qu.1.topic=ParametrizeCircle@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=Parametrize2.5Circle@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(2,8);
$pa=plotmaple("plot([$a*sin(t),$a*cos(t),t=Pi/2..Pi],x=-$a..$a,y=-$a..$a,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.1.1.uid=b557f6e0-a814-4892-baf3-f0bb4ec557b4@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Circle;
@
qu.1.1.question=<p>Which of the following pairs of parametric equations draws the circle of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></mstyle></math> clockwise to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>?</p>
<p>$pa</p>@
qu.1.1.answer=1@
qu.1.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&pi;</mi></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&pi;</mi></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=Parametrize3/4Circle@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(2,8);
$pa=plotmaple("plot([$a*cos(t),$a*sin(t),t=0..3*Pi/2],x=-$a..$a,y=-$a..$a,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.1.2.uid=91ccd135-02fd-48eb-98a9-a02b6a138011@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Circle;
@
qu.1.2.question=<p>Which of the following pairs of parametric equations draws the circle of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></mstyle></math> counter-clockwise to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mstyle></math>?</p>
<p>$pa</p>@
qu.1.2.answer=1@
qu.1.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.fixed=@

qu.2.topic=ParEquDer@

qu.2.1.question=<p>Find<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>, where $displayf and $displayg.</p>@
qu.2.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=simplify(diff(diff($g,t)/diff($f,t),t)/diff($f,t))@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=2ndDer@
qu.2.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dy</mi><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></mfenced><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>= $step</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><msup><mi>y</mi><mrow><mi></mi></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi><msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&prime;</mi></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></mfenced><mrow><mfrac><mi>dx</mi><mrow><mi>dt</mi></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>= $step2</p>
<p>&nbsp;=$ans</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(2,5);
$n=rint(2,10);
$z1=rint(4);
$z2=rint(4);
condition:not(eq($z1,$z2));
$f=switch($z1,"$a*t^$n","sin($a*t)","cos($a*t)","exp($a*t)");
$g=switch($z2,"$a*t^$n","sin($a*t)", "cos($a*t)","exp($a*t)");
$M=maple("
MathML[ExportPresentation](x=$f),
MathML[ExportPresentation](y=$g),
MathML[ExportPresentation]((diff($g,t))/(diff($f,t))),
MathML[ExportPresentation](diff(diff($g,t)/diff($f,t),t)/diff($f,t)),
MathML[ExportPresentation](simplify(diff(diff($g,t)/diff($f,t),t)/diff($f,t)))
");
$displayf=switch(0,$M);
$displayg=switch(1,$M);
$step=switch(2,$M);
$step2=switch(3,$M);
$ans=switch(4,$M);@
qu.2.1.uid=29b3f7eb-ef4a-4115-b712-f4e6c2e90681@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Derivatives;
@

qu.2.2.question=<p>Find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></mstyle></math> where $displayf and $displayg.</p>@
qu.2.2.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.2.allow2d=1@
qu.2.2.maple_answer=diff($g,t)/diff($f,t)@
qu.2.2.type=formula@
qu.2.2.mode=Maple@
qu.2.2.name=(dy/dt)/(dx/dt)@
qu.2.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></mfenced><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$ans</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(2,5);
$n=rint(2,10);
$an=$a*$n;
$n1=$n-1;
$z1=rint(8);
$z2=rint(8);
condition:not(eq($z1,$z2));
$f=switch($z1,"$a*t^$n","sin($a*t)","tan($a*t)","ln($a*t+$n)",
"sinh($a*t)","cosh($a*t)","cos($a*t)","exp($a*t)");
$df=switch($z1,"$an*t^$n1","$a*cos($a*t)","$a*(sec($a*t))^2","$a/($a*t+$n)","$a*cosh($a*t)","$a*sinh($a*t)","-$a*sin($a*t)","$a*e^($a*t)");
$g=switch($z2,"$a*t^$n","sin($a*t)","tan($a*t)","ln($a*t+$n)",
"sinh($a*t)","cosh($a*t)","cos($a*t)","exp($a*t)");
$dg=switch($z2,"$an*t^$n1","$a*cos($a*t)","$a*(sec($a*t))^2","$a/($a*t+$n)","$a*cosh($a*t)","$a*sinh($a*t)","-$a*sin($a*t)","$a*e^($a*t)");
$M=maple("
MathML[ExportPresentation](x=$f),
MathML[ExportPresentation](y=$g),
MathML[ExportPresentation](($dg)/($df))
");
$displayf=switch(0,$M);
$displayg=switch(1,$M);
$ans=switch(2,$M);@
qu.2.2.uid=64b0c5ae-9353-4fc8-a315-a5f054094558@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Derivatives;
@

