qu.1.topic=n Sub-intervals@

qu.1.1.mode=Inline@
qu.1.1.name=n, RtEndPt@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$c=rint(0,3);
$d=rint(0,3);
$a=rint(-3,0);
$b=rint($a+2,$a+8,2);
$f="x^2+($c)*x+($d)";
$DX="(($b)-($a))/n";
$dx="(($b)-($a))/10";
$M=maple("
c:=eval($f,x=($a)+i*($DX))*($DX):
a:=int($f,x=($a)..($b)):
convert(c,string),
convert(a,string),
MathML[ExportPresentation]($f)
");
$ANSc=switch(0,$M);
$A=switch(1,$M);
$F=switch(2,$M);
$plot=plotmaple("plot([[x, $f, x =($a) .. ($b)], [$a, x, x = 0 .. eval($f,x=$a)],
[($a)+5*($dx),x,x=0..eval($f,x=($a)+6*($dx))],
[($a)+6*($dx),x,x=0..eval($f,x=($a)+6*($dx))],
[$b,x,x=0..eval($f,x=$b)],[x,eval($f,x=($a)+6*($dx)),x=($a)+5*($dx)..($a)+6*($dx)]],
x=$a-1..$b+1,thickness=2, color=[black,black,red,red,black,red]),
plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.1.uid=829e868f-89b3-46c9-9878-4f1cabcdd1f7@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Reimann Sums;
  Sub-Topic=Infinite;
@
qu.1.1.weighting=1,1,1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.maple_answer=(($b)-($a))/n@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.libname=@
qu.1.1.part.1.mode=Maple@
qu.1.1.part.1.allow2d=1@
qu.1.1.part.1.plot=@
qu.1.1.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.1.part.1.type=formula@
qu.1.1.part.2.name=sro_id_2@
qu.1.1.part.2.maple_answer=($a)+(($b)-($a))/n*i@
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.question=(Unset)@
qu.1.1.part.2.libname=@
qu.1.1.part.2.mode=Maple@
qu.1.1.part.2.allow2d=1@
qu.1.1.part.2.plot=@
qu.1.1.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.1.part.2.type=formula@
qu.1.1.part.3.name=sro_id_3@
qu.1.1.part.3.maple_answer=$ANSc@
qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.question=(Unset)@
qu.1.1.part.3.libname=@
qu.1.1.part.3.mode=Maple@
qu.1.1.part.3.allow2d=1@
qu.1.1.part.3.plot=@
qu.1.1.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.1.part.3.type=formula@
qu.1.1.part.4.name=sro_id_4@
qu.1.1.part.4.maple_answer=int($f,x=$a..$b)@
qu.1.1.part.4.editing=useHTML@
qu.1.1.part.4.question=(Unset)@
qu.1.1.part.4.libname=@
qu.1.1.part.4.mode=Maple@
qu.1.1.part.4.allow2d=1@
qu.1.1.part.4.plot=@
qu.1.1.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.1.part.4.type=formula@
qu.1.1.question=<p>$plot</p><p>Evaluate the definite integral <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></munderover></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mi>&Delta;</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></math> , using <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>n</mi></mrow></math> equal subintervals and right endpoints<strong> </strong>(so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></math>).</p><p>&nbsp;</p><p>(a) The subinterval length is<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> <1></p><p>&nbsp;</p><p><span> </span></p><p><span>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><2></span></p><p>&nbsp;</p><p><span><span> </span></span></p><p><span>(c) The general term in the Riemann Sum</span> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> <3><span> </span></p><p>&nbsp;<br /><span>(d) Now, using <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>c</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math>, </span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mi>i</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></span></p><p><span>and then taking the limit as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></math>, find the exact value of </span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></munderover></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mi>&Delta;</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></math>.</p><p><span><4><span> </span></span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=n, LeftEndPt@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$c=rint(0,3);
$d=rint(0,3);
$a=rint(-3,0);
$b=rint($a+2,$a+8,2);
$f="x^2+($c)*x+($d)";
$DX="(($b)-($a))/n";
$dx="(($b)-($a))/10";
$F=maple("printf(MathML[ExportPresentation]($f))");


