qu.1.topic=CtyTheory@

qu.1.1.mode=Multiple Selection@
qu.1.1.name=fog@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=@
qu.1.1.uid=5068b3f8-fb82-4979-a990-8f1ca72e7eda@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Sub-Topic=Theory;
  Topic=Continuity and Differentiability;
  Difficulty=Easy;
@
qu.1.1.question=<p>Click beside each true statement. If a statement is not true, be prepared to give a counter-example on the next midterm!</p>@
qu.1.1.answer=1, 2@
qu.1.1.choice.1=If  <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is continuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>a</mi></mrow></munder><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>b</mi></mrow></math>, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&amp;period;</mo></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.1.1.choice.2=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>g</mi></mrow></math> is continuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is continuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow></math>, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&compfn;</mo><mi>g</mi></mrow></math> is continuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&amp;period;</mo></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'></math>@
qu.1.1.choice.3=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow></math>, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mo lspace='0.1666667em' rspace='0.1666667em'>&compfn;</mo><mi>g</mi></mrow></math> cannot be continuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>a</mi></mrow></math>@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Selection@
qu.1.2.name=AlternateDefnOfCty@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=968b64c7-5ea9-41d8-a1ed-c74e8b957fa5@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Theory;
  Difficulty=Easy;
@
qu.1.2.question=<p>If<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mn>0</mn></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>h</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> , then we can conclude that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is continuous at</p>
<p>(Click beside each correct statement. We will need this formulation of continuity when we prove <strong>THE PRODUCT RULE FOR DERIVATIVES!</strong>)</p>@
qu.1.2.answer=1@
qu.1.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>@
qu.1.2.choice.2=0@
qu.1.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>h</mi></mrow></math>@
qu.1.2.choice.4= everywhere@
qu.1.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>h</mi></mrow></math>@
qu.1.2.fixed=@

qu.1.3.mode=Multiple Selection@
qu.1.3.name=DefnOfCty@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=@
qu.1.3.uid=dadc8e79-ffc1-4d17-a7da-5465b9e63d8c@
qu.1.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Theory;
  Difficulty=Easy;
@
qu.1.3.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is continuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>a</mi></mrow></math>, then</p>
<p>(Click beside each correct statement.)</p>@
qu.1.3.answer=1, 2, 3, 4, 5@
qu.1.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>a</mi></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> exists@
qu.1.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow></math> exists@
qu.1.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>a</mi></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow></math> @
qu.1.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi>a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>@
qu.1.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is continuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi></mrow></math> from both the left and the right.@
qu.1.3.fixed=@

qu.2.topic=discontinuous because@

qu.2.1.mode=Non Permuting Multiple Selection@
qu.2.1.name=NoVal@
qu.2.1.comment=@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(-10,10);
condition:ne($a,0);
$m=maple("randomize();
f:=RandomTools[Generate](choose({(x^2-($a)^2)/(x-($a)),
(x^3-($a)^3)/(x-($a)),sin(x-($a))/(x-($a))})):
if $a=abs($a) then convert(f,string), -($a) else convert(f,string), -($a) end if
");
$f=switch(0,$m);
$F=mathml($f);
$c=switch(1,$m);@
qu.2.1.uid=ba51570d-098e-4e35-9059-a9d00bed718a@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Topic=Continuity and Differentiabilty;
  Sub-Topic=Discontinuous reason;
  Course=Introduction to Calculus I;
@
qu.2.1.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$F is discontinuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>____ because _________________</p>
<p>(More than one selection may be correct. Keep in mind that if a function tends to infinity as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>a</mi></mrow></math>, the limit does not exist at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>.)</p>@
qu.2.1.answer=1@
qu.2.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math> does not exist.@
qu.2.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$c</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></math> does not exist.@
qu.2.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> does not exist.@
qu.2.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$c</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$c</mi></mrow></munder></mrow><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> does not exist.@
qu.2.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder></mrow><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/><mo lspace='0.2777778em' rspace='0.2777778em'>&NotEqual;</mo></mrow><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder></mrow><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> @
qu.2.1.fixed=@

qu.2.2.mode=Multiple Selection@
qu.2.2.name=NoLimNoVal@
qu.2.2.comment=@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(-10,10);
condition:ne($a,0);
$m=maple("f:=ln(abs(x-($a))):
if $a=abs($a) then convert(f,string),-($a) else convert(f,string),-($a) end if");
$f=switch(0,$m);
$c=switch(1,$m);
$F=mathml($f);@
qu.2.2.uid=a8590f1d-3592-4a18-ac23-1af36a4febce@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Discontinuous reason;
  Difficulty=Easy;
@
qu.2.2.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$F is discontinuous at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>____ because _________________</p>
<p>(More than one selection may be correct. Keep in mind that if a function tends to infinity as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>a</mi></mrow></math>, the limit does not exist at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>.)</p>@
qu.2.2.answer=1, 3@
qu.2.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math> does not exist.@
qu.2.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$c</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></math> does not exist.@
qu.2.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$a</mi></mrow></munder></mrow><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> does not exist.@
qu.2.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$c</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi mathvariant='normal'>$c</mi></mrow></munder></mrow><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> does not exist.@
qu.2.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> because <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder></mrow><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/><mo lspace='0.2777778em' rspace='0.2777778em'>&NotEqual;</mo></mrow><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder></mrow><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> @
qu.2.2.fixed=@