qu.3.topic=ParametrizeEllipse@

qu.3.1.mode=Non Permuting Multiple Choice@
qu.3.1.name=Parametrize1.25Ellipse@
qu.3.1.comment=@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(2,8);
$b=rint(2,8);
$as=$a^2;
$bs=$b^2;
condition:ne($a,$b);
$pa=plotmaple("plot([-$a*cos(t),$b*sin(t),t=-Pi/2..Pi],x=-$a..$a,y=-$b..$b,scaling=constrained,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.3.1.uid=2e1a952d-0f16-48eb-bad0-0678d55dabbd@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Ellipse;
@
qu.3.1.question=<p>Which of the following pairs of parameric equations draws the&nbsp; ellipse <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow><mi mathvariant='normal'>$as</mi></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mi mathvariant='normal'>$bs</mi></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math></p>
<p>clockwise from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></mstyle></math>?</p>
<p>$pa<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math></p>@
qu.3.1.answer=1@
qu.3.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&pi;</mi></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.3.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&pi;</mi></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.3.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&pi;</mi></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.3.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.3.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></mstyle></math>@
qu.3.1.fixed=@

qu.3.2.mode=Non Permuting Multiple Choice@
qu.3.2.name=Parametrize1/2Ellipse@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(2,8);
$b=rint(2,8);
$as=$a^2;
$bs=$b^2;
condition:ne($a,$b);
$pa=plotmaple("plot([$a*cos(t),$b*sin(t),t=Pi/2..3*Pi/2],x=-$a..$a,y=-$b..$b,scaling=constrained,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.3.2.uid=0ed732ad-f0d2-406f-8800-f5652ad16dbb@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Ellipse;
@
qu.3.2.question=<p>Which of the following pairs of parametric equations draws the ellipse<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math></p>
<p>counter-clockwise from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mstyle></math>?</p>
<p>$pa</p>@
qu.3.2.answer=1@
qu.3.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.3.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></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'/></mrow></mstyle></math>@
qu.3.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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></mrow></mstyle></math>@
qu.3.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&pi;</mi></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.3.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.3.2.fixed=@

qu.4.topic=ParEquXIntercepts@

qu.4.1.question=<p>State the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math> intercept(s) using set notation, that is, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math>, for the parametric equations</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayx and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;$displayy.</p>
<p>(Some answers involve "ln". Don't use absolute value signs in your answer if they are not necessary. Use exp(x) for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>)</p>@
qu.4.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.4.1.allow2d=0@
qu.4.1.maple_answer={$va,$vb}@
qu.4.1.type=maple@
qu.4.1.mode=Maple@
qu.4.1.name=x intercepts@
qu.4.1.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math>intercepts, set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>&nbsp;and solve.</p>
<p>$displayy <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>&nbsp;</p>
<p>therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>.</p>
<p>Corresponding to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>&nbsp;we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math>&nbsp;intercept <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$va</mi></mrow></mstyle></math>.</p>
<p>Corresponding to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>intercept <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$vb</mi></mrow></mstyle></math>.</p>
<p>Enter your answer as</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='{' close='}' separators=','><mrow><mi mathvariant='normal'>$va</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$vb</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(1,10);
$b=rint(1,10);
condition:not(eq($a,$b));
$z1=rint(5);
$z2=rint(3);
$x=switch($z1,"t^2","sqrt(t)","sin(t)","ln(abs(t))","exp(t)");
$y=switch($z2,"t^2-($a+$b)*t+$a*$b","(t-$b)*sinh(t-$a)",
"(t-$b)*(t-$a)^(1/3)");
$M=maple("
MathML[ExportPresentation]($x),
MathML[ExportPresentation]($y),
convert(eval(subs(t=$a,$x)),string),
convert(eval(subs(t=$b,$x)),string)
");
$displayx=switch(0,$M);
$displayy=switch(1,$M);
$va=switch(2,$M);
$vb=switch(3,$M);@
qu.4.1.uid=fcf18097-d49e-4951-a5ed-11e030161cb3@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=x intercepts;
@

