved using various methods such as Gauss-Jordan, Cramer's rule, or row operations to reach the row echelon form.</p>
<p>The solutions are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$a, <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>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$b, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$c.</p>
<p>&nbsp;</p>
<p>The Hessian is found by differentiating the first order conditions again, and arranging them in a matrix.</p>
<p>$Hpretty</p>
<p>To determine the nature of the Hessian we look at the principal minors, the determinants of the whole matrix, of the matrix formed by dropping the last row and column, and the top left entry.</p>
<p>The latter is $H1, the 2x2 determinant is det $H2</p>
<p>and finally the determinant of the whole matrix is det $Hpretty.</p>
<p>You can verify that these are all positive, which means the that Hessian is positive definite, and thus the solution we found is a minimum.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=range(-9,9,1);
$b=range(-9,9,1);
$c=range(-9,9,1);
$Z=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
V:=Matrix([[X-($a)],[Y-($b)],[Z-($c)]]):
V1:=m1.V:
V2:=(LinearAlgebra[Transpose](V1)).V1:
F:=V2[1,1]:
F1:=expand(F):
E:=MathML[ExportPresentation](F1):
H:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y),diff(diff($F,X),Z)],[diff(diff($F,Y),X),diff(diff($F,Y),Y),diff(diff($F,Y),Z)],[diff(diff($F,Z),X),diff(diff($F,Z),Y),diff(diff($F,Z),Z)]]):
H1:=diff(diff($F,X),X):
H2:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y)],[diff(diff($F,Y),X),diff(diff($F,Y),Y)]]):
G:=MathML[ExportPresentation](H):
G1:=MathML[ExportPresentation](H1):
G2:=MathML[ExportPresentation](H2):
JX:=diff($F,X):
JY:=diff($F,Y):
JZ:=diff($F,Z):
Jx:=MathML[ExportPresentation](JX):
Jy:=MathML[ExportPresentation](JY):
Jz:=MathML[ExportPresentation](JZ):
F,E,G,G1,G2,convert(H,string), convert(H2,string),convert(JX,string),convert(JY,string),convert(JZ,string),convert(V,string),convert(V1,string),convert(m1,string),convert(V2,string),Jx,Jy,Jz
");
$E=switch(1,$Z);
$H=switch(5,$Z);
$Hpretty=switch(2,$Z);
$JX=switch(7,$Z);
$JY=switch(8,$Z);
$JZ=switch(9,$Z);
$H1=switch(3,$Z);
$H2=switch(4,$Z);
$Jx=switch(14,$Z);
$Jy=switch(15,$Z);
$Jz=switch(16,$Z);@
qu.3.2.uid=a5177cc2-db21-476e-95ed-8c343f54160c@
qu.3.2.info=  Course=Introductory Mathematical Economics;
  Topic=Second Order Conditions;
  Sub-Topic=Unconstrained Optimization;
  Difficulty=Hard;
  Author=Asha Sadanand;
  Feature=Many Steps;
@
qu.3.2.weighting=1,1,1,1,1,1,1,1,1,1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.maple_answer=$JX@
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.question=(Unset)@
qu.3.2.part.1.libname=@
qu.3.2.part.1.mode=Maple@
qu.3.2.part.1.allow2d=1@
qu.3.2.part.1.plot=@
qu.3.2.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.2.part.1.type=formula@
qu.3.2.part.2.name=sro_id_2@
qu.3.2.part.2.maple_answer=$JY@
qu.3.2.part.2.editing=useHTML@
qu.3.2.part.2.question=(Unset)@
qu.3.2.part.2.libname=@
qu.3.2.part.2.mode=Maple@
qu.3.2.part.2.allow2d=1@
qu.3.2.part.2.plot=@
qu.3.2.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.2.part.2.type=formula@
qu.3.2.part.3.name=sro_id_3@
qu.3.2.part.3.maple_answer=$JZ@
qu.3.2.part.3.editing=useHTML@
qu.3.2.part.3.question=(Unset)@
qu.3.2.part.3.libname=@
qu.3.2.part.3.mode=Maple@
qu.3.2.part.3.allow2d=1@
qu.3.2.part.3.plot=@
qu.3.2.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.2.part.3.type=formula@
qu.3.2.part.4.editing=useHTML@
qu.3.2.part.4.question=(Unset)@
qu.3.2.part.4.name=sro_id_4@
qu.3.2.part.4.answer=$a@
qu.3.2.part.4.mode=Formula@
qu.3.2.part.5.editing=useHTML@
qu.3.2.part.5.question=(Unset)@
qu.3.2.part.5.name=sro_id_5@
qu.3.2.part.5.answer=$b@
qu.3.2.part.5.mode=Formula@
qu.3.2.part.6.editing=useHTML@
qu.3.2.part.6.question=(Unset)@
qu.3.2.part.6.name=sro_id_6@
qu.3.2.part.6.answer=$c@