$plot=plotmaple("plot([[x, $f, x =($a) .. ($b)], [$a, x, x = 0 ..eval($f,x=$a)], [($a)+5*($dx),x,x=0..eval($f,x=($a)+5*($dx))],
[($a)+6*($dx),x,x=0..eval($f,x=($a)+5*($dx))],
[$b,x,x=0..eval($f,x=$b)],
[x,eval($f,x=($a)+5*($dx)),x=($a)+5*($dx)..($a)+6*($dx)]],
x=$a-1..$b+1,thickness=2, color=[black,black,red,red,black,red]),
plotdevice='gif', plotoptions='height=250,width=250'");@
qu.1.2.uid=27355853-5f44-4cb6-b31d-cf0f370721de@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Reimann Sums;
  Sub-Topic=Infinite;
@
qu.1.2.weighting=1,1,1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.maple_answer=(($b)-($a))/n@
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=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.2.part.1.type=formula@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.maple_answer=($a)+(($b)-($a))/n*(i-1)@
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.question=(Unset)@
qu.1.2.part.2.libname=@
qu.1.2.part.2.mode=Maple@
qu.1.2.part.2.allow2d=1@
qu.1.2.part.2.plot=@
qu.1.2.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.2.part.2.type=formula@
qu.1.2.part.3.name=sro_id_3@
qu.1.2.part.3.maple_answer=eval($f,x=($a)+(i-1)*($DX))*($DX)@
qu.1.2.part.3.editing=useHTML@
qu.1.2.part.3.question=(Unset)@
qu.1.2.part.3.libname=@
qu.1.2.part.3.mode=Maple@
qu.1.2.part.3.allow2d=1@
qu.1.2.part.3.plot=@
qu.1.2.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.2.part.3.type=formula@
qu.1.2.part.4.name=sro_id_4@
qu.1.2.part.4.maple_answer=int($f,x=$a..$b)@
qu.1.2.part.4.editing=useHTML@
qu.1.2.part.4.question=(Unset)@
qu.1.2.part.4.libname=@
qu.1.2.part.4.mode=Maple@
qu.1.2.part.4.allow2d=1@
qu.1.2.part.4.plot=@
qu.1.2.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.2.part.4.type=formula@
qu.1.2.question=<p>$plot</p><p>Evaluate the definite integral<strong> </strong><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></munderover></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mi>&Delta;</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></math>, using <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>n</mi></mrow></math> equal subintervals and left endpoints<strong> </strong>(so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msub></mrow></math>).</p><p>&nbsp;</p><p>(a) The subinterval length is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> <1>.</p><p>&nbsp;</p><p><span>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><2></span></p><p>&nbsp;</p><p><span><span> </span></span></p><p><span>(c) The general term in the Riemann Sum</span> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> <3><span> </span></p><p><br /><span>(d) Now, using <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>c</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math>, </span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mi>i</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></span></p><p><span>and then taking the limit as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></math>, find the exact value of </span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></munderover></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mi>&Delta;</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></math>.</p><p><span><4><span> </span></span></p>@

qu.2.topic=Finite Left@

qu.2.1.mode=Inline@
qu.2.1.name=n=3, LeftEndPt@
qu.2.1.comment=@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$c=rint(0,2);
$a=rint(-3,0);
$b=rint($a+3,$a+10,3);
$f="x^2+($c)*x+1";
$M=maple("
ansc:=eval($f,x=($a)+(i-1)*((($b)-($a))/3)):
if eval($f,x=$b)> eval($f,x=($b)-((($b)-($a))/3)) 
then q:=$b: else q:=(($b)-((($b)-($a))/3)): end if:

convert(ansc,string),
convert(q,string),
MathML[ExportPresentation](y=$f)

");
$ANSc=switch(0,$M);
$F=switch(2,$M);
$q=switch(1,$M);