qu.4.2.question=<p>State the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math> intercept(s) using set notation, that is, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math>, for the parametric equations</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayx and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;$displayy.</p>
<p>(Some answers involve "ln". Don't use absolute value signs in your answer if they are not necessary. Use exp(x) for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>)</p>@
qu.4.2.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.4.2.allow2d=0@
qu.4.2.maple_answer={$va,$vb}@
qu.4.2.type=maple@
qu.4.2.mode=Maple@
qu.4.2.name=x intercepts (-a, b)@
qu.4.2.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math>intercepts, set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>&nbsp;and solve.</p>
<p>$displayy <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>&nbsp;</p>
<p>therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>.</p>
<p>Corresponding to&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>&nbsp;we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math>&nbsp;intercept <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$va</mi></mrow></mstyle></math>.</p>
<p>Corresponding to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>intercept <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$vb</mi></mrow></mstyle></math>.</p>
<p>Enter your answer as</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='{' close='}' separators=','><mrow><mi mathvariant='normal'>$va</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$vb</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(1,10);
$b=rint(1,10);
condition:not(eq($a,$b));
$z1=rint(4);
$z2=rint(3);
$x=switch($z1,"t^2","sin(t)","ln(abs(t))","exp(t)");
$y=switch($z2,"t^2-($b-$a)*t-$a*$b","(t-$b)*sinh(t+$a)",
"(t-$b)*(t+$a)^(1/3)");
$M=maple("
MathML[ExportPresentation]($x),
MathML[ExportPresentation]($y),
convert(eval(subs(t=-$a,$x)),string),
convert(eval(subs(t=$b,$x)),string)
");
$displayx=switch(0,$M);
$displayy=switch(1,$M);
$va=switch(2,$M);
$vb=switch(3,$M);@
qu.4.2.uid=047bf29b-2987-4cb0-9779-24d39ef57a92@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=x intercepts;
@

qu.5.topic=ParEquYIntercepts@

qu.5.1.question=<p>State the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math> intercept(s) using set notation, that is, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math>, for the parametric equations</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayx and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;$displayy.</p>
<p>(Some questions involve "ln". Don't use absolute value if it is not necessary. Remember to use exp(x) for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>)</p>@
qu.5.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.5.1.allow2d=0@
qu.5.1.maple_answer={$va,$vb}@
qu.5.1.type=maple@
qu.5.1.mode=Maple@
qu.5.1.name=y intercepts@
qu.5.1.comment=<p>To find y intercepts, set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>and solve.</p>
<p>$displayx = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math></p>
<p>Therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>&nbsp;or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>.</p>
<p>Corresponding to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>we have y intercept <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$va</mi></mrow></mstyle></math>.</p>
<p>Corresponding to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>we have y intercept <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$vb</mi></mrow></mstyle></math>.</p>
<p>Enter your answer as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='{' close='}' separators=','><mrow><mi mathvariant='normal'>$va</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$vb</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(1,10);
$b=rint(1,10);
condition:not(eq($a,$b));
$z1=rint(5);
$z2=rint(3);
$y=switch($z1,"t^2","sqrt(t)","ln(abs(t))","sin(t)","exp(t)");
$x=switch($z2,"t^2-($a+$b)*t+$a*$b","(t-$b)*sinh(t-$a)",
"(t-$b)*(t-$a)^(1/3)");
$M=maple("
MathML[ExportPresentation]($y),
MathML[ExportPresentation]($x),
convert(eval(subs(t=$a,$y)),string),
convert(eval(subs(t=$b,$y)),string)
");
$displayy=switch(0,$M);
$displayx=switch(1,$M);
$va=switch(2,$M);
$vb=switch(3,$M);@
qu.5.1.uid=26d574d4-5172-4743-b368-80e72762dcc8@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=y intercepts;
@