qu.3.2.part.6.mode=Formula@
qu.3.2.part.7.name=sro_id_7@
qu.3.2.part.7.maple_answer=printf("$Hpretty");@
qu.3.2.part.7.editing=useHTML@
qu.3.2.part.7.question=(Unset)@
qu.3.2.part.7.libname=@
qu.3.2.part.7.mode=Maple@
qu.3.2.part.7.allow2d=2@
qu.3.2.part.7.plot=@
qu.3.2.part.7.maple=ans:=$H:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.3.2.part.7.type=maple@
qu.3.2.part.8.grader=exact@
qu.3.2.part.8.name=sro_id_8@
qu.3.2.part.8.editing=useHTML@
qu.3.2.part.8.answer.5=Negative semi-definite@
qu.3.2.part.8.display.permute=true@
qu.3.2.part.8.answer.4=Positive semi-definite@
qu.3.2.part.8.answer.3=Indefinite@
qu.3.2.part.8.question=(Unset)@
qu.3.2.part.8.answer.2=Negative definite@
qu.3.2.part.8.answer.1=Positive definite@
qu.3.2.part.8.mode=List@
qu.3.2.part.8.display=menu@
qu.3.2.part.8.credit.5=0.0@
qu.3.2.part.8.credit.4=0.0@
qu.3.2.part.8.credit.3=0.0@
qu.3.2.part.8.credit.2=0.0@
qu.3.2.part.8.credit.1=1.0@
qu.3.2.part.9.grader=exact@
qu.3.2.part.9.name=sro_id_9@
qu.3.2.part.9.editing=useHTML@
qu.3.2.part.9.display.permute=false@
qu.3.2.part.9.answer.3=an inconclusive@
qu.3.2.part.9.question=(Unset)@
qu.3.2.part.9.answer.2=a sufficient@
qu.3.2.part.9.answer.1=a necessary@
qu.3.2.part.9.mode=List@
qu.3.2.part.9.display=menu@
qu.3.2.part.9.credit.3=0.0@
qu.3.2.part.9.credit.2=1.0@
qu.3.2.part.9.credit.1=0.0@
qu.3.2.part.10.grader=exact@
qu.3.2.part.10.name=sro_id_10@
qu.3.2.part.10.editing=useHTML@
qu.3.2.part.10.display.permute=false@
qu.3.2.part.10.answer.3=that indicates neither a maximum nor a minimum@
qu.3.2.part.10.question=(Unset)@
qu.3.2.part.10.answer.2=for a minimum@
qu.3.2.part.10.answer.1=for a maximum@
qu.3.2.part.10.mode=List@
qu.3.2.part.10.display=menu@
qu.3.2.part.10.credit.3=0.0@
qu.3.2.part.10.credit.2=1.0@
qu.3.2.part.10.credit.1=0.0@
qu.3.2.question=<p>Consider the function $E. To optimize this function we find the first order conditions for <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>, <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>Y</mi></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><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>Please give the first order conditions in the order <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>,<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>Y</mi></mrow></mstyle></math> and lastly <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>Z</mi></mrow></mstyle></math>.</p><p><span>&nbsp;</span><1><span> </span>=0</p><p><span>&nbsp;</span><2><span> </span>=0</p><p><span>&nbsp;</span><3><span> </span>=0.</p><p>&nbsp;</p><p>Solving, we find <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='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span> </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='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><5><span>, and&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>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><6><span>.&nbsp;</span> </span></p><p>&nbsp;</p><p>Find the Hessian for this function. <span>&nbsp;</span><7><span> </span>.</p><p>&nbsp;</p><p>The Hessian is <span>&nbsp;</span><8><span> </span>, and this is<span>&nbsp; </span><9><span>&nbsp;</span>condition&nbsp;&nbsp; <10><span> </span>.</p>@

qu.3.3.mode=Blanks@
qu.3.3.name=Use of Second Order Condition@
qu.3.3.comment=@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=@
qu.3.3.uid=645dd3fe-d584-43b1-a9ed-7bffddeb2ec0@
qu.3.3.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Second Order Conditions;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.3.question=In an optimization problem, the  <1>  <2>  <3> indicate whether the solution to the first order conditions is a maximum, and minimum or neither. @
qu.3.3.blank.1=second@
qu.3.3.blank.2=order@
qu.3.3.blank.3=conditions@
qu.3.3.extra=@
qu.3.3.format.input=text@

qu.3.4.mode=Inline@
qu.3.4.name=Hessian matrix -- indefinite@
