qu.1.topic=Curve Sketching@

qu.1.1.mode=Inline@
qu.1.1.name=Critical Numbers (undefined)@
qu.1.1.comment=<p>The derivative of the function is zero at <em>x = </em>$ANS, hence $ANS is a critical number.</p>
<p>The function has an endpoint at <em>x = $c, </em>hence $c is a critical number.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$c=rint(2,10);
$func="x*sqrt($c-x)";
$m = maple("
MathML[ExportPresentation]($func),
convert(2/3*$c, string)
");
$disp = switch(0,$m);
$ANS = switch(1,$m);@
qu.1.1.uid=45daf35d-ddef-4022-96c0-b1de4b8d8102@
qu.1.1.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Curve Sketching;
  Sub-Topic=Critical Numbers;
@
qu.1.1.weighting=1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.maple_answer=$c@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.libname=@
qu.1.1.part.1.mode=Maple@
qu.1.1.part.1.allow2d=1@
qu.1.1.part.1.plot=@
qu.1.1.part.1.maple=grade:=0;
if evalb(simplify($ANSWER - $RESPONSE)=0) then
grade := 1:
elif evalb(simplify($ANS - $RESPONSE)=0) then
grade := 1:
end if; 
grade @
qu.1.1.part.1.type=formula@
qu.1.1.part.2.name=sro_id_2@
qu.1.1.part.2.maple_answer=$ANS@
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=grade:=0;
if evalb(simplify($ANSWER - $RESPONSE)=0) then
grade := 1:
elif evalb(simplify($c - $RESPONSE)=0) then
grade := 1:
end if; 
grade @
qu.1.1.part.2.type=formula@
qu.1.1.question=<p>The critical numbers of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp are?</p><p>If you find just one critical number, leave the second box blank.</p><p><1><span>&nbsp;</span></p><p><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.2.mode=Inline@
qu.1.2.name=Local Extrema@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(5);
$func=switch($a,"x^3-3*x^2","9*x^2-x^3","4*x^3-2*x^2+4","-2*x^3-4*x^2","-x^3+4*x^2");
$disp=maple("MathML[ExportPresentation]($func)");
$ANSmax=switch($a,'0','6','0','0','8/3');
$ANSmin=switch($a,'2','0','1/3','-8/6','0');@
qu.1.2.uid=416548c5-8b68-4894-82e1-90b63eb4f531@
qu.1.2.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Curve Sketching;
  Sub-Topic=Local max/min;
@
qu.1.2.weighting=1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.maple_answer=$ANSmax@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.libname=@
qu.1.2.part.1.mode=Maple@
qu.1.2.part.1.allow2d=1@
qu.1.2.part.1.plot=@
qu.1.2.part.1.maple=is(($ANSWER)-($RESPONSE) =0);@
qu.1.2.part.1.type=formula@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.maple_answer=$ANSmin@
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.question=<p>Find all local extrema for the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp.</p><p><br />local max at x =&nbsp; <1><span>&nbsp;</span></p><p>&nbsp;</p><p>local min at x = &nbsp; <2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.3.mode=Inline@
qu.1.3.name=Increasing Functions@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=rint(4);
$func=switch($a,"3*x^4-12*x","2*x*exp(-2*x)","x*exp(-(1/4)*x)","(1/3)*x^3+(5/2)*x^2+6*x+4");
$disp=maple("MathML[ExportPresentation]($func)");
$ANS=switch($a,'"[1,infinity)"','"(-infinity,1/2]"','"(-infinity,4]"','"(-infinity,-3]U[-2,infinity)"');@
qu.1.3.uid=f9a2995e-e5fa-4594-8967-78f413819f60@
qu.1.3.info=  Author=Steve Crane, Gord Clement;
  Topic=Curve Sketching;
  Course=Introductory Calculus for the Biological Sciences;
  Sub-Topic=Intervals of Increase/Decrease;
@
qu.1.3.weighting=1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.maple_answer=show($ANS)@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.1.3.part.1.mode=Maple@
qu.1.3.part.1.allow2d=0@
qu.1.3.part.1.plot=@
qu.1.3.part.1.maple=grade("$RESPONSE",$ANS)@
qu.1.3.part.1.type=maple@