qu.5.2.question=<p>State the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math> intercept(s) using set notation, that is, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math>, for the parametric equations</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayx and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;$displayy.</p>
<p>(Some questions involve "ln". Don't use absolute value if it is not necessary. Remember to use exp(x) for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>)</p>@
qu.5.2.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.5.2.allow2d=0@
qu.5.2.maple_answer={$va,$vb}@
qu.5.2.type=maple@
qu.5.2.mode=Maple@
qu.5.2.name=y intercepts (-a, b)@
qu.5.2.comment=<p>To find y intercepts, set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>and solve.</p>
<p>$displayx = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math></p>
<p>Therefore&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>&nbsp;or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>.</p>
<p>Corresponding to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>we have y intercept <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$va</mi></mrow></mstyle></math>.</p>
<p>Corresponding to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>we have y intercept <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$vb</mi></mrow></mstyle></math>.</p>
<p>Enter your answer as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='{' close='}' separators=','><mrow><mi mathvariant='normal'>$va</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$vb</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$a=rint(1,10);
$b=rint(1,10);
condition:not(eq($a,$b));
$z1=rint(4);
$z2=rint(3);
$y=switch($z1,"t^2","sin(t)","ln(abs(t))","exp(t)");
$x=switch($z2,"t^2-($b-$a)*t-$a*$b","(t-$b)*sinh(t+$a)",
"(t-$b)*(t+$a)^(1/3)");
$M=maple("
MathML[ExportPresentation]($y),
MathML[ExportPresentation]($x),
convert(eval(subs(t=-$a,$y)),string),
convert(eval(subs(t=$b,$y)),string)
");
$displayy=switch(0,$M);
$displayx=switch(1,$M);
$va=switch(2,$M);
$vb=switch(3,$M);@
qu.5.2.uid=0cf0763e-1e1a-4aae-93fc-2b405fad614c@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=y intercepts;
@

qu.6.topic=PEIncDec@

qu.6.1.mode=Multiple Choice@
qu.6.1.name=PEdecreasingA greater-than 0@
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);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.6.1.uid=d69698c7-c7fe-4685-8e87-5d9b6d8d9887@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Increasing/decreasing;
@
qu.6.1.question=<p>A pair of parametric equations <strong>is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math></strong>. The first derivative is given by $displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>intervals is the graph <strong>DECREASING</strong>, that is, when is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>decreasing as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math> increases?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be decreasing on an interval even if its derivative is 0 or undefined somewhere in the interval!</p>@
qu.6.1.answer=4@
qu.6.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mstyle></math>@
qu.6.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced></mrow></math>@
qu.6.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.6.1.fixed=@

qu.6.2.mode=Multiple Choice@
qu.6.2.name=PEincreasing(fractionexponents)@
qu.6.2.comment=@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(3,7,2);
$n=rint(3,5,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^(1/$m)*(t-$b)^(1/$n)/(t-$c)^(2/$p)";
$displayf=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.6.2.uid=f1ff575f-91bb-41ff-88da-8d2f87854bbd@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Increasing/decreasing;
@
qu.6.2.question=<p>A pair of parametric equations <strong>is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math></strong>. The first derivative is given by $displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>intervals is the graph <strong>INCREASING</strong>, that is, when is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math> increasing as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math> increases?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be increasing on an interval even if its derivative is 0 or undefined somewhere in the interval!</p>@
qu.6.2.answer=1@
qu.6.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.6.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mrow></mrow><mfenced><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=']' separators=','><mrow><mo separator='true' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&infin;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.6.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo></mrow></math>@
qu.6.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced></mrow></math>@
qu.6.2.fixed=@

qu.6.3.mode=Multiple Choice@
qu.6.3.name=PEincreasingA greater-than 0@
qu.6.3.comment=@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.6.3.uid=9edd1e51-c678-47a3-8b7d-501e9160157f@
qu.6.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Increasing/decreasing;
@
qu.6.3.question=<p>A pair of parametric equations <strong>is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>.</strong> The first derivative is given by $displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>intervals is the graph <strong>INCREASING</strong>, that is, when is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math> increasing as<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math>increases?</p>
<p><strong>Note/Hint</strong>: A function can still be increasing on an interval even if its derivative is 0 or undefined somewhere in the interval!</p>@
qu.6.3.answer=1@
qu.6.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.6.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=']' separators=','><mrow><mo separator='true' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&InvisibleTimes;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&infin;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.6.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.6.3.fixed=@