qu.3.topic=CtsFromLeftRight@

qu.3.1.mode=Inline@
qu.3.1.name=Floor Left@
qu.3.1.comment=<p>The plot below shows the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;is continuous from&nbsp;just the left at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math>.</p>
<p>$plot</p>@
qu.3.1.editing=useHTML@
qu.3.1.hint.1=Calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and compare these&nbsp;values to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi></mrow></mfenced></mrow></math>.@
qu.3.1.hint.2=Investigate<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>$am</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>$ap</mi></mrow></mfenced></mrow></math>.@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(-2,2);
$ap=$a+.1;
$am=$a-.1;
$b=switch(rint(3),-3,-2,-1);
$f=maple("randomize();
RandomTools[Generate](choose({floor(($b)*x)}))");
$F=maple("printf(MathML[ExportPresentation]($f))");
$plot=plotmaple("plot($f,x=-abs($a)-1..abs($a)+1,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.1.uid=d93568a0-2cb0-4aff-8f87-bcd6a8429291@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continous from left or right;
  Difficulty=Easy;
@
qu.3.1.weighting=1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.grader=exact@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.display.permute=true@
qu.3.1.part.1.answer.4=continuous from neither the left nor the right at x = $a.@
qu.3.1.part.1.answer.3=continuous from right at x = $a.@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.answer.2=continuous from the left at x = $a.@
qu.3.1.part.1.answer.1=continuous from both the left and right at x = $a.@
qu.3.1.part.1.mode=List@
qu.3.1.part.1.display=menu@
qu.3.1.part.1.credit.4=0.0@
qu.3.1.part.1.credit.3=0.0@
qu.3.1.part.1.credit.2=1.0@
qu.3.1.part.1.credit.1=0.0@
qu.3.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F is&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p><p><span><strong>Note: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>floor</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;is Maple's name for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mfenced open='[' close=']' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow></math>, the "greatest integer less than or equal to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>".</strong></span></p>@