qu.1.3.question=<p>State the interval(s)&nbsp;for which&nbsp;the function&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp is increasing?</p><p>Enter your answers&nbsp;using interval notation.</p><p>Use 'U' for union and 'infinity' for infinity.</p><p>Note:&nbsp; Increasing is not equivalent to <em>f ' > 0. </em></p><p>For instance, <em>x<sup>2</sup> </em>has positive derivative&nbsp;on (0, infinity)&nbsp;but is increasing on [0, infinity).</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Decreasing Functions@
qu.1.4.comment=@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=rint(3);
$func=switch($a,"2*x^2*exp(x)","6*x^4-4*x^3+2","10*x*exp(-x)");
$disp=maple("MathML[ExportPresentation]($func)");
$ANS=switch($a,'"[-2,0]"','"(-infinity,1/2]"','"[1,infinity)"');@
qu.1.4.uid=ee22cb32-1cb3-43a0-bc09-d6970539d083@
qu.1.4.info=  Author=Steve Crane, Gord Clement;
  Topic=Curve Sketching;
  Sub-Topic=Intervals of increase/decrease;
  Course=Introductory Calculus for the Biological Sciences;
@
qu.1.4.weighting=1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.maple_answer=show($ANS)@
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.1.4.part.1.mode=Maple@
qu.1.4.part.1.allow2d=0@
qu.1.4.part.1.plot=@
qu.1.4.part.1.maple=grade("$RESPONSE",$ANS)@
qu.1.4.part.1.type=maple@
qu.1.4.question=<p>State&nbsp;interval(s) for which the function&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp is decreasing?</p><p><br />Enter your answers using interval notation.</p><p>Use 'U' for union and 'infinity' for infinity.</p><p>Note:&nbsp; Decreasing is not equivalent to f '&nbsp;< 0.</p><p>For instance,&nbsp;<em>x<sup>2</sup></em> has negative&nbsp;derivative on (-infinity, 0) but is decreasing on (-infinity, 0].</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Inflection points (x^2)@
qu.1.5.comment=@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$a=rint(1,4);
$b=rint(1,5);
condition: ne($a,$b);
$c=rint(6,9);
$func="-$a*x^5+$b*x^4-$c*x";
$m = maple("
MathML[ExportPresentation]($func),
convert(3/5*$b/$a,string)
");
$disp = switch(0,$m);
$ANS = switch(1,$m);@
qu.1.5.uid=174ed3f3-fed6-4478-a709-525eb473db81@
qu.1.5.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Curve Sketching;
  Sub-Topic=Points of Inflection;
@
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=$ANS@
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.libname=@
qu.1.5.part.1.mode=Maple@
qu.1.5.part.1.allow2d=1@
qu.1.5.part.1.plot=@
qu.1.5.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.5.part.1.type=formula@
qu.1.5.question=<p>The point(s) of inflection for the curve <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp occur at which <em>x</em>-values.</p><p>If you find more than one <em>x-</em>value, enter them separated by commas.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.1.6.mode=Inline@
qu.1.6.name=Amount of Rain (max)@
qu.1.6.comment=@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$a=rint(2,9);
$func="$a*t^2-t^3";
$m = maple("
MathML[ExportPresentation]($func),
convert(2*$a/3,string)
");
$disp=switch(0,$m);
$ANS=switch(1,$m);@
qu.1.6.uid=cf5f03bd-e817-4ce1-b3d8-8bec8dcf0bf7@
qu.1.6.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Curve Sketching;
  Sub-Topic=Local max/min;
@
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=$ANS@
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.libname=@
qu.1.6.part.1.mode=Maple@
qu.1.6.part.1.allow2d=1@
qu.1.6.part.1.plot=@
qu.1.6.part.1.maple=is(abs(($ANSWER)-($RESPONSE)) < 0.1);@
qu.1.6.part.1.type=formula@
qu.1.6.question=<p>Assume the amount of rainfall <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced></mrow></mstyle></math> (in cm) in Guelph during the 6 month rainy season is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$disp 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><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mn>6</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>(t in months)</p><p>When is the amount of rainfall the greatest during the 6 month rainy season?</p><p>Do not round your answer (i.e.,&nbsp;If you find <em>t = 4/3,</em> enter <em>4/3 </em>not 1 or 2).</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Inflection points@