qu.6.4.mode=Multiple Choice@
qu.6.4.name=PEincreasingA less-than 0@
qu.6.4.comment=@
qu.6.4.editing=useHTML@
qu.6.4.solution=@
qu.6.4.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="-$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.6.4.uid=65298320-8b9c-4238-8084-1b013d63a001@
qu.6.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Increasing/decreasing;
@
qu.6.4.question=<p>A pair of parametric equations <strong>is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math></strong>. The first derivative is given by $displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi></mrow></mstyle></math> intervals is the graph INCREASING, that is, when is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math> increasing as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math> increases?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be increasing on an interval even if its derivative is 0 or undefined somewhere in the interval!</p>@
qu.6.4.answer=1@
qu.6.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mrow></mfenced></mrow></math>@
qu.6.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo></mrow></math>@
qu.6.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mstyle></math>@
qu.6.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo></mrow></mrow></math>@
qu.6.4.fixed=@

qu.6.5.mode=Multiple Choice@
qu.6.5.name=PEdecreasingA less-than 0@
qu.6.5.comment=@
qu.6.5.editing=useHTML@
qu.6.5.solution=@
qu.6.5.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="-$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.6.5.uid=e2af8055-babe-4419-90ea-97ba6b42ee00@
qu.6.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Increasing/decreasing;
@
qu.6.5.question=<p>A pair of parametric equations is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>. The first derivative is given by $displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math> intervals is the graph <strong>DECREASING</strong>, that is, when is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math> decreasing as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math> increases?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be decreasing on an interval even if its derivative is 0 or undefined somewhere in the interval!</p>@
qu.6.5.answer=1@
qu.6.5.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.6.5.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.5.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced></mrow></math>@
qu.6.5.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced></mrow></math>@
qu.6.5.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.6.5.fixed=@

qu.6.6.mode=Multiple Choice@
qu.6.6.name=PEdecreasing(fractionexponents)@
qu.6.6.comment=@
qu.6.6.editing=useHTML@
qu.6.6.solution=@
qu.6.6.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(3,7,2);
$n=rint(3,5,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^(1/$m)*(t-$b)^(2/$n)/(t-$c)^(1/$p)";
$displayf=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.6.6.uid=a290bf21-a340-48bc-a96c-124ae655b798@
qu.6.6.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Increasing/decreasing;
@
qu.6.6.question=<p>A pair of parametric equations <strong>is defined for all real numbers</strong> <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>. The first derivative is given by $displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi></mrow></mstyle></math> intervals is the graph <strong>DECREASING</strong>, that is, when is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>decreasing as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math> increases?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be decreasing on an interval even if its derivative is 0 or undefined somewhere in the interval!</p>@
qu.6.6.answer=5@
qu.6.6.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.6.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.6.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mstyle></math>@
qu.6.6.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.6.6.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced></mrow></math>@
qu.6.6.fixed=@

qu.7.topic=HorVerTangents@

qu.7.1.mode=Multiple Choice@
qu.7.1.name=VTat$b@
qu.7.1.comment=<p>Vertical tangents occur when <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&plusmn;</mo></mrow><mrow><mi>&infin;</mi></mrow></mrow></mstyle></math>.</p>@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.7.1.uid=5012fa3b-7995-475a-a59e-efc60f9aa712@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Horizontal/Vertical tangents;
@
qu.7.1.question=<p>A pair of parametric equations using parameter <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='italic' >t</mi></mrow></math> is defined for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='italic' >t</mi><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&isin;</mo></mrow><mrow><mi    mathvariant='italic' >&reals;</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&period;</mo></mrow></mrow></math>The first derivative is given by $displayf. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='italic' >t</mi><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$c</mi></mrow></math> , we have&nbsp;</p>@
qu.7.1.answer=1@
qu.7.1.choice.1=a vertical tangent@
qu.7.1.choice.2=a horizontal tangent@
qu.7.1.choice.3=a horizontal asymptote@
qu.7.1.choice.4=a vertical asymptote@
qu.7.1.choice.5=an undefined relation since we are dividing by 0@
qu.7.1.fixed=@