$plot=plotmaple("plot([[x, $f, x =($a) .. ($b)], 
[$a, x, x = 0 .. eval($f,x=$a)],
[($a)+((($b)-($a))/3),x,x=0..max(eval($f,x=($a)),eval($f,x=($a)+((($b)-($a))/3)))],
[($a)+2*((($b)-($a))/3),x,x=0..max(eval($f,x=($a)+((($b)-($a))/3)),eval($f,x=($a)+2*((($b)-($a))/3)))],
[$b,x,x=0..eval($f,x=$q)],
[x,eval($f,x=($a)),x=($a)..($a)+((($b)-($a))/3)],
[x,eval($f,x=($a)+((($b)-($a))/3)),x=($a)+((($b)-($a))/3)..($a)+2*((($b)-($a))/3)],
[x,eval($f,x=($a)+2*((($b)-($a))/3)),x=($a)+2*((($b)-($a))/3)..($b)]],
x=$a-1..$b+1,thickness=2, color=[black,black,black,black,black,red,red,red]),
plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.1.uid=e71f4a22-649f-405b-8afa-532c173637df@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Reimann Sums;
  Sub-Topic=Finite;
@
qu.2.1.weighting=1,1,1,1@
qu.2.1.numbering=alpha@
qu.2.1.part.1.name=sro_id_1@
qu.2.1.part.1.maple_answer=(($b)-($a))/3@
qu.2.1.part.1.editing=useHTML@
qu.2.1.part.1.question=(Unset)@
qu.2.1.part.1.libname=@
qu.2.1.part.1.mode=Maple@
qu.2.1.part.1.allow2d=1@
qu.2.1.part.1.plot=@
qu.2.1.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.2.1.part.1.type=formula@
qu.2.1.part.2.editing=useHTML@
qu.2.1.part.2.question=(Unset)@
qu.2.1.part.2.name=sro_id_2@
qu.2.1.part.2.answer=($a)+(($b)-($a))/3*(i-1)@
qu.2.1.part.2.mode=Formula@
qu.2.1.part.3.editing=useHTML@
qu.2.1.part.3.question=(Unset)@
qu.2.1.part.3.name=sro_id_3@
qu.2.1.part.3.answer=$ANSc@
qu.2.1.part.3.mode=Formula@
qu.2.1.part.4.name=sro_id_4@
qu.2.1.part.4.maple_answer=((($b)-($a))/3)*sum(eval($f,x=$a+(i-1)*((($b)-($a))/3)),i=1..3)@
qu.2.1.part.4.editing=useHTML@
qu.2.1.part.4.question=(Unset)@
qu.2.1.part.4.libname=@
qu.2.1.part.4.mode=Maple@
qu.2.1.part.4.allow2d=1@
qu.2.1.part.4.plot=@
qu.2.1.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.2.1.part.4.type=formula@
qu.2.1.question=<p>$plot</p><p>Find the Riemann Sum <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> for the function $F between&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. Use three equal subintervals and left endpoints<strong> </strong>(so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msub></mrow></math>).</p><p>&nbsp;</p><p>(a) Subinterval length <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> <1></p><p>&nbsp;</p><p><span> </span></p><p><span>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><2></span></p><p>&nbsp;</p><p><span><span> </span></span></p><p><span><span>(c) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><3></span></span></p><p>&nbsp;</p><p><span><span><span> </span></span></span></p><p><span>(d) The Riemann Sum</span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><4><span> </span></p>@

qu.2.2.mode=Inline@
qu.2.2.name=n=4, LeftEndPt@
qu.2.2.comment=@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$c=rint(0,2);
$a=rint(-3,0);
$b=rint($a+2,$a+8,2);
$f="x^2+($c)*x+1";
$dx="(($b)-($a))/4";
$M=maple("
ansc:=eval($f,x=($a)+(i-1)*($dx)):
if eval($f,x=$b)> eval($f,x=($b)-($dx)) 
then q:=$b: else q:=(($b)-($dx)): end if:
convert(ansc,string),
convert(q,string),
MathML[ExportPresentation](y=$f)
");
$ANSc=switch(0,$M);
$F=switch(2,$M);
$q=switch(1,$M);
$plot=plotmaple("plot([[x, $f, x =($a) .. ($b)], 
[$a, x, x = 0.. eval($f,x=$a)],
[($a)+($dx),x,x=0..max(eval($f,x=($a)),eval($f,x=($a)+($dx)))],
[($a)+2*($dx),x,x=0..max(eval($f,x=($a)+($dx)),eval($f,x=($a)+2*($dx)))],
[($a)+3*($dx),x,x=0..max(eval($f,x=($a)+2*($dx)),eval($f,x=($a)+3*($dx)))],
[$b,x,x=0..eval($f,x=$q)],
[x,eval($f,x=($a)),x=($a)..($a)+($dx)],
[x,eval($f,x=($a)+($dx)),x=($a)+($dx)..($a)+2*($dx)],
[x,eval($f,x=($a)+2*($dx)),x=($a)+2*($dx)..($a)+3*($dx)],
[x,eval($f,x=($a)+3*($dx)),x=($a)+3*($dx)..($b)]],
x=$a-1..$b+1,thickness=2, color=[black,black,black,black,black,black,red,red,red,red]),
plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.2.uid=060e698f-91cd-4b73-9d13-f9763ac799f4@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Reimann Sums;
  Sub-Topic=Finite;
@
qu.2.2.weighting=1,1,1,1@
qu.2.2.numbering=alpha@
qu.2.2.part.1.name=sro_id_1@
qu.2.2.part.1.maple_answer=(($b)-($a))/4@
qu.2.2.part.1.editing=useHTML@
qu.2.2.part.1.question=(Unset)@
qu.2.2.part.1.libname=@
qu.2.2.part.1.mode=Maple@
qu.2.2.part.1.allow2d=1@
qu.2.2.part.1.plot=@
qu.2.2.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.2.2.part.1.type=formula@
qu.2.2.part.2.name=sro_id_2@
qu.2.2.part.2.maple_answer=($a)+(($b)-($a))/4*(i-1)@
qu.2.2.part.2.editing=useHTML@
qu.2.2.part.2.question=(Unset)@
qu.2.2.part.2.libname=@
qu.2.2.part.2.mode=Maple@
qu.2.2.part.2.allow2d=1@
qu.2.2.part.2.plot=@
qu.2.2.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.2.2.part.2.type=formula@
qu.2.2.part.3.editing=useHTML@
qu.2.2.part.3.question=(Unset)@
qu.2.2.part.3.name=sro_id_3@
qu.2.2.part.3.answer=$ANSc@
qu.2.2.part.3.mode=Formula@
qu.2.2.part.4.name=sro_id_4@
qu.2.2.part.4.maple_answer=$dx*sum(eval($f,x=$a+(i-1)*$dx),i=1..4)@
qu.2.2.part.4.editing=useHTML@
qu.2.2.part.4.question=(Unset)@
qu.2.2.part.4.libname=@
qu.2.2.part.4.mode=Maple@
qu.2.2.part.4.allow2d=1@
qu.2.2.part.4.plot=@
qu.2.2.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.2.2.part.4.type=formula@
qu.2.2.question=<p>$plot</p><p>Find the Riemann Sum <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> for the function $F between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. Use four equal subintervals and left endpoints<strong> </strong>(so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msub></mrow></math>).</p><p>&nbsp;</p><p>(a) Subinterval length <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> <1></p><p>&nbsp;</p><p><span> </span></p><p><span>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><2><span> </span></span></p><p>&nbsp;</p><p><span><span>(c) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><3></span></span></p><p>&nbsp;</p><p><span><span><span> </span></span></span></p><p><span>(d) The Riemann Sum</span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><4><span> </span></p>@

