qu.1.topic=Differentials/Linear Approximation@

qu.1.1.mode=Inline@
qu.1.1.name=Differentials (Approximations)@
qu.1.1.comment=<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><mi>f</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'>&ap;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=[15,24,35,48];
$picka=switch(rint(6),$a);
$b=$picka+1;
$f="x->x^(1/2)";
$Derv="x->1/2/x^(1/2)";
$ANS=maple("(($f)($b)+(($Derv)($b))*($picka-$b))");@
qu.1.1.uid=bc0ac866-46f9-4267-954f-d0198005dd89@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.maple_answer=$ANS@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.libname=@
qu.1.1.part.1.mode=Maple@
qu.1.1.part.1.allow2d=1@
qu.1.1.part.1.plot=@
qu.1.1.part.1.maple=is(abs(($ANSWER)-($RESPONSE)) < 0.05);@
qu.1.1.part.1.type=formula@
qu.1.1.question=<p>Use differentials to approximate the value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mi>$picka</mi></mrow></msqrt></mrow></mstyle></math>.</p><p>Give your answer to at least 2 decimal places.</p><p>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Differentials (Approximation - Cube)@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(2,7);
$b=[0.001,0.002,0.003,0.004];
$pickb=switch(rint(4),$b);
$c=$a+$pickb;
$ANS1=3*$a^2*($c-$a);
$ANS2=12*$a*($c-$a);@
qu.1.2.uid=4d501184-cf95-4464-898a-f5ec29a16ff6@
qu.1.2.info=  Author=Steve Crane;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Differentials, Linear Approximation and Taylor Polynomials;
  Sub-Topic=Differentials;
@
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=$ANS1@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.libname=@
qu.1.2.part.1.mode=Maple@
qu.1.2.part.1.allow2d=1@
qu.1.2.part.1.plot=@
qu.1.2.part.1.maple=is(abs(($ANSWER)-($RESPONSE)) <0.05);@
qu.1.2.part.1.type=formula@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.maple_answer=$ANS2@
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(abs(($ANSWER)-($RESPONSE)) < 0.05);@
qu.1.2.part.2.type=formula@
qu.1.2.question=<p>If the length of the sides of a cube changes from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>l</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>to $c <em>cm</em>&nbsp;, then use differentials to determine the approximate change in (a) the volume and (b) the surface area of the cube in cm<sup>3</sup>.</p><p>Note: The units are&nbsp;given&nbsp;for you after the input boxes. <strong>Omit units from your answer.</strong></p><p>Give your answers to at least 2 decimal places.</p><p>&nbsp;(a)&nbsp;&nbsp;Approx. change in volume = <span>&nbsp;</span><1><span>&nbsp; cm<sup>3</sup></span></p><p>&nbsp;<span>(b)&nbsp;Approx. change in volume =&nbsp; <2><span>&nbsp; cm<sup>2</sup></span></span></p><p>&nbsp;</p>@

qu.1.3.mode=Inline@
qu.1.3.name=Taylor approximation@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$ANS="1+2*(x-4)+(4/2)*(x-4)^2+(8/6)*(x-4)^3";@
qu.1.3.uid=ef80c6a8-4b50-4453-85c3-9e3e089979f6@
qu.1.3.info=  Author=Steve Crane;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Differentials, Linear Approximation and Taylor Polynomials;
  Sub-Topic=Taylor Polynomials;
@
qu.1.3.weighting=1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.maple_answer=$ANS@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.libname=@
qu.1.3.part.1.mode=Maple@
qu.1.3.part.1.allow2d=1@
qu.1.3.part.1.plot=@
qu.1.3.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.3.part.1.type=formula@
qu.1.3.question=<p>The third order Taylor polynomial approximation about <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>4</mn></mrow></mstyle></math>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><msup><mi>e</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>8</mn></mrow></msup></mrow></mstyle></math>is:</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Differentials@
qu.1.4.comment=<p>Recall that the differential is defined 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>df</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><mi>dx</mi></mrow></mstyle></math></p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=rint(1,9);
$b=rint(1,9);
$c=rint(1,14);
$dlist=['100','200','300','400','500','600','700'];
$d=switch(rint(7),$dlist);
$e=rint(1,4);
$funca="$a*x^2+sin(exp(x))";
$funcb="sqrt($b*x-$c)";
$ques1=switch(rint(3),"$funca","$funcb");