qu.3.2.mode=Inline@
qu.3.2.name=CtsLeft@
qu.3.2.comment=<p>The plot below shows the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;is continuous from&nbsp;just the left at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math>.</p>
<p>$plot</p>@
qu.3.2.editing=useHTML@
qu.3.2.hint.1=Calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and compare these&nbsp;values to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>.@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(-3,5);
$b=rint(1,5);
condition:ne($b,0);
$m=maple("randomize():
g:=RandomTools[Generate](choose({0,sin(x-($a))-($b),cos(x-($a))-($b),
tan(x-($a))-($b),(x-($a))-($b),(x-($a))^2-($b),(x-($a))^3-($b),
exp(x)-($b)})):
h:=RandomTools[Generate](choose({1,sin(x-($a))+($b),cos(x-($a))+($b),
tan(x-($a))+($b),(x-($a))+($b),(x-($a))^2+($b),(x-($a))^3+($b)} )):
f:=piecewise(x <= $a, g, x >$a, h):
convert(f,string), MathML[ExportPresentation](f)
");
$f=switch(0,$m);
$F=switch(1,$m);
$plot=plotmaple("p1:=plot($f,x=$a-1..$a+1,y=-10..10,thickness=2,discont=true):
p2:=plots[pointplot]([[$a,subs(x=$a,$f)]],symbol = solidcircle, symbolsize = 15):
plots[display]([p1,p2]),plotdevice='gif', plotoptions='height=250,width=250'
");@
qu.3.2.uid=56ab0593-b261-428c-aa97-987db5a49eb1@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continous from left or right;
  Difficulty=Easy;
@
qu.3.2.weighting=1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.grader=exact@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.display.permute=true@
qu.3.2.part.1.answer.4=continuous from neither the left nor the right at x = $a.@
qu.3.2.part.1.answer.3=continuous from right at x = $a.@
qu.3.2.part.1.question=(Unset)@
qu.3.2.part.1.answer.2=continuous from the left at x = $a.@
qu.3.2.part.1.answer.1=continuous from both the left and right at x = $a.@
qu.3.2.part.1.mode=List@
qu.3.2.part.1.display=menu@
qu.3.2.part.1.credit.4=0.0@
qu.3.2.part.1.credit.3=0.0@
qu.3.2.part.1.credit.2=1.0@
qu.3.2.part.1.credit.1=0.0@
qu.3.2.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F is&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.3.mode=Inline@
qu.3.3.name=CtsBoth@
qu.3.3.comment=<p>The plot below shows the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;is continuous from both the left and right at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math>.</p>
<p>$plot</p>@
qu.3.3.editing=useHTML@
qu.3.3.hint.1=Calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and compare these&nbsp;values to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>.@
qu.3.3.solution=@
qu.3.3.algorithm=$a=rint(-3,5);
$b=rint(1,5);
condition:ne($b,0);
$z1=rint(7);
$z2=rint(7);
condition:ne($z1,$z2);
$g=switch($z1,"sin(x-($a))+($b)","tan(x-($a))+($b)",
"(x-($a))+($b)","(x-($a))^2+($b)","(x-($a))^3+($b)",
"($b)","cos(x-($a))+($b)-1");
$h=switch($z2,"sin(x-($a))+($b)","tan(x-($a))+($b)",
"(x-($a))+($b)","(x-($a))^2+($b)","(x-($a))^3+($b)",
"($b)","cos(x-($a))+($b)-1");
$f=maple("piecewise(x <= $a, '$g', x >$a, $h)");
$F=maple("printf(MathML[ExportPresentation]($f))");
$ANS=B;
$plot=plotmaple("plot($f,x=$a-2..$a+2,y=-10..10,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.3.uid=0a6fb3e8-fd6a-46f1-b424-fe78e8be5705@
qu.3.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continous from left or right;
  Difficulty=Easy;
@
qu.3.3.weighting=1@
qu.3.3.numbering=alpha@
qu.3.3.part.1.grader=exact@
qu.3.3.part.1.name=sro_id_1@
qu.3.3.part.1.editing=useHTML@
qu.3.3.part.1.display.permute=true@
qu.3.3.part.1.answer.4=continuous from neither the left nor the right at x = $a.@
qu.3.3.part.1.answer.3=continuous from right at x = $a.@
qu.3.3.part.1.question=(Unset)@
qu.3.3.part.1.answer.2=continuous from the left at x = $a.@
qu.3.3.part.1.answer.1=continuous from both the left and right at x = $a.@
qu.3.3.part.1.mode=List@
qu.3.3.part.1.display=menu@
qu.3.3.part.1.credit.4=0.0@
qu.3.3.part.1.credit.3=0.0@
qu.3.3.part.1.credit.2=0.0@
qu.3.3.part.1.credit.1=1.0@
qu.3.3.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F is&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.4.mode=Inline@
qu.3.4.name=|x-a|/|x-a| CtsNeither@
qu.3.4.comment=<p>The plot below shows the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;is not&nbsp;continuous from&nbsp;the left&nbsp;nor&nbsp;the right at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math>, since <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>&nbsp;does not exist.</p>
<p>$plot</p>@
qu.3.4.editing=useHTML@
qu.3.4.hint.1=Calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and compare these&nbsp;values to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>.@
qu.3.4.solution=@
qu.3.4.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition:ne($b,0);
$m=maple("f:=eval(abs(x-($a))/(x-($a))):