qu.3.topic=Finite Right@

qu.3.1.mode=Inline@
qu.3.1.name=n=4, RtEndPt@
qu.3.1.comment=@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$c=rint(0,2);
$a=rint(-3,1);
$b=rint($a+2,$a+8,2);
$f="x^2+($c)*x+1";
$dx="(($b)-($a))/4";
$M=maple("
if eval($f,x=$a)> eval($f,x=($a)+($dx)) 
then q:=$a: else q:=(($a)+($dx)): end if:

convert(q,string),
MathML[ExportPresentation](y=$f)

");
$F=switch(1,$M);
$q=switch(0,$M);
$plot=plotmaple("plot([[x, $f, x =($a) .. ($b)], 
[$a, x, x = 0.. eval($f,x=$q)],
[($a)+($dx),x,x=0..max(eval($f,x=($a)+($dx)),eval($f,x=($a)+2*($dx)))],
[($a)+2*($dx),x,x=0..max(eval($f,x=($a)+2*($dx)),eval($f,x=($a)+3*($dx)))],
[($a)+3*($dx),x,x=0..max(eval($f,x=($a)+3*($dx)),eval($f,x=($a)+4*($dx)))],
[$b,x,x=0..eval($f,x=$b)],
[x,eval($f,x=($a)+($dx)),x=($a)..($a)+($dx)],
[x,eval($f,x=($a)+2*($dx)),x=($a)+($dx)..($a)+2*($dx)],
[x,eval($f,x=($a)+3*($dx)),x=($a)+2*($dx)..($a)+3*($dx)],
[x,eval($f,x=$b),x=($a)+3*($dx)..($b)]],
x=$a-1..$b+1,thickness=2, color=[black,black,black,black,black,black,red,red,red,red]),
plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.1.uid=41a9bfff-31d8-4d16-b879-6862768cd0df@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Topic=Reimann Sums;
  Sub-Topic=Finite;
  Course=Introduction to Calculus I;
@
qu.3.1.weighting=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=(($b)-($a))/4@
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=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.1.part.1.type=formula@
qu.3.1.part.2.name=sro_id_2@
qu.3.1.part.2.maple_answer=($a)+(($b)-($a))/4*i@
qu.3.1.part.2.editing=useHTML@
qu.3.1.part.2.question=(Unset)@
qu.3.1.part.2.libname=@
qu.3.1.part.2.mode=Maple@
qu.3.1.part.2.allow2d=1@
qu.3.1.part.2.plot=@
qu.3.1.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.1.part.2.type=formula@
qu.3.1.part.3.name=sro_id_3@
qu.3.1.part.3.maple_answer=eval($f,x=($a)+i*($dx))@
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=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.1.part.3.type=formula@
qu.3.1.part.4.name=sro_id_4@
qu.3.1.part.4.maple_answer=$dx*sum(eval($f,x=$a+i*$dx),i=1..4)@
qu.3.1.part.4.editing=useHTML@
qu.3.1.part.4.question=(Unset)@
qu.3.1.part.4.libname=@
qu.3.1.part.4.mode=Maple@
qu.3.1.part.4.allow2d=1@
qu.3.1.part.4.plot=@
qu.3.1.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.1.part.4.type=formula@
qu.3.1.question=<p>$plot</p><p>Find the Riemann Sum <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> for the function $F between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. Use four equal subintervals and right endpoints<strong> </strong>(so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></math>).</p><p>(a) Subinterval length <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> <1></p><p>&nbsp;</p><p><span> </span></p><p><span>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><2></span></p><p>&nbsp;</p><p><span><span> </span></span></p><p><span><span>(c) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><3><span> </span></span></span></p><p><br /><span>(d) The Riemann Sum</span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><4><span> </span></p>@

