qu.1.topic=Derivatives@

qu.1.1.mode=Inline@
qu.1.1.name=Derivative (constant^function)@
qu.1.1.comment=<p>Chain Rule: if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>a</mi><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup></mrow></mstyle></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>a</mi><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(2,9);
$b=rint(2,9);
condition:ne($a,$b);
$f="($a)^(x^($b))";
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.1.uid=0e20b87d-4357-4586-945c-849e44bf517d@
qu.1.1.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Chain Rule;
@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.maple_answer=diff(($f),x)@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.libname=@
qu.1.1.part.1.mode=Maple@
qu.1.1.part.1.allow2d=1@
qu.1.1.part.1.plot=@
qu.1.1.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.1.part.1.type=formula@
qu.1.1.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F then&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>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> is:</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Quotient Rule Derivative@
qu.1.2.comment=<p><em><font size="3">If a funciton is of the form <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>h</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math> then the derivative is <img style="width: 292px; height: 50px" align="middle" width="321" height="68" alt="" src="__BASE_URI__Pictures/quotientrule.png" />.</font></em></p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(1,9);
$b=rint(-8,6);
$c=rint(-5,10);
$d=rint(1,11);
$e=rint(-8,5);
$h=rint(-7,6);
condition:ne($a,$d);
$f=switch(rint(9),"($a)*x+($b)","($a)*x^2+($b)*x+($c)","($a)*x^3+($b)*x+($c)","($d)*x^3+($b)*x^2+($a)");
$g=switch(rint(9),"($d)*x+($e)","($d)*x^2+($e)*x+($h)","($d)*x^3+($e)*x+($h)");

$Func=maple("printf(MathML[ExportPresentation](($f)/($g)))");
$ANS=maple("diff(($f)/($g),x)");@
qu.1.2.uid=cbe98d4b-5e6f-42e9-bc0a-aa262648911c@
qu.1.2.weighting=1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.maple_answer=$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=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.2.part.1.type=formula@
qu.1.2.question=<p>Using the quotient rule, find the derivative of the following 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>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Func</p><p>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.3.mode=Inline@
qu.1.3.name=Implicit Derivative@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=rint(2,9);
$b=rint(2,7);
$c=rint(2,4);
$f=switch(rint(4),"($a)*x^3+x*y^2+($b)*y=$y^3","sin(($a)*x^2*y)=x+y^2",
"cos(($a)*x*y)=x+y","x^4*y^3+x^2=($a)*x-y");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.3.uid=ba50df7c-0188-4dbe-9787-0fb70a0e13e4@
qu.1.3.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Implicit Differentiation;
@
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=implicitdiff($f,y,x)@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.libname=@
qu.1.3.part.1.mode=Maple@
qu.1.3.part.1.allow2d=1@
qu.1.3.part.1.plot=@
qu.1.3.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.3.part.1.type=formula@
qu.1.3.question=<p>Use implicit differentiation to derive the following:</p><p>$F</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Third Derivative - Trig ii@
qu.1.4.comment=@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=rint(2,8);
$func1=switch(rint(2),"$a*x","x^2");
$func2=switch(rint(2),"cos(x)","sin(x)");
$func="($func1)*($func2)";
$disp=maple("MathML[ExportPresentation]($func)");@
qu.1.4.uid=1e7da7d3-86fc-460c-9274-1908ca60df3a@
qu.1.4.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivative;
  Sub-Topic=Third Derivative;
@
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=diff($func,[x,x,x])@
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.libname=@
qu.1.4.part.1.mode=Maple@
qu.1.4.part.1.allow2d=1@
qu.1.4.part.1.plot=@
qu.1.4.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.4.part.1.type=formula@
qu.1.4.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> (the <strong>third</strong> derivative of <em>f</em> with respect to <em>x</em>) is</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Combination of Rules - Derivatives@