$funcc="$d*x^(-1/2)";
$funcd="$c*x^4*sin($e*x)";
$ques2=switch(rint(3),"$funcc","$funcd");
$m = maple("
MathML[ExportPresentation]($ques1),
MathML[ExportPresentation]($ques2)
");
$m1=switch(0,$m);
$m2=switch(1,$m);@
qu.1.4.uid=81441045-fc49-4ce3-9c8e-c32145cb9613@
qu.1.4.weighting=1,1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.maple_answer=diff($ques1,x)@
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.libname=@
qu.1.4.part.1.mode=Maple@
qu.1.4.part.1.allow2d=1@
qu.1.4.part.1.plot=@
qu.1.4.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.4.part.1.type=formula@
qu.1.4.part.2.name=sro_id_2@
qu.1.4.part.2.maple_answer=diff($ques2,x)@
qu.1.4.part.2.editing=useHTML@
qu.1.4.part.2.question=(Unset)@
qu.1.4.part.2.libname=@
qu.1.4.part.2.mode=Maple@
qu.1.4.part.2.allow2d=1@
qu.1.4.part.2.plot=@
qu.1.4.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.4.part.2.type=formula@
qu.1.4.question=<p>Find the differential <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>df</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> for the following functon:</p><p>Note that the required 'd<em>x'</em>, at the end of the answers is written in for you after the input box. <strong>Omit dx from your answer.&nbsp;</strong></p><p>Note 2: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></mrow></mstyle></math>can be represented by exp(x).&nbsp; For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>4</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn></mrow></msup></mrow></mstyle></math> would be entered as 4*exp(5*x+2).</p><p>1) $m1</p><p><span>&nbsp;</span><1><span> d<em>x</em><br /></span></p><p>&nbsp;</p><p><span>2) $m2</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;d<em>x</em></span></span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Differentials (Approximations x^1/4)@
qu.1.5.comment=<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><mi>f</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'>&ap;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$a=[17,82];
$picka=switch(rint(3),$a);
$b=$picka-1;
$f="x->x^(1/4)";
$ANS = maple("
df:=unapply(diff(($f)(x),x),x):
(($f)($b)+(df($b))*($picka-$b))");@
qu.1.5.uid=31125dee-c4db-48c1-a0d7-f97348bd6a20@
qu.1.5.weighting=1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.maple_answer=$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(abs(($ANSWER)-($RESPONSE)) < 0.05);@
qu.1.5.part.1.type=formula@
qu.1.5.question=<p>Use differentials to approximate the value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mroot><mrow><mi>$picka</mi></mrow><mrow><mn>4</mn></mrow></mroot></mrow></mstyle></math>.</p><p>&nbsp;Round your answer to 2 ddecimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Taylor approximation iii@
qu.1.6.comment=@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$a=rint(1,5);
$b=rint(2,9);
$c=rint(5,12);
$d=rint(2,7);
$e=rint(2,4);
$func="$a+$b*x+$c*x^2+$d*x^3+$e*x^4";
$disp=maple("MathML[ExportPresentation]($func)");
$f=rint(2);
$num=switch($f,3,4);
$ANS=switch($f,6*$d,24*$e);@
qu.1.6.uid=236b6800-757c-49d0-8a8a-7d4defb79a31@
qu.1.6.info=  Author=Steve Crane;
  Course=Introductory Calculus for the Biological Sciences;
  Topic=Differentials, Linear Approximation and Taylor Polynomials;
  Sub-Topic=Taylor Polynomials;
@
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=$ANS@
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.libname=@
qu.1.6.part.1.mode=Maple@
qu.1.6.part.1.allow2d=1@
qu.1.6.part.1.plot=@
qu.1.6.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.6.part.1.type=formula@
qu.1.6.question=<p>If&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> = $disp is a Taylor polynomial determined by a function <em>f</em> about the point<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>, determine&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>f</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$num</mi></mrow></mfenced></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mn>0</mn></mrow></mfenced></mrow></mstyle></math>.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Differentials (Approximation - Sphere)@
qu.1.7.comment=@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$a=rint(2,7);
$b=[0.001,0.002,0.003,0.004];
$pickb=switch(rint(4),$b);
$c=$a+$pickb;
$ANS1=4*3.14*($a)^2*($c-$a);