convert(f,string),MathML[ExportPresentation](f) ");
$f=switch(0,$m);
$F=switch(1,$m);
$plot=plotmaple("plot($f,x=$a-1..$a+1,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.4.uid=e4e4c02a-23ee-4d18-a6d4-ce5ae6c292c6@
qu.3.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continous from left or right;
  Difficulty=Easy;
@
qu.3.4.weighting=1@
qu.3.4.numbering=alpha@
qu.3.4.part.1.grader=exact@
qu.3.4.part.1.name=sro_id_1@
qu.3.4.part.1.editing=useHTML@
qu.3.4.part.1.display.permute=true@
qu.3.4.part.1.answer.4=continuous from neither the left nor the right at x = $a.@
qu.3.4.part.1.answer.3=continuous from right at x = $a.@
qu.3.4.part.1.question=(Unset)@
qu.3.4.part.1.answer.2=continuous from the left at x = $a.@
qu.3.4.part.1.answer.1=continuous from both the left and right at x = $a.@
qu.3.4.part.1.mode=List@
qu.3.4.part.1.display=menu@
qu.3.4.part.1.credit.4=1.0@
qu.3.4.part.1.credit.3=0.0@
qu.3.4.part.1.credit.2=0.0@
qu.3.4.part.1.credit.1=0.0@
qu.3.4.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F is&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.5.mode=Inline@
qu.3.5.name=CtsRight@
qu.3.5.comment=<p>The plot below shows the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;is continuous from&nbsp;just the right at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math>.</p>
<p>$plot</p>@
qu.3.5.editing=useHTML@
qu.3.5.hint.1=Calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and compare these&nbsp;values to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>.@
qu.3.5.solution=@
qu.3.5.algorithm=$a=rint(-3,5);
$b=rint(1,5);
condition:ne($b,0);
$m=maple("randomize():
g:=RandomTools[Generate](choose({0,sin(x-($a))-($b),cos(x-($a))-($b),
tan(x-($a))-($b),(x-($a))-($b),(x-($a))^2-($b),(x-($a))^3-($b),
exp(x)-($b)})):
h:=RandomTools[Generate](choose({1,sin(x-($a))+($b),cos(x-($a))+($b),
tan(x-($a))+($b),(x-($a))+($b),(x-($a))^2+($b),(x-($a))^3+($b)} )):
f:=piecewise(x < $a, $g, x >= $a, $h):
convert(f,string), MathML[ExportPresentation](f)
");
$f=switch(0,$m);
$F=switch(1,$m);
$plot=plotmaple("p1:=plot($f,x=$a-1..$a+1,y=-10..10,thickness=2,discont=true):
p2:=plots[pointplot]([[$a,subs(x=$a,$f)]],symbol = solidcircle, symbolsize = 15):
plots[display]([p1,p2]),plotdevice='gif', plotoptions='height=250,width=250'
");@
qu.3.5.uid=5f87dc05-ee45-438d-9ee9-c4a671d400fc@
qu.3.5.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continous from left or right;
  Difficulty=Easy;
@
qu.3.5.weighting=1@
qu.3.5.numbering=alpha@
qu.3.5.part.1.grader=exact@
qu.3.5.part.1.name=sro_id_1@
qu.3.5.part.1.editing=useHTML@
qu.3.5.part.1.display.permute=true@
qu.3.5.part.1.answer.4=continuous from neither the left nor the right at x = $a.@
qu.3.5.part.1.answer.3=continuous from right at x = $a.@
qu.3.5.part.1.question=(Unset)@
qu.3.5.part.1.answer.2=continuous from the left at x = $a.@
qu.3.5.part.1.answer.1=continuous from both the left and right at x = $a.@
qu.3.5.part.1.mode=List@
qu.3.5.part.1.display=menu@
qu.3.5.part.1.credit.4=0.0@
qu.3.5.part.1.credit.3=1.0@
qu.3.5.part.1.credit.2=0.0@
qu.3.5.part.1.credit.1=0.0@
qu.3.5.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F is&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.6.mode=Inline@
qu.3.6.name=Floor Right@
qu.3.6.comment=<p>The plot below shows the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;is continuous from&nbsp;just the&nbsp;right at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math>.</p>
<p>$plot</p>@
qu.3.6.editing=useHTML@
qu.3.6.hint.1=Calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;and compare these&nbsp;values to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>.@
qu.3.6.hint.2=Investigate <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$am</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ap</mi></mrow></mfenced></mrow></math>.@
qu.3.6.solution=@
qu.3.6.algorithm=$a=rint(-2,2);
$b=switch(rint(3),1,2,3);
$f=maple("randomize();
RandomTools[Generate](choose({floor(($b)*x),floor(x)-x,
x-floor(x)}))");
$F=maple("printf(MathML[ExportPresentation]($f))");
$plot=plotmaple("plot($f,x=-abs($a)-1..abs($a)+1,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$ap=$a+0.1;
$am=$a-0.1;@
qu.3.6.uid=2ab16c62-3a34-4ba4-9c72-9a1c2775adb7@
qu.3.6.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continous from left or right;
  Difficulty=Easy;
@
qu.3.6.weighting=1@
qu.3.6.numbering=alpha@
qu.3.6.part.1.grader=exact@
qu.3.6.part.1.name=sro_id_1@
qu.3.6.part.1.editing=useHTML@
qu.3.6.part.1.display.permute=true@
qu.3.6.part.1.answer.4=continuous from neither the left nor the right at x = $a.@
qu.3.6.part.1.answer.3=continuous from right at x = $a.@
qu.3.6.part.1.question=(Unset)@
qu.3.6.part.1.answer.2=continuous from the left at x = $a.@
qu.3.6.part.1.answer.1=continuous from both the left and right at x = $a.@
qu.3.6.part.1.mode=List@
qu.3.6.part.1.display=menu@
qu.3.6.part.1.credit.4=0.0@
qu.3.6.part.1.credit.3=1.0@
qu.3.6.part.1.credit.2=0.0@
qu.3.6.part.1.credit.1=0.0@
qu.3.6.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$F is&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span>Note: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>floor</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;is Maple's name for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mfenced open='[' close=']' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow></math>, the "greatest integer less than or equal to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>".</span></p>@

qu.4.topic=IntervalsCty@

qu.4.1.mode=Multiple Selection@
qu.4.1.name=IntCty@
qu.4.1.comment=@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(-5,-2);
$b=rint(3,6);
$c=rint(-1,2);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
$A=rint(2,6);
$f=maple("$A/(x-($a))^(1/$m)*(x-($b))^$n/(x-($c))^(1/$p)");
$displayf=maple("printf(MathML[ExportPresentation]($f))");@
qu.4.1.uid=0b421fc7-2179-4e88-9c0b-53238654e607@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Continuity and Differentiability;
  Sub-Topic=Intervals of Continuity;
@
qu.4.1.question=<p>Select the interval or intervals on which the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayf is continuous.</p>@
qu.4.1.answer=1, 2, 3@
qu.4.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo separator='true' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.4.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></math>@
qu.4.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mi>&infin;</mi></mrow></mrow></mfenced></mrow></math>@
qu.4.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&infin;</mi></mrow></mfenced></mrow></math>@
qu.4.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>@
qu.4.1.choice.6=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mi mathvariant='normal'>$c</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>@
qu.4.1.fixed=@