qu.3.4.comment=<p>The first order conditions are found by differentiating the 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> first, then 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>Y</mi></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math> and finally 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>Z</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>to find:</p>
<p>$Jx =0,</p>
<p>$Jy =0,</p>
<p>$Jz =0.</p>
<p>These equations can be solved using various methods such as Gauss-Jordan, Cramer's rule, or row operations to reach the row echelon form.</p>
<p>The solutions are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$a, <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>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$b, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$c.</p>
<p>&nbsp;</p>
<p>The Hessian is found by differentiating the first order conditions again, and arranging them in a matrix.</p>
<p>$Hpretty</p>
<p>To determine the nature of the Hessian we look at the principal minors, the determinants of the whole matrix, of the matrix formed by dropping the last row and column, and the top left entry.</p>
<p>The latter is $H1, the 2x2 determinant is det $H2</p>
<p>and finally the determinant of the whole matrix is det $Hpretty.</p>
<p>You can verify that these are neither all positive nor alternating in sign starting with negative, which means the that Hessian is indefinite, and thus the solution we found is neither a maximum nor a minimum.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$a=range(-9,9,1);
$b=range(-9,9,1);
$c=range(-9,9,1);
$d=2*(range(0,1,1))-1;
$e=2*(range(0,1,1))-1;
$f=((-1)^(($d)*($e)))*($d);
$Z=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
A:=Matrix([[($d),0,0],[0,($e),0],[0,0,($f)]]):
V:=Matrix([[X-($a)],[Y-($b)],[Z-($c)]]):
V1:=m1.V:
V2:=(LinearAlgebra[Transpose](V1)).A.V1:
F:=(-1)*(V2[1,1]):
F1:=expand(F):
E:=MathML[ExportPresentation](F1):
H:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y),diff(diff($F,X),Z)],[diff(diff($F,Y),X),diff(diff($F,Y),Y),diff(diff($F,Y),Z)],[diff(diff($F,Z),X),diff(diff($F,Z),Y),diff(diff($F,Z),Z)]]):
H1:=diff(diff($F,X),X):
H2:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y)],[diff(diff($F,Y),X),diff(diff($F,Y),Y)]]):
G:=MathML[ExportPresentation](H):
G1:=MathML[ExportPresentation](H1):
G2:=MathML[ExportPresentation](H2):
JX:=diff($F,X):
JY:=diff($F,Y):
JZ:=diff($F,Z):
Jx:=MathML[ExportPresentation](JX):
Jy:=MathML[ExportPresentation](JY):
Jz:=MathML[ExportPresentation](JZ):
F,E,G,G1,G2,convert(H,string), convert(H2,string),convert(JX,string),convert(JY,string),convert(JZ,string),convert(V,string),convert(V1,string),convert(m1,string),convert(V2,string),Jx,Jy,Jz
");
$E=switch(1,$Z);
$Hpretty=switch(2,$Z);
$JX=switch(7,$Z);
$JY=switch(8,$Z);
$JZ=switch(9,$Z);
$H=switch(5,$Z);
$H1=switch(3,$Z);
$H2=switch(4,$Z);
$Jx=switch(14,$Z);
$Jy=switch(15,$Z);
$Jz=switch(16,$Z);@
qu.3.4.uid=d25c781b-8db4-48e2-a017-72623858fb3f@
qu.3.4.info=  Course=Introductory Mathematical Economics;
  Topic=Second Order Conditions;
  Sub-Topic=Unconstrained Optimization;
  Difficulty=Hard;
  Author=Asha Sadanand;
  Feature=Many Steps;
@
qu.3.4.weighting=1,1,1,1,1,1,1,1,1,1@
qu.3.4.numbering=alpha@
qu.3.4.part.1.name=sro_id_1@
qu.3.4.part.1.maple_answer=$JX@
qu.3.4.part.1.editing=useHTML@
qu.3.4.part.1.question=(Unset)@
qu.3.4.part.1.libname=@
qu.3.4.part.1.mode=Maple@
qu.3.4.part.1.allow2d=1@
qu.3.4.part.1.plot=@
qu.3.4.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.4.part.1.type=formula@
qu.3.4.part.2.name=sro_id_2@
qu.3.4.part.2.maple_answer=$JY@
qu.3.4.part.2.editing=useHTML@
qu.3.4.part.2.question=(Unset)@
qu.3.4.part.2.libname=@
qu.3.4.part.2.mode=Maple@
qu.3.4.part.2.allow2d=1@
qu.3.4.part.2.plot=@
qu.3.4.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.4.part.2.type=formula@
qu.3.4.part.3.name=sro_id_3@