qu.1.5.comment=<p><em><font size="3">Chain Rule: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math> then the derivative is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>. </font></em></p>
<p><em><font size="3">Quotient&nbsp;Rule: If <img style="width: 116px; height: 45px" align="middle" width="142" height="61" alt="" src="__BASE_URI__Pictures/goverh.png" />&nbsp; then the derivative is <img align="middle" width="321" height="68" alt="" src="__BASE_URI__Pictures/quotientrule.png" />&nbsp;.<br />
</font></em></p>@
qu.1.5.editing=useHTML@
qu.1.5.hint.1=You will need to use two derivative rules for this question.@
qu.1.5.hint.2=You will need the chain rule and the product rule.@
qu.1.5.solution=@
qu.1.5.algorithm=$a=rint(2,9);
$b=rint(-8,6);
$c=rint(-5,10);
$d=rint(1,11);
$e=rint(4,11);
$i=rint(-3,6);
$j=rint(-10,10);
condition:ne($a,$d);
$f=switch(rint(9),"($a)*x+($b)","($a)*x^2+($b)*x+($c)","($a)*x^3+($b)*x+($c)","($d)*x^3+($b)*x^2+($a)");
$g=switch(rint(9),"($a)*x+($e)","($b)*x^2+($j)*x+($i)","($i)*x^2+($e)*x+($b)","($i)*x^3+($a)*x^2+($j)");
$h="($f)^($e)";
$Func=maple("printf(MathML[ExportPresentation](($h)/($g)))");@
qu.1.5.uid=784287df-db33-4638-a6e5-152683e32d9e@
qu.1.5.weighting=1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.maple_answer=diff((($g)*($h)),x)@
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.libname=@
qu.1.5.part.1.mode=Maple@
qu.1.5.part.1.allow2d=1@
qu.1.5.part.1.plot=@
qu.1.5.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.5.part.1.type=formula@
qu.1.5.question=<p>Find the derivative of the following function:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Func</p><p>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Combination of Rules - Derivatives ii@
qu.1.6.comment=<p>Chain Rule: if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>a</mi><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup></mrow></mstyle></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>a</mi><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.1.6.editing=useHTML@
qu.1.6.hint.1=You will need to use two different derivative rules for this question.@
qu.1.6.hint.2=You will need the chain rule and the product rule.@
qu.1.6.solution=@
qu.1.6.algorithm=$a=rint(2,9);
$b=rint(2,9);
$c=rint(2,10);
condition:ne($a,$b);
$f="($a)^(x^($b))";
$g=switch(rint(2),"($c*x+1)^(1/2)","($c*x+1)^(3/2)");
$F=maple("printf(MathML[ExportPresentation](($f)*($g)))");@
qu.1.6.uid=66c0add7-119f-410c-89f0-b155a9e57cf6@
qu.1.6.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Combination of Rules;
@
qu.1.6.weighting=1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.maple_answer=diff((($f)*($g)),x)@
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.libname=@
qu.1.6.part.1.mode=Maple@
qu.1.6.part.1.allow2d=1@
qu.1.6.part.1.plot=@
qu.1.6.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.6.part.1.type=formula@
qu.1.6.question=<p>If <em>f(x) = </em>$F then&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>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> is:</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Second Derivative (e^trig)@
qu.1.7.comment=@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$func1=switch(rint(4),"cos(x)","sin(x)","-cos(x)","-sin(x)");
$realfunc="exp($func1)";
$disp=maple("MathML[ExportPresentation]($realfunc)");@
qu.1.7.uid=1eebc117-892f-42a2-b8d4-02b99ccaea78@
qu.1.7.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Second Derivative;
@
qu.1.7.weighting=1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.maple_answer=diff($realfunc,[x,x])@
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.libname=@
qu.1.7.part.1.mode=Maple@
qu.1.7.part.1.allow2d=1@
qu.1.7.part.1.plot=@
qu.1.7.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.7.part.1.type=formula@
qu.1.7.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><msup><mi>dx</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>equals</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Third Derivative - Trig@
qu.1.8.comment=<p>Chain Rule: if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$a=rint(2,6);
$b=rint(2,4);
$f=switch(rint(3),"$a*sin($b*x)","-$a*cos($b*x)");
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.8.uid=1d2d8064-2632-4a08-92fb-7e844f278ad4@
qu.1.8.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Third Derivative;
@