qu.4.2.mode=Non Permuting Multiple Selection@
qu.4.2.name=IntCtySqrtLeft@
qu.4.2.comment=@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(-5,-1);
$b=rint(1,5);
$m=rint(2,6,2);
$n=rint(3,5,2);
$A=rint(2,6);
$f=maple("$A*(x-($a))^(1/$m)*(x-($b))^(1/$n)");
$displayf=mathml("$f");@
qu.4.2.uid=0f094186-5ca5-4507-bf8b-901c0d8ff890@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Continuity and Differentiability;
  Sub-Topic=Intervals of Continuity;
@
qu.4.2.question=<p>Select the interval or intervals on which the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayf is continuous.</p>@
qu.4.2.answer=3@
qu.4.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo separator='true' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.4.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>@
qu.4.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lsqb;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow></math>@
qu.4.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&infin;</mi></mrow></mfenced></mrow></math>@
qu.4.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>@
qu.4.2.fixed=@

qu.4.3.mode=Multiple Selection@
qu.4.3.name=IntCtySqrt{a,b]@
qu.4.3.comment=@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$a=rint(-5,-1);
$b=rint(1,5);
$m=rint(2,6,2);
$n=rint(2,6,2);
$A=rint(2,6);
$f=maple("$A*(x-($a))^(1/$m)*(($b)-x)^(1/$n)");
$displayf=maple("printf(MathML[ExportPresentation]($f))");@
qu.4.3.uid=0daf1e46-8ccd-4b1f-b2d7-d94599b296c5@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Continuity and Differentiabilty;
  Sub-Topic=Intervals of Continuity;
@
qu.4.3.question=<p>Select the interval or intervals on which the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayf is continuous.</p>@
qu.4.3.answer=6@
qu.4.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo separator='true' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.4.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math>@
qu.4.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lsqb;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow></math>@
qu.4.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&infin;</mi></mrow></mfenced></mrow></math>@
qu.4.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>@
qu.4.3.choice.6=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>@
qu.4.3.choice.7=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>@
qu.4.3.fixed=@

qu.4.4.mode=Multiple Selection@
qu.4.4.name=IntCtySqrtRight@
qu.4.4.comment=@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=$a=rint(-5,-1);
$b=rint(1,5);
$m=rint(2,6,2);
$n=rint(3,5,2);
$A=rint(2,6);
$f=maple("$A*(x-($a))^(1/$n)*(x-($b))^(1/$m)");
$displayf=mathml("$f");@
qu.4.4.uid=d78f89e7-ba7e-42ff-aab5-d4cc7941bb9b@
qu.4.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Intervals of Continuity;
  Difficulty=Easy;
@
qu.4.4.question=<p>Select the interval or intervals on which the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayf is continuous.</p>@
qu.4.4.answer=6@
qu.4.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo separator='true' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.4.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>@
qu.4.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lsqb;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow></math>@
qu.4.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&infin;</mi></mrow></mfenced></mrow></math>@
qu.4.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>@
qu.4.4.choice.6=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>@
qu.4.4.fixed=@

qu.5.topic=IVT@

qu.5.1.question=<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$k</mi></mrow></math>. Note that&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Fa <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Fb.</p>
<p>Find a value <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math> (guaranteed by the Intermediate Value Theorem) that satisfies <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$k</mi></mrow></math>.</p>
<p>NOTE: If your answer is, for example, (-9)^(1/3), enter -9^(1/3). (Remember that the cubed root of -1 is -1!)</p>
<p>(For this to work, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> must be continuous on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>, which it is!)</p>
<p>$plot</p>
<p>In this plot, the red line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></math>$Fa, the green line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></math>$Fb, and the blue line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$k</mi></mrow></math>. The curve <strong>ALWAYS</strong>&nbsp;crosses the blue line. We are finding a <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>value (there is always at least one) where this happens.</p>@
qu.5.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.5.1.allow2d=1@
qu.5.1.maple_answer=$ANS@
qu.5.1.type=formula@
qu.5.1.mode=Maple@
qu.5.1.name=IVT@
qu.5.1.comment=@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(-4,-1);
$b=rint(2,4);
$z=rint(3);
$f=switch($z, "x^3" , "surd(x,3)" , "exp(x)");
$M=maple("fa:=eval($f,x=$a):
fb:=eval($f,x=$b):
convert(fa,string),convert(fb,string),MathML[ExportPresentation]($f),MathML[ExportPresentation](fa),MathML[ExportPresentation](fb)
");
$F=switch(2,$M);
$fa=switch(0,$M);
$fb=switch(1,$M);
$Fa=switch(3,$M);
$Fb=switch(4,$M);
$k=int((($fa)+($fb))/2);
$ANS=switch($z, "surd($k,3)", "$k^3",  "ln($k)");
$plot=plotmaple("plot([$f,$fa,$fb,$k],x=-3..3,y=$fa-1..$fb+1,thickness=2,
color=[black,red,green,blue]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.1.uid=8cf4a1fb-db52-47cb-8611-9aa8dbb06913@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Intermediate Value Theorem;
@