qu.7.2.mode=Multiple Choice@
qu.7.2.name=HTat-$a,$b@
qu.7.2.comment=<p>Horizontal tangents occur when <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>.</p>@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.7.2.uid=c0340b57-b8a6-4ffa-adc8-df8d9faa02bf@
qu.7.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus II;
  Difficulty=Easy;
  Topic=Parametric equations;
  Sub-Topic=Horizontal/Vertical tangents;
@
qu.7.2.question=<p>A pair of parametric equations using parameter <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='italic' >t</mi></mrow></math> is defined for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='italic' >t</mi><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&isin;</mo></mrow><mrow><mi    mathvariant='italic' >&reals;</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&period;</mo></mrow></mrow></math>The first derivative is given by $displayf. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>&minus;</mi><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='italic' >t</mi><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$b</mi></mrow></math> , we have</p>
<p>&nbsp;</p>@
qu.7.2.answer=1@
qu.7.2.choice.1=horizontal tangents.@
qu.7.2.choice.2=vertical tangents.@
qu.7.2.choice.3=a horizontal tangent (<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>) and a vertical tangent (<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>).@
qu.7.2.choice.4=a vertical tangent (<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>) and a horizontal tangent (<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>).@
qu.7.2.choice.5=horizontal asymptotes.@
qu.7.2.fixed=@

qu.8.topic=PEasymptotes@

qu.8.1.question=<p>Vertical asymptotes are finite <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math> values, that is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math> approaches either plus or minus infinity.</p>
<p>List in set notation, that is, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math>, the value(s) of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>a</mi></mrow></mstyle></math> for the relation given by</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayx and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;$displayy.</p>@
qu.8.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.8.1.allow2d=0@
qu.8.1.maple_answer={$va,$vb}@
qu.8.1.type=maple@
qu.8.1.mode=Maple@
qu.8.1.name=VA@
qu.8.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>t</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>x</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mn>0</mn></mrow></mstyle></math></p>
<p>Therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>is a vertical asymptote.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>t</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi mathvariant='normal'>$a</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p>Therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mrow></mstyle></math> is a vertical asymptote.</p>
<p>Enter your answer as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='{' close='}' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.8.1.editing=useHTML@
qu.8.1.solution=@
qu.8.1.algorithm=$a=rint(1,10);
$x="1/t";
$y="ln(t-$a)";
$displayx=mathml("$x");
$displayy=mathml("$y");
$va="1/$a";
$vb=0;@
qu.8.1.uid=fc15622b-6205-4c36-b5da-2eda0e8a7bea@
qu.8.1.info=  Author=Jack Weiner, Gord Clement;
  Difficuly=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Vertical asymptotes;
@

qu.8.2.question=<p>Horizontal asymptotes are finite&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math> values, that is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi></mrow></mstyle></math>approaches either plus or minus infinity.</p>
<p>List in set notation, that is, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math>, the value(s) of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>a</mi></mrow></mstyle></math> for the relation given by</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayx and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayy.</p>@
qu.8.2.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.8.2.allow2d=0@
qu.8.2.maple_answer={$va}@
qu.8.2.type=maple@
qu.8.2.mode=Maple@
qu.8.2.name=HA@
qu.8.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>t</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>y</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi mathvariant='normal'>$va</mi></mrow></mstyle></math></p>
<p>Therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$va</mi></mrow></mstyle></math> is a vertical asymptote.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>&nbsp;never tends to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow></mrow></mstyle></math>, therefore this is our only vertical asymptote.</p>
<p>Enter your answer as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='{' close='}' separators=','><mrow><mi mathvariant='normal'>$va</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.8.2.editing=useHTML@
qu.8.2.solution=@
qu.8.2.algorithm=$a=rint(1,10);
$y="2*$a+ 1/(t+$a)";
$x="e^(t)";
$displayx=mathml("$x");
$displayy=mathml("$y");
$va=2*$a;@
qu.8.2.uid=bfebbbfa-e1fc-46ff-88d9-f7e1909533d1@
qu.8.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Horizontal asymptotes;
@

qu.9.topic=PEConcave@

qu.9.1.mode=Multiple Choice@
qu.9.1.name=PEConcaveUp(fractionexponents)@
qu.9.1.comment=@
qu.9.1.editing=useHTML@
qu.9.1.solution=@
qu.9.1.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(3,7,2);
$n=rint(3,5,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^(1/$m)*(t-$b)^(1/$n)/(t-$c)^(2/$p)";
$displayf=maple("printf(MathML[ExportPresentation]($f))");@
qu.9.1.uid=c6b95f2e-ee74-47a2-93dc-4ea72146aef4@
qu.9.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Concave up/down;
@
qu.9.1.question=<p>A pair of parametric equations is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math> The second derivative is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>d</mi><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><msup><mi>dx</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi></mrow></mstyle></math>intervals is the graph concave up?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be concave up on an interval even if its second derivative is 0 or undefined somewhere in the interval!</p>@
qu.9.1.answer=1@
qu.9.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.9.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mrow></mrow><mfenced><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=']' separators=','><mrow><mo separator='true' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&infin;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.9.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo></mrow></math>@
qu.9.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced></mrow></math>@
qu.9.1.fixed=@