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=diff(($f),[x,x,x])@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.question=(Unset)@
qu.1.8.part.1.libname=@
qu.1.8.part.1.mode=Maple@
qu.1.8.part.1.allow2d=1@
qu.1.8.part.1.plot=@
qu.1.8.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.8.part.1.type=formula@
qu.1.8.question=<p>If <em>f(x)= </em>$F then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> (the<strong> third</strong> derivative of <em>f</em> with respect to <em>x</em>) is:</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.9.mode=Inline@
qu.1.9.name=Logarithmic Differentiation@
qu.1.9.comment=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$disp, we use logarithmic differentiation to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo></mrow></mstyle></math>.&nbsp;</p>
<p>Start by taking the <em>ln</em> of both sides.</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$func1=switch(rint(2),"sin(x)","cos(x)");
$a=rint(2,7);
$func2="exp($a*x)";
$func="($func1)^($func2)";
$disp=maple("MathML[ExportPresentation]($func)");@
qu.1.9.uid=0774147f-c47f-4753-9c3a-5612b9fffd94@
qu.1.9.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Log differentiation;
@
qu.1.9.weighting=1@
qu.1.9.numbering=alpha@
qu.1.9.part.1.name=sro_id_1@
qu.1.9.part.1.maple_answer=diff($func,x)@
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.question=(Unset)@
qu.1.9.part.1.libname=@
qu.1.9.part.1.mode=Maple@
qu.1.9.part.1.allow2d=1@
qu.1.9.part.1.plot=@
qu.1.9.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.9.part.1.type=formula@
qu.1.9.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$disp then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></mstyle></math>equals</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.10.mode=Inline@
qu.1.10.name=Equation of a Tangent Line@
qu.1.10.comment=@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$a=['5','10','15','20'];
$picka=switch(rint(4),$a);
$b=rint(1,6);
$func="$x->$picka*ln(x)";
$fd="$picka*ln(x)";
$pd="$picka*ln($b)";
$m = maple("
deriv := diff(($func)(x),x):
slope := subs(x=$b, deriv):
y1 := $picka*ln($b)-$b:
MathML[ExportPresentation]($fd),
MathML[ExportPresentation]($pd),
convert(slope*x + y1,string)
");
$disp=switch(0,$m);
$dispd=switch(1,$m);
$ANS=switch(2, $m);@
qu.1.10.uid=a1864005-5c20-4cf5-b5d1-5f74d7a4e29b@
qu.1.10.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Equation of Tangent Line;
@
qu.1.10.weighting=1@
qu.1.10.numbering=alpha@
qu.1.10.part.1.name=sro_id_1@
qu.1.10.part.1.maple_answer=$ANS@
qu.1.10.part.1.editing=useHTML@
qu.1.10.part.1.question=(Unset)@
qu.1.10.part.1.libname=@
qu.1.10.part.1.mode=Maple@
qu.1.10.part.1.allow2d=1@
qu.1.10.part.1.plot=@
qu.1.10.part.1.maple=is(abs(($ANSWER)-($RESPONSE)) < 0.1);@
qu.1.10.part.1.type=formula@
qu.1.10.question=<p>Find the&nbsp;equation of the tangent line to&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>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp at the point&nbsp;<font size="2"><em>($b,</em></font>$dispd)?</p><p>&nbsp;<em>y = </em><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.11.mode=Inline@
qu.1.11.name=Second Derivative (constant^function)@
qu.1.11.comment=<p>Chain Rule: if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>a</mi><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup></mrow></mstyle></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>a</mi><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=$a=rint(2,5);
$b=rint(2,5);
condition:ne($a,$b);
$f="($a)^(x^($b))";
$F=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.11.uid=35259c6e-6fb6-4782-b39c-1a8c0cb7006a@
qu.1.11.info=  Author=Steve Crane, Gord Clement;
  Topic=Derivatives;
  Sub-Topic=Second Derivative;
  Course=Introductory Calculus for the Biological Sciences;
@
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=diff($f, [x,x])@
qu.1.11.part.1.editing=useHTML@
qu.1.11.part.1.question=(Unset)@
qu.1.11.part.1.libname=@
qu.1.11.part.1.mode=Maple@
qu.1.11.part.1.allow2d=1@
qu.1.11.part.1.plot=@
qu.1.11.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.11.part.1.type=formula@
qu.1.11.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F then&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>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> (the second derivative with respect to x) is:</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.12.mode=Inline@