qu.1.7.comment=<p>Inflection points occur where the second derivative changes sign.&nbsp;For this function, this occurs&nbsp;at <em>x = $ANS</em> and at <em>x = </em>0.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$a=rint(2,13);
$b=rint(2,14);
$c=rint(5,13);
condition: ne($a,$b);
$func="-$a*x^4+$b*x^3-$c";
$m = maple("
MathML[ExportPresentation]($func),
convert($b/(2*$a),string)
");
$disp=switch(0,$m);
$ANS=switch(1,$m);@
qu.1.7.uid=17172032-0225-45c4-b3d8-f282c2600fa7@
qu.1.7.info=  Author=Steve Crane, Gord Clement;
  Topic=Curve Sketching;
  Sub-Topic=Points of Inflection;
  Course=Introductory Calculus for the Biological Sciences;
@
qu.1.7.weighting=1,1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.maple_answer=$ANS@
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.libname=@
qu.1.7.part.1.mode=Maple@
qu.1.7.part.1.allow2d=1@
qu.1.7.part.1.plot=@
qu.1.7.part.1.maple=grade := 0; 
if evalb(simplify($ANS-$RESPONSE) = 0) then grade := 1
elif evalb(simplify(0-$RESPONSE) = 0) then grade := 1 end if; grade@
qu.1.7.part.1.type=formula@
qu.1.7.part.2.name=sro_id_2@
qu.1.7.part.2.maple_answer=0@
qu.1.7.part.2.editing=useHTML@
qu.1.7.part.2.question=(Unset)@
qu.1.7.part.2.libname=@
qu.1.7.part.2.mode=Maple@
qu.1.7.part.2.allow2d=1@
qu.1.7.part.2.plot=@
qu.1.7.part.2.maple=grade := 0; 
if evalb(simplify($ANS-$RESPONSE) = 0) then grade := 1
elif evalb(simplify(0-$RESPONSE) = 0) then grade := 1 end if; grade@
qu.1.7.part.2.type=formula@
qu.1.7.question=<p>The <em>x-</em>values of the inflection points&nbsp;for the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp are?</p><p>If you find just one inflection point, leave the second box blank.</p><p><br /><1><span>&nbsp;</span></p><p><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.8.mode=Inline@
qu.1.8.name=Critical Numbers@
qu.1.8.comment=<p>The derivative of this function is zero at <em>x = $ANS </em>and <em>x = 0.</em></p>@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$a=rint(2,9);
$b=rint(2,9);
$c=rint(5,13);
condition: ne($a,$b);
$func="$a*x^3-$b*x^2+$c";
$m = maple("
MathML[ExportPresentation]($func),
convert(2/3*$b/$a,string)
");
$disp=switch(0,$m);
$ANS = switch(1,$m);@
qu.1.8.uid=81564477-fc91-4a26-aefb-160be4e72b79@
qu.1.8.info=  Author=Steve Crane, Gord Clement;
  Topic=Curve Sketching;
  Sub-Topic=Critical Numbers;
  Course=Introductory Calculus for the Biological Sciences;
@
qu.1.8.weighting=1,1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.maple_answer=$ANS@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.question=(Unset)@
qu.1.8.part.1.libname=@
qu.1.8.part.1.mode=Maple@
qu.1.8.part.1.allow2d=1@
qu.1.8.part.1.plot=@
qu.1.8.part.1.maple=grade := 0;
 if evalb(simplify($ANSWER-$RESPONSE) = 0) then 
grade := 1 
elif evalb(simplify(0-$RESPONSE) = 0) then 
grade := 1 end if;
grade@
qu.1.8.part.1.type=formula@
qu.1.8.part.2.name=sro_id_2@
qu.1.8.part.2.maple_answer=0@
qu.1.8.part.2.editing=useHTML@
qu.1.8.part.2.question=(Unset)@
qu.1.8.part.2.libname=@
qu.1.8.part.2.mode=Maple@
qu.1.8.part.2.allow2d=1@
qu.1.8.part.2.plot=@
qu.1.8.part.2.maple=grade := 0;
 if evalb(simplify($ANSWER-$RESPONSE) = 0) then 
grade := 1 
elif evalb(simplify($ANS-$RESPONSE) = 0) then 
grade := 1 end if;
grade@
qu.1.8.part.2.type=formula@
qu.1.8.question=<p>The critical numbers of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp are?</p><p>If you find just one critical number, leave the second box blank.</p><p><1><span>&nbsp;</span></p><p><2><span>&nbsp;</span></p><p>&nbsp;</p>@