qu.6.topic=HorVerTangents@

qu.6.1.mode=Multiple Choice@
qu.6.1.name=VTat$b@
qu.6.1.comment=@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(3,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$M=maple("f:=$A*(x+$a)^$m*(x-$b)^$n/(x-$c)^(1/$p):
convert(f,string),MathML[ExportPresentation](dy/dx=$f)");
$f=switch(0,$M);
$displayf=switch(1,$M);@
qu.6.1.uid=e0978c42-516e-487d-bb60-27dba5db2734@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Topic=Continuity and Differentiability;
  Sub-Topic=Vertical Tangents;
  Course=Introduction to Calculus I;
@
qu.6.1.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>is defined for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mi>&reals;</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></math>The first derivative is given by $displayf. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$c</mi></mrow></math> , we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math></p>@
qu.6.1.answer=1@
qu.6.1.choice.1=a vertical tangent.@
qu.6.1.choice.2=a vertical asymptote.@
qu.6.1.choice.3=a horizontal asymptote.@
qu.6.1.choice.4=a horizontal tangent.@
qu.6.1.choice.5=an undefined relation since we are dividing by 0.@
qu.6.1.fixed=@

qu.6.2.mode=Multiple Choice@
qu.6.2.name=HTat-$a,$b@
qu.6.2.comment=@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=$a=rint(1,3);
$b=rint(4,6);
$c=rint(7,9);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(3,5,2);
$A=rint(2,6);
$M=maple("f:=$A*(x-($a))^$m*(x-($b))^$n/(x-($c))^(1/$p):
convert(f,string),MathML[ExportPresentation](dy/dx=f),convert(int(f,x),string)");
$f=switch(0,$M);
$displayf=switch(1,$M);
$F=switch(2,$M);@
qu.6.2.uid=d2354564-89ec-475a-8956-d2da8694155d@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Horizontal Tangents;
  Difficulty=Medium;
@
qu.6.2.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>is defined for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mi>&reals;</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></math>The first derivative is given by $displayf. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></math> , we have</p>@
qu.6.2.answer=1@
qu.6.2.choice.1=horizontal tangents.@
qu.6.2.choice.2=vertical tangents.@
qu.6.2.choice.3=a horizontal tangent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math> and a vertical tangent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>.@
qu.6.2.choice.4=a vertical tangent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></math> and a horizontal tangent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>.@
qu.6.2.choice.5=horizontal asymptotes.@
qu.6.2.fixed=@

qu.7.topic=DefnofDer@

qu.7.1.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>f</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mn>0</mn></mrow></munder></mrow></math>$F, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>. . .</p>@
qu.7.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.7.1.allow2d=1@
qu.7.1.maple_answer=$ANS@
qu.7.1.type=formula@
qu.7.1.mode=Maple@
qu.7.1.name=DefofDer,h->0@
qu.7.1.comment=@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=$z=rint(5);
$f=switch($z,"((x+h)^2-x^2)/h","(sin(x+h)-sin(x))/h",
"(exp(x+h)-exp(x))/h","(sqrt(x+h)-sqrt(x))/h","(ln(x+h)-ln(x))/h");
$F=maple("printf(MathML[ExportPresentation]($f))");
$ANS=switch($z,"x^2","sin(x)","exp(x)", "sqrt(x)", "log(x)");@
qu.7.1.uid=4b3ea3a7-22b6-4bfc-9ce9-7ac4ce3c69dc@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Sub-Topic=Theory;
  Topic=Continuity and Differentiability;
  Difficulty=Easy;
@

qu.7.2.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>f</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>a</mi></mrow></munder></mrow></math>$F, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>. . .</p>@
qu.7.2.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.7.2.allow2d=1@
qu.7.2.maple_answer=$ANS@
qu.7.2.type=formula@
qu.7.2.mode=Maple@
qu.7.2.name=DefofDer,x->a@
qu.7.2.comment=@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=$z=rint(5);
$f=switch($z,"((x)^2-a^2)/(x-a)","(sin(x)-sin(a))/(x-a)",
"(exp(x)-exp(a))/(x-a)","(sqrt(x)-sqrt(a))/(x-a)","(ln(x)-ln(a))/(x-a)");
$F=maple("printf(MathML[ExportPresentation]($f))");
$ANS=switch($z, "x^2", "sin(x)", "exp(x)", "sqrt(x)", "log(x)");@
qu.7.2.uid=3ccfc00f-d4f3-4479-b47c-260e7bde2225@
qu.7.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Theory;
  Difficulty=Easy;
@

qu.8.topic=slope of tangent/normal@

qu.8.1.question=<p>Find the slope of the tangent to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>.</p>@
qu.8.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.8.1.allow2d=1@
qu.8.1.maple_answer=$ANS@
qu.8.1.type=formula@
qu.8.1.mode=Maple@
qu.8.1.name=slope of tangent@
qu.8.1.comment=<p>$plot</p>
<p>The curve <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F is red and the tangent line to the curve at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> is green.</p>@
qu.8.1.editing=useHTML@
qu.8.1.solution=@
qu.8.1.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$f=switch(rint(8),"x^2+($a)","sin(Pi*($a)*x)", "cos(Pi*($a)*x)",
"x^3+($a)", "1/sqrt(x+abs($a)+1)", "1/(x+($a))");
$M=maple("
MathML[ExportPresentation]($f),
convert(diff($f,x), string),
convert(eval($f,x=($a)), string),
convert(eval(diff($f,x),x=($a)), string)");
$F=switch(0,$M);
$fp=switch(1,$M);