qu.1.12.name=Product Rule Derivative@
qu.1.12.comment=<p><em><font size="3">If a funciton is of the form <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>h</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> then the derivative is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>h</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>h</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>.</font></em></p>@
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$a=rint(1,9);
$b=rint(-8,6);
$c=rint(-5,10);
$d=rint(1,11);
$e=rint(-8,5);
$h=rint(-7,6);
condition:ne($a,$d);
$f=switch(rint(9),"($a)*x+($b)","($a)*x^2+($b)*x+($c)","($a)*x^3+($b)*x+($c)");
$g=switch(rint(9),"($d)*x+($e)","($d)*x^2+($e)*x+($h)","($d)*x^3+($e)*x+($h)");
$F=maple("printf(MathML[ExportPresentation]($f))");
$G=maple("printf(MathML[ExportPresentation]($g))");@
qu.1.12.uid=1de738a1-35f9-4861-b74d-44a2a569b984@
qu.1.12.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Product Rule;
@
qu.1.12.weighting=1@
qu.1.12.numbering=alpha@
qu.1.12.part.1.name=sro_id_1@
qu.1.12.part.1.maple_answer=diff(($f)*($g),x)@
qu.1.12.part.1.editing=useHTML@
qu.1.12.part.1.question=(Unset)@
qu.1.12.part.1.libname=@
qu.1.12.part.1.mode=Maple@
qu.1.12.part.1.allow2d=1@
qu.1.12.part.1.plot=@
qu.1.12.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.12.part.1.type=formula@
qu.1.12.question=<p>Using the product rule, what is the derivative of the following 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>($F)($G)</p><p>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.13.mode=Inline@
qu.1.13.name=Product Rule Derivative ii@
qu.1.13.comment=<p>This question could be evaluated by using the product rule, or you could expand first and be left with a basic derivative.</p>
<p>Recall that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>b</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>b</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><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>.</p>@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$a=rint(1,6);
$b=rint(2,4);
condition: ne($a,$b);
$g="(x^($b)+$a)*(x^($b)-$a)";
$m = maple("
MathML[ExportPresentation]($g),
convert(diff($g,x),string)
");
$df=switch(0,$m);
$ANS=switch(1,$m);@
qu.1.13.uid=f6df5852-fd5f-48dd-b441-7e6a697dd4d8@
qu.1.13.weighting=1@
qu.1.13.numbering=alpha@
qu.1.13.part.1.name=sro_id_1@
qu.1.13.part.1.maple_answer=$ANS@
qu.1.13.part.1.editing=useHTML@
qu.1.13.part.1.question=(Unset)@
qu.1.13.part.1.libname=@
qu.1.13.part.1.mode=Maple@
qu.1.13.part.1.allow2d=1@
qu.1.13.part.1.plot=@
qu.1.13.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.13.part.1.type=formula@
qu.1.13.question=<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$df</p><p>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>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> .</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.14.mode=Inline@
qu.1.14.name=Chain Rule Derivative@
qu.1.14.comment=<p><em><font size="3">If a funciton is of the form <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math> then the derivative is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>.</font></em></p>@
qu.1.14.editing=useHTML@
qu.1.14.solution=@
qu.1.14.algorithm=$a=rint(1,9);
$b=rint(-8,6);
$c=rint(-5,10);
$d=rint(1,11);
$e=rint(4,11);
condition:ne($a,$d);
$f=switch(rint(9),"($a)*x+($b)","($a)*x^2+($b)*x+($c)","($a)*x^3+($b)*x+($c)","($d)*x^3+($b)*x^2+($a)");
$g="($f)^($e)";
$Func=maple("printf(MathML[ExportPresentation]($g))");@
qu.1.14.uid=374ccf17-7b83-49b9-a507-f843ea8f8e8b@
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=diff(($g),x)@
qu.1.14.part.1.editing=useHTML@
qu.1.14.part.1.question=(Unset)@
qu.1.14.part.1.libname=@
qu.1.14.part.1.mode=Maple@
qu.1.14.part.1.allow2d=1@
qu.1.14.part.1.plot=@
qu.1.14.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.14.part.1.type=formula@
qu.1.14.question=<p>Using the chain rule, find the derivative of the following function: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Func</p><p>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.15.mode=Inline@
qu.1.15.name=Power Rule Derivative@