qu.3.4.part.3.maple_answer=$JZ@
qu.3.4.part.3.editing=useHTML@
qu.3.4.part.3.question=(Unset)@
qu.3.4.part.3.libname=@
qu.3.4.part.3.mode=Maple@
qu.3.4.part.3.allow2d=1@
qu.3.4.part.3.plot=@
qu.3.4.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.3.4.part.3.type=formula@
qu.3.4.part.4.editing=useHTML@
qu.3.4.part.4.question=(Unset)@
qu.3.4.part.4.name=sro_id_4@
qu.3.4.part.4.answer=$a@
qu.3.4.part.4.mode=Formula@
qu.3.4.part.5.editing=useHTML@
qu.3.4.part.5.question=(Unset)@
qu.3.4.part.5.name=sro_id_5@
qu.3.4.part.5.answer=$b@
qu.3.4.part.5.mode=Formula@
qu.3.4.part.6.editing=useHTML@
qu.3.4.part.6.question=(Unset)@
qu.3.4.part.6.name=sro_id_6@
qu.3.4.part.6.answer=$c@
qu.3.4.part.6.mode=Formula@
qu.3.4.part.7.name=sro_id_7@
qu.3.4.part.7.maple_answer=printf("$Hpretty");@
qu.3.4.part.7.editing=useHTML@
qu.3.4.part.7.question=(Unset)@
qu.3.4.part.7.libname=@
qu.3.4.part.7.mode=Maple@
qu.3.4.part.7.allow2d=2@
qu.3.4.part.7.plot=@
qu.3.4.part.7.maple=ans:=$H:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.3.4.part.7.type=maple@
qu.3.4.part.8.grader=exact@
qu.3.4.part.8.name=sro_id_8@
qu.3.4.part.8.editing=useHTML@
qu.3.4.part.8.answer.5=Negative semi-definite@
qu.3.4.part.8.display.permute=true@
qu.3.4.part.8.answer.4=Positive semi-definite@
qu.3.4.part.8.answer.3=Indefinite@
qu.3.4.part.8.question=(Unset)@
qu.3.4.part.8.answer.2=Negative definite@
qu.3.4.part.8.answer.1=Positive definite@
qu.3.4.part.8.mode=List@
qu.3.4.part.8.display=menu@
qu.3.4.part.8.credit.5=0.0@
qu.3.4.part.8.credit.4=0.0@
qu.3.4.part.8.credit.3=1.0@
qu.3.4.part.8.credit.2=0.0@
qu.3.4.part.8.credit.1=0.0@
qu.3.4.part.9.grader=exact@
qu.3.4.part.9.name=sro_id_9@
qu.3.4.part.9.editing=useHTML@
qu.3.4.part.9.display.permute=true@
qu.3.4.part.9.answer.3=neither necessary nor sufficient@
qu.3.4.part.9.question=(Unset)@
qu.3.4.part.9.answer.2=sufficient@
qu.3.4.part.9.answer.1=necessary@
qu.3.4.part.9.mode=List@
qu.3.4.part.9.display=menu@
qu.3.4.part.9.credit.3=1.0@
qu.3.4.part.9.credit.2=0.0@
qu.3.4.part.9.credit.1=0.0@
qu.3.4.part.10.grader=exact@
qu.3.4.part.10.name=sro_id_10@
qu.3.4.part.10.editing=useHTML@
qu.3.4.part.10.display.permute=true@
qu.3.4.part.10.answer.3=an inconclusive result@
qu.3.4.part.10.question=(Unset)@
qu.3.4.part.10.answer.2=a minimum@
qu.3.4.part.10.answer.1=a maximum@
qu.3.4.part.10.mode=List@
qu.3.4.part.10.display=menu@
qu.3.4.part.10.credit.3=1.0@
qu.3.4.part.10.credit.2=0.0@
qu.3.4.part.10.credit.1=0.0@
qu.3.4.question=<p>Consider the function $E. To optimize this function we find the first order conditions for <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>, <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>Y</mi></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><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>Please give the first order conditions in the order <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>,<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>Y</mi></mrow></mstyle></math> and lastly <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>Z</mi></mrow></mstyle></math>.</p><p><span>&nbsp;</span><1><span> </span>=0</p><p><span>&nbsp;</span><2><span> </span>=0</p><p><span>&nbsp;</span><3><span> </span>=0.</p><p>&nbsp;</p><p>Solving, we find <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='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span> </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='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><5><span>, and&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>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><6><span>.&nbsp;</span> </span></p><p>&nbsp;</p><p>Find the Hessian for this function. <span>&nbsp;</span><7><span> </span>.</p><p>&nbsp;</p><p>The Hessian is <span>&nbsp;</span><8><span> </span>, and this is<span> a </span><9><span>&nbsp;</span>condition for&nbsp;&nbsp; <10><span> </span>.</p>@