qu.3.2.mode=Inline@
qu.3.2.name=n=3, RtEndPt@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$c=rint(0,2);
$a=rint(-3,1);
$b=rint($a+3,$a+10,3);
$f="x^2+($c)*x+1";
$dx="(($b)-($a))/3";
$M=maple("
if eval($f,x=$a)> eval($f,x=($a)+($dx)) 
then q:=$a: else q:=(($a)+($dx)): end if:

convert(q,string),
MathML[ExportPresentation](y=$f)
");
$F=switch(1,$M);
$q=switch(0,$M);
$plot=plotmaple("plot([[x, $f, x =($a) .. ($b)], 
[$a, x, x = 0 ..eval($f,x=$q)],
[($a)+($dx),x,x=0..max(eval($f,x=($a)+($dx)),eval($f,x=($a)+2*($dx)))],
[($a)+2*($dx),x,x=0..max(eval($f,x=($a)+2*($dx)),eval($f,x=($a)+3*($dx)))],
[$b,x,x=0..eval($f,x=$b)],
[x,eval($f,x=($a)+($dx)),x=($a)..($a)+($dx)],
[x,eval($f,x=($a)+2*($dx)),x=($a)+($dx)..($a)+2*($dx)],
[x,eval($f,x=($a)+3*($dx)),x=($a)+2*($dx)..($b)]],
x=$a-1..$b+1,thickness=2, color=[black,black,black,black,black,red,red,red]),
plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.2.uid=b37ceaa5-5c83-4ec5-aeb3-c6156c04d08a@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Reimann Sums;
  Sub-Topic=Finite;
@
qu.3.2.weighting=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=(($b)-($a))/3@
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=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.2.part.1.type=formula@
qu.3.2.part.2.name=sro_id_2@
qu.3.2.part.2.maple_answer=($a)+(($b)-($a))/3*i@
qu.3.2.part.2.editing=useHTML@
qu.3.2.part.2.question=(Unset)@
qu.3.2.part.2.libname=@
qu.3.2.part.2.mode=Maple@
qu.3.2.part.2.allow2d=1@
qu.3.2.part.2.plot=@
qu.3.2.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.2.part.2.type=formula@
qu.3.2.part.3.name=sro_id_3@
qu.3.2.part.3.maple_answer=eval($f,x=($a)+i*($dx))@
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=is(simplify(($ANSWER)-($RESPONSE)) = 0);@
qu.3.2.part.3.type=formula@
qu.3.2.part.4.name=sro_id_4@
qu.3.2.part.4.maple_answer=$dx*sum(eval($f,x=$a+i*$dx),i=1..3)@
qu.3.2.part.4.editing=useHTML@
qu.3.2.part.4.question=(Unset)@
qu.3.2.part.4.libname=@
qu.3.2.part.4.mode=Maple@
qu.3.2.part.4.allow2d=1@
qu.3.2.part.4.plot=@
qu.3.2.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.2.part.4.type=formula@
qu.3.2.question=<p>$plot</p><p>Find the Riemann Sum <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> for the function $F between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. Use three equal subintervals and right endpoints<strong> </strong>(so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></math>).</p><p>&nbsp;</p><p>(a) Subinterval length <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math> <1><span> </span></p><p>&nbsp;</p><p><span>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><2><span> </span></span></p><p>&nbsp;</p><p><span><span>(c) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><3></span></span></p><p>&nbsp;</p><p><span><span><span> </span></span></span></p><p><span>(d) The Riemann Sum</span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&Delta;</mi></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math><span> </span><4><span> </span></p>@