qu.1.15.comment=<p>The power rule says that the derivative 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><mi>c</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math>is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>nc</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$a=range(2,10);
$c=range(3,9);
$d=range(2,14);
$e=range(-12,-4);
$f=range(-9,-2);
$g=range(-8,-4);
condition: ne($a,$c),ne($c,$d),ne($e,$f),ne($f,$g),ne($e,$g),ne($a,$d);
$funcf="$a*x^$d";
$funcg="$d*x^$c";
$funch="$f*x^($g)";
$funcj="$c*x^($f)";
$funck="$e*x^$c";
$f1=switch(rint(5),"$funcf","$funcg","$funch","$funcj","$funck");
$f2=switch(rint(5),"$funcf","$funcg","$funch","$funcj","$funck");
condition:ne($f1,$f2);
$m = maple("
MathML[ExportPresentation]($f1),
MathML[ExportPresentation]($f2)
");
$df=switch(0,$m);
$dh=switch(1,$m);@
qu.1.15.uid=de30a4ab-1300-46ef-a0ef-e40cb4274b07@
qu.1.15.weighting=1,1@
qu.1.15.numbering=alpha@
qu.1.15.part.1.name=sro_id_1@
qu.1.15.part.1.maple_answer=diff($f1,x)@
qu.1.15.part.1.editing=useHTML@
qu.1.15.part.1.question=(Unset)@
qu.1.15.part.1.libname=@
qu.1.15.part.1.mode=Maple@
qu.1.15.part.1.allow2d=1@
qu.1.15.part.1.plot=@
qu.1.15.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.15.part.1.type=formula@
qu.1.15.part.2.name=sro_id_2@
qu.1.15.part.2.maple_answer=diff($f2,x)@
qu.1.15.part.2.editing=useHTML@
qu.1.15.part.2.question=(Unset)@
qu.1.15.part.2.libname=@
qu.1.15.part.2.mode=Maple@
qu.1.15.part.2.allow2d=1@
qu.1.15.part.2.plot=@
qu.1.15.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.15.part.2.type=formula@
qu.1.15.question=<p>Using the power rule for derivatives, find the derivatives of the following functions.</p><p>&nbsp;i) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$df</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>ii)&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$dh</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p><span>&nbsp;</span><span><br /></span></p>@

qu.1.16.mode=Inline@
qu.1.16.name=Combination of Rules - Derivatives@
qu.1.16.comment=<p><em><font size="3">Chain Rule: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math> then the derivative is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>. </font></em></p>
<p><em><font size="3">Product Rule: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>h</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> then the derivative is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>h</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>h</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>.<br />
</font></em></p>@
qu.1.16.editing=useHTML@
qu.1.16.hint.1=You will need to use two different rules.@
qu.1.16.hint.2=You will need the chain rule and the product rule.@
qu.1.16.solution=@
qu.1.16.algorithm=$a=rint(1,9);
$b=rint(-8,6);
$c=rint(-5,10);
$d=rint(1,11);
$e=rint(4,11);
$i=rint(-3,6);
$j=rint(-10,10);
condition:ne($a,$d);
$f=switch(rint(9),"($a)*x+($b)","($a)*x^2+($b)*x+($c)","($a)*x^3+($b)*x+($c)","($d)*x^3+($b)*x^2+($a)");
$g=switch(rint(9),"($a)*x+($e)","($b)*x^2+($j)*x+($i)","($i)*x^3+($e)*x+($b)","($i)*x^3+($a)*x^2+($j)");
$h="($f)^($e)";
$Func=maple("printf(MathML[ExportPresentation](($g)*($h)))");@
qu.1.16.uid=fbe4fe70-c87b-43ca-830e-8ffd984e995a@
qu.1.16.weighting=1@
qu.1.16.numbering=alpha@
qu.1.16.part.1.name=sro_id_1@
qu.1.16.part.1.maple_answer=diff((($g)*($h)),x)@
qu.1.16.part.1.editing=useHTML@
qu.1.16.part.1.question=(Unset)@
qu.1.16.part.1.libname=@
qu.1.16.part.1.mode=Maple@
qu.1.16.part.1.allow2d=1@
qu.1.16.part.1.plot=@
qu.1.16.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.16.part.1.type=formula@
qu.1.16.question=<p>Find the derivative of the following function:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Func</p><p>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.17.mode=Inline@
qu.1.17.name=Cell Culture (Derivatives)@
qu.1.17.comment=<p>To find the instantaneous rate of change, take the derivative.</p>@
qu.1.17.editing=useHTML@
qu.1.17.solution=@
qu.1.17.algorithm=$a=rint(2,7);
$b=rint(1,3);
$func="t->$b*t+($a)/t";
$funcd="$b*t+($a)/t";
$m = maple("
MathML[ExportPresentation]($funcd),
convert(diff(($func)(t),t),string)
");
$disp=switch(0,$m);
$ANS=switch(1,$m);@
qu.1.17.uid=7541d727-7579-4fa0-a957-cbe14b515e59@
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=$ANS@
qu.1.17.part.1.editing=useHTML@
qu.1.17.part.1.question=(Unset)@
qu.1.17.part.1.libname=@
qu.1.17.part.1.mode=Maple@
qu.1.17.part.1.allow2d=1@