$fa=switch(2,$M);

$ANS=switch(3,$M);

$plot=plotmaple("plot([$f,($ANS)*(x-($a))+($fa)],x=($a)-1..($a)+1,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.8.1.uid=6b7be89a-1625-4243-9d0a-b88356992be0@
qu.8.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Continuity and Differentiability;
  Sub-Topic=Slope of Tangent;
@

qu.8.2.question=<p>Find the slope of the normal to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>.</p>@
qu.8.2.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.8.2.allow2d=1@
qu.8.2.maple_answer=$ANS@
qu.8.2.type=formula@
qu.8.2.mode=Maple@
qu.8.2.name=slope of normal@
qu.8.2.comment=<p>$plot</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F is the black curve. The tangent (when you can distinguish it from the curve) is in red and the normal in green. They should look perpendicular<strong> </strong>to one another.</p>@
qu.8.2.editing=useHTML@
qu.8.2.solution=@
qu.8.2.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$f=switch(rint(8),"x^2+($a)","sin(Pi*($a)*x)",
"x^3+($a)", "1/sqrt(x+abs($a)+1)", "1/(x+($a))");
$M=maple("
MathML[ExportPresentation]($f), convert(diff($f,x),string), convert(eval($f,x=($a)),string),convert(-1/eval(diff($f,x),x=($a)),string)");
$F=switch(0,$M);
$fp=switch(1,$M);
$fa=switch(2,$M);
$ANS=switch(3,$M);
$plot=plotmaple("plot([$f,-1/($ANS)*(x-($a))+($fa),($ANS)*(x-($a))+($fa)],
x=($a)-4..($a)+4,y=($fa)-2..($fa)+2,thickness=2,color=[black,red, green],scaling=constrained),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.8.2.uid=4f1f2fa5-a4a7-42d7-8c6e-f0ed865828de@
qu.8.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Slope of Tangent;
  Difficulty=Medium;
@

qu.9.topic=Branch Cty&Diff@

qu.9.1.mode=Multiple Choice@
qu.9.1.name=Cts&DiffLeft@
qu.9.1.comment=<p>$plot</p>@
qu.9.1.editing=useHTML@
qu.9.1.solution=@
qu.9.1.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$f="x^2-($a)^2+($b)-5";
$g="2*($a)*x-2*($a)^2+($b)";
$m=maple("f:=piecewise(x<=$a,$f,x>$a,$g):
convert(f,string),MathML[ExportPresentation](f)
");
$F=switch(0,$m);
$displayF=switch(1,$m);
$plot=plotmaple("plot($F,x=-5.1..5.1,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.1.uid=cd550975-3acf-41af-872a-bedefab5da25@
qu.9.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continuity and Differentiability;
  Difficulty=Easy;
@
qu.9.1.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayF, then at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is</p>
<p>&nbsp;</p>
<p>(Note that the option "differentiable but <strong>NOT </strong>continuous" is <strong>NOT</strong> a choice. Why not?)</p>@
qu.9.1.answer=5@
qu.9.1.choice.1=differentiable and therefore continuous as well.@
qu.9.1.choice.2=continuous but differentiable only from the right.@
qu.9.1.choice.3=continuous but differentiable only from the left.@
qu.9.1.choice.4=discontinuous from both the left and the right.@
qu.9.1.choice.5=continuous and differentiable from the left only.@
qu.9.1.choice.6=continuous and differentiable from the right only.@
qu.9.1.fixed=@

qu.9.2.mode=Multiple Choice@
qu.9.2.name=Cts&DiffRt@
qu.9.2.comment=<p>$plot</p>@
qu.9.2.editing=useHTML@
qu.9.2.solution=@
qu.9.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$f="x^2-($a)^2+($b)-5";
$g="2*($a)*x-2*($a)^2+($b)";
$m=maple("f:=piecewise(x<$a,$f,x>=$a,$g):
convert(f,string), MathML[ExportPresentation](f)
");
$F=switch(0,$m);
$displayF=switch(1,$m);
$plot=plotmaple("plot($F,x=-6..6,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.2.uid=f8eb77ff-aa82-42d9-92c2-adeb18872993@
qu.9.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continuity and Differentiability;
  Difficulty=Easy;
@
qu.9.2.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayF, then at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is</p>
<p>(Note that the option "differentiable but <strong>NOT </strong>continuous" is <strong>NOT</strong> a choice. Why not?)</p>@
qu.9.2.answer=6@
qu.9.2.choice.1=differentiable and therefore continuous as well.@
qu.9.2.choice.2=continuous but differentiable only from the right.@
qu.9.2.choice.3=continuous but differentiable only from the left.@
qu.9.2.choice.4=discontinuous from both the left and the right.@
qu.9.2.choice.5=continuous and differentiable from the left only.@
qu.9.2.choice.6=continuous and differentiable from the right only.@
qu.9.2.fixed=@