$ANS2=8*3.14*($a)*($c-$a);@
qu.1.7.uid=78c8b577-14be-4082-a6ac-ea894b0487cc@
qu.1.7.info=  Author=Steve Crane, Gord Clement;
  Sub-Topic=Differentials;
  Topic=Differentials, Linear Approximation and Taylor Polynomials;
  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=$ANS1@
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.libname=@
qu.1.7.part.1.mode=Maple@
qu.1.7.part.1.allow2d=1@
qu.1.7.part.1.plot=@
qu.1.7.part.1.maple=is(abs(($ANSWER)-($RESPONSE)) <0.05);@
qu.1.7.part.1.type=formula@
qu.1.7.part.2.name=sro_id_2@
qu.1.7.part.2.maple_answer=$ANS2@
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=is(abs(($ANSWER)-($RESPONSE)) < 0.05);@
qu.1.7.part.2.type=formula@
qu.1.7.question=<p>If the radius of a sphere changes from r = $a to r = $c cm.&nbsp; Use differentials to determine the approximate change in (a) the volume and (b) the surface area.</p><p>Note 1: The units have been entered for you after the input box. <strong>Omit units from your answer.</strong></p><p>Note 2: Use 3.14 for Pi. Give your answer to at least two decimal places.</p><p>&nbsp;(a) &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<span> </span><1><span>&nbsp; cm<sup>3</sup></span></p><p>&nbsp;</p><p><span>(b)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp; <2><span>&nbsp;cm<sup>2</sup></span></span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Taylor approximation ii@
qu.1.8.comment=@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$a=rint(2);
$b=switch($a,"5*x^3+sin(2*x-2)","ln(x+1)");
$c=switch($a,"1","0");
$disp=maple("MathML[ExportPresentation]($b)");
$ANS=switch($a,"5+17*(x-1)+15*(x-1)^2+(11/3)*(x-1)^3","x-(1/2)*x^2+(1/3)*x^3");@
qu.1.8.uid=a49ae772-a0cc-4a54-9d56-5772e2cbf880@
qu.1.8.info=  Author=Steve Crane, Gord Clement;
  Topic=Differentials, Linear Approximation and Taylor Polynomials;
  Sub-Topic=Taylor Polynomials;
  Course=Introductory Calculus for the Biological Sciences;
@
qu.1.8.weighting=1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.maple_answer=$ANS@
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qu.1.8.part.1.question=(Unset)@
qu.1.8.part.1.libname=@
qu.1.8.part.1.mode=Maple@
qu.1.8.part.1.allow2d=1@
qu.1.8.part.1.plot=@
qu.1.8.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.1.8.part.1.type=formula@
qu.1.8.question=<p>The third order Taylor polynomial approximation about <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$c</mi></mrow></mstyle></math>of the function $disp is:</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.9.mode=Inline@
qu.1.9.name=Linear Approximation (MC)@
qu.1.9.comment=<p>Note that this is the tangent line to the graph of <em>f </em>at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=@
qu.1.9.uid=edcd7874-04a2-482f-ad8e-4326af6e8255@
qu.1.9.info=  Author=Steve Crane;
  Sub-Topic=Linear Approximation;
  Topic=Differentials, Linear Approximation and Taylor Polynomials;
  Course=Introductory Calculus for the Biological Sciences;
@
qu.1.9.weighting=1@
qu.1.9.numbering=alpha@
qu.1.9.part.1.name=sro_id_1@
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ap;</mo><mi>f</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>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>@
qu.1.9.part.1.fixed=@
qu.1.9.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ap;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>@
qu.1.9.part.1.question=null@
qu.1.9.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ap;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><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><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>@
qu.1.9.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><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'>&ap;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mstyle></math>@
qu.1.9.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><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'>&ap;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mstyle></math>@
qu.1.9.part.1.mode=Multiple Choice@
qu.1.9.part.1.display=vertical@
qu.1.9.part.1.answer=4@
qu.1.9.question=<p>Which of the following represents the linear approximation of <em>f</em> at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mstyle></math>.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