qu.1.17.part.1.plot=@
qu.1.17.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.17.part.1.type=formula@
qu.1.17.question=<p>The size of a cell culture at time <em>t</em> is given by the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$disp, <em>t > 0.1</em>.</p><p>Find the function that&nbsp;gives the instantaneous rate of change of the size of the cell culture&nbsp;for time&nbsp;<em>t > 0.1.</em></p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span>&nbsp;</span><span><br /></span></p>@

qu.1.18.mode=Inline@
qu.1.18.name=Derivative - Finding f'(c)@
qu.1.18.comment=<p><em><font size="3">If a funciton is of the form <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>h</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> then the derivative is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>h</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>h</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>.</font></em></p>@
qu.1.18.editing=useHTML@
qu.1.18.solution=@
qu.1.18.algorithm=$a=switch(rint(4),1,3);
$b=rint(2,10);
$c=rint(2,10);
$d=switch(rint(2),4,9);
condition:ne($b,$c);
$f="x^($a/2)";
$g="($b*x-$c)";
$df=maple("MathML[ExportPresentation](($f)*($g))");@
qu.1.18.uid=c3b7e0ef-8979-4b9f-a226-357c35e23f41@
qu.1.18.info=  Author=Steve Crane, Gord Clement;
  Topic=Derivatives;
  Sub-Topic=Derivative at a point;
  Course=Introductory Calculus for the Biological Sciences;
@
qu.1.18.weighting=1@
qu.1.18.numbering=alpha@
qu.1.18.part.1.name=sro_id_1@
qu.1.18.part.1.maple_answer=simplify(subs(x=$d,diff(($f)*($g),x)))@
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=is(abs(($ANSWER)-($RESPONSE)) <0.05);@
qu.1.18.part.1.type=formula@
qu.1.18.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$df, then <img style="width: 36px; height: 39px" align="middle" width="43" height="48" alt="" src="__BASE_URI__Pictures/fprime.png" /><font size="4"><font size="3">$d</font>)</font><font size="5"> </font>:</p><p>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.19.mode=Inline@
qu.1.19.name=Second Derivative@
qu.1.19.comment=<p>Chain Rule: if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.1.19.editing=useHTML@
qu.1.19.solution=@
qu.1.19.algorithm=$a=rint(1,9);
$b=rint(2,6);
$c=rint(-5,10);
$d=rint(1,11);
$e=rint(-8,5);
$h=rint(-7,-1);
condition:ne($a,$d);
$f=switch(rint(9),"($a)*x+($b)","($d)*x+($c)","($h)*x+($e)");
$g=switch(rint(3),"3/2","5/2","7/2");
$i="($f)^($g)";
$F=maple("printf(MathML[ExportPresentation]($i))");@
qu.1.19.uid=31641309-7b32-4bf0-996c-0811cd6cfcfb@
qu.1.19.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Second Derivatives;
@
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=diff(($i),[x,x])@
qu.1.19.part.1.editing=useHTML@
qu.1.19.part.1.question=(Unset)@
qu.1.19.part.1.libname=@
qu.1.19.part.1.mode=Maple@
qu.1.19.part.1.allow2d=1@
qu.1.19.part.1.plot=@
qu.1.19.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.19.part.1.type=formula@
qu.1.19.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> (the <strong>second</strong> derivative of <em>f </em>with respect to<em> x</em>) is</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.20.mode=Inline@
qu.1.20.name=Second Derivative - Product Rule@
qu.1.20.comment=<p>This question uses a combination of Chain Rule and Product Rule.</p>@
qu.1.20.editing=useHTML@
qu.1.20.solution=@
qu.1.20.algorithm=$a=rint(2,10);
$b=rint(2,5);
condition: ne($a,$b);
$d=rint(4,12);
$f="$a*x^$b";
$g="exp($d*x)";
$F=maple("printf(MathML[ExportPresentation]($f*$g))");@
qu.1.20.uid=32003b62-b3fd-4620-8319-d6f9766b23e5@
qu.1.20.info=  Author=Steve Crane, Gord Clement;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Derivatives;
  Sub-Topic=Second Derivative;
@
qu.1.20.weighting=1@
qu.1.20.numbering=alpha@
qu.1.20.part.1.name=sro_id_1@
qu.1.20.part.1.maple_answer=diff(($f*$g),[x,x])@
qu.1.20.part.1.editing=useHTML@
qu.1.20.part.1.question=(Unset)@
qu.1.20.part.1.libname=@
qu.1.20.part.1.mode=Maple@
qu.1.20.part.1.allow2d=1@
qu.1.20.part.1.plot=@
qu.1.20.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.20.part.1.type=formula@
qu.1.20.question=<p>If <em>f(x) = </em>$F then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> (the <strong>second</strong> derivative of <em>f</em> with respect to<em> x</em>) is</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