qu.9.3.mode=Multiple Choice@
qu.9.3.name=Cts&Diff@
qu.9.3.comment=<p>$plot</p>@
qu.9.3.editing=useHTML@
qu.9.3.solution=@
qu.9.3.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$f="x^2-($a)^2+($b)";
$g="2*($a)*x-2*($a)^2+($b)";
$m=maple("f:=piecewise(x<$a,$f,x>=$a,$g):
convert(f,string), MathML[ExportPresentation](f)");
$F=switch(0,$m);
$displayF=switch(1,$m);
$plot=plotmaple("plot($F,x=-6..6,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.3.uid=290b53f5-6da5-4e9a-933a-8f20e7fc8e87@
qu.9.3.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayF, then at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is</p>
<p>&nbsp;</p>
<p>(Note that the option "differentiable but <strong>NOT </strong>continuous" is <strong>NOT</strong> a choice. Why not?)</p>@
qu.9.3.answer=1@
qu.9.3.choice.1=differentiable and therefore continuous as well.@
qu.9.3.choice.2=continuous but differentiable only from the right.@
qu.9.3.choice.3=continuous but differentiable only from the left.@
qu.9.3.choice.4=discontinuous from both the left and the right.@
qu.9.3.choice.5=continuous and differentiable from the left only.@
qu.9.3.choice.6=continuous and differentiable from the right only.@
qu.9.3.fixed=@

qu.9.4.mode=Multiple Choice@
qu.9.4.name=DisctsSin(1/x)@
qu.9.4.comment=<p>$plot</p>@
qu.9.4.editing=useHTML@
qu.9.4.solution=@
qu.9.4.algorithm=$f="sin(1/x)";
$m = maple("f:=piecewise(x<0,$f,x=0,0,x>0,sin(1/x)):
convert(f,string), MathML[ExportPresentation](f)");
$F=switch(0,$m);
$displayF=switch(1,$m);
$plot=plotmaple("plot($F,x=-6..6,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.4.uid=a7f6ec68-f7e5-4050-944f-a60649891f13@
qu.9.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continuity and Differentiability;
  Difficulty=Easy;
@
qu.9.4.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayF, then at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mn>0</mn></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is</p>
<p>&nbsp;</p>
<p>(Note that the option "differentiable but <strong>NOT </strong>continuous" is <strong>NOT</strong> a choice. Why not?)</p>@
qu.9.4.answer=4@
qu.9.4.choice.1=differentiable and therefore continuous as well.@
qu.9.4.choice.2=continuous but differentiable only from the right.@
qu.9.4.choice.3=continuous but differentiable only from the left.@
qu.9.4.choice.4=discontinuous from both the left and the right.@
qu.9.4.choice.5=continuous and differentiable from the left only.@
qu.9.4.choice.6=continuous and differentiable from the right only.@
qu.9.4.fixed=@

qu.9.5.mode=Multiple Choice@
qu.9.5.name=Discts@
qu.9.5.comment=<p>$plot</p>@
qu.9.5.editing=useHTML@
qu.9.5.solution=@
qu.9.5.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$f="x^2-($a)^2+($b)-5";
$g="2*($a)*x-2*($a)^2+($b)";
$m=$b-2;
$n=maple("f:=piecewise(x<$a,$f,x=$a,$m,x>$a,$g):
convert(f,string), MathML[ExportPresentation](f)");
$F=switch(0,$n);
$displayF=switch(1,$n);
$plot=plotmaple("plot($F,x=-6..6,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.5.uid=47181e48-f4f8-478a-b67b-5136e02ab992@
qu.9.5.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Continuity and Differentiability;
  Sub-Topic=Continuity and Differentiability;
  Difficulty=Easy;
@
qu.9.5.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$displayF, then at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is</p>
<p>&nbsp;</p>
<p>(Note that the option "differentiable but <strong>NOT </strong>continuous" is <strong>NOT</strong> a choice. Why not?)</p>@
qu.9.5.answer=4@
qu.9.5.choice.1=differentiable and therefore continuous as well.@
qu.9.5.choice.2=continuous but differentiable only from the right.@
qu.9.5.choice.3=continuous but differentiable only from the left.@
qu.9.5.choice.4=discontinuous from both the left and the right.@
qu.9.5.choice.5=continuous and differentiable from the left only.@
qu.9.5.choice.6=continuous and differentiable from the right only.@
qu.9.5.fixed=@