qu.9.2.mode=Multiple Choice@
qu.9.2.name=PEConcaveDown(fractionexponents)@
qu.9.2.comment=@
qu.9.2.editing=useHTML@
qu.9.2.solution=@
qu.9.2.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(3,7,2);
$n=rint(3,5,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^(1/$m)*(t-$b)^(2/$n)/(t-$c)^(1/$p)";
$displayf=maple("printf(MathML[ExportPresentation]($f))");@
qu.9.2.uid=b1162f94-b470-44db-91f4-bada15ce4fc5@
qu.9.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Concave up/down;
@
qu.9.2.question=<p>A pair of parametric equations is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math> The second derivative is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>d</mi><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><msup><mi>dx</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi></mrow></mstyle></math> intervals is the graph concave down?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be concave down on an interval even if its second derivative is 0 or undefined somewhere in the interval!</p>@
qu.9.2.answer=5@
qu.9.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mstyle></math>@
qu.9.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced></mrow></math>@
qu.9.2.fixed=@

qu.9.3.mode=Multiple Choice@
qu.9.3.name=PEConcaveDownA greater-than 0@
qu.9.3.comment=@
qu.9.3.editing=useHTML@
qu.9.3.solution=@
qu.9.3.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation]($f))");@
qu.9.3.uid=faf525ef-c89d-44f0-bc55-c6a4eaaf1748@
qu.9.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Concave up/down;
@
qu.9.3.question=<p>A pair of parametric equations is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>. The second derivative is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>d</mi><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><msup><mi>dx</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi></mrow></mstyle></math> intervals is the graph concave down?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be concave down on an interval even if its second derivative is 0 or undefined somewhere in the interval!</p>@
qu.9.3.answer=4@
qu.9.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mstyle></math>@
qu.9.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced></mrow></math>@
qu.9.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.9.3.fixed=@

qu.9.4.mode=Multiple Choice@
qu.9.4.name=PEConcaveDownA less-than 0@
qu.9.4.comment=@
qu.9.4.editing=useHTML@
qu.9.4.solution=@
qu.9.4.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="-$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation]($f))");@
qu.9.4.uid=c9041251-3a90-4d92-83a2-066dfd9d9ef2@
qu.9.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Concave up/down;
@
qu.9.4.question=<p align="left">A pair of parametric equations is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>. The second derivative is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>d</mi><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><msup><mi>dx</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi></mrow></mstyle></math>intervals is the graph concave down?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be concave down on an interval even if its second derivative is 0 or undefined somewhere in the interval!</p>@
qu.9.4.answer=1@
qu.9.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.9.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mstyle></math>@
qu.9.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced></mrow></math>@
qu.9.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.9.4.fixed=@

qu.9.5.mode=Multiple Choice@
qu.9.5.name=PEConcaveUpA less-than 0@
qu.9.5.comment=@
qu.9.5.editing=useHTML@
qu.9.5.solution=@
qu.9.5.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="-$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation]($f))");@
qu.9.5.uid=1d4a9659-6906-4609-823e-f4ec80441b9b@
qu.9.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Concave up/down;
@
qu.9.5.question=<p>A pair of parametric equations is defined for all real numbers <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>. The second derivative is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>d</mi><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><msup><mi>dx</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi></mrow></mstyle></math> intervals is the graph concave up?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be concave up on an interval even if its second derivative is 0 or undefined somewhere in the interval!</p>@
qu.9.5.answer=1@
qu.9.5.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mrow></mfenced></mrow></math>@
qu.9.5.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mstyle></math>@
qu.9.5.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo></mrow></math>@
qu.9.5.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.5.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo></mrow></mrow></math>@
qu.9.5.fixed=@

qu.9.6.mode=Multiple Choice@
qu.9.6.name=PEConcaveUpA greater-than 0@
qu.9.6.comment=@
qu.9.6.editing=useHTML@
qu.9.6.solution=@
qu.9.6.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(1,5,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(t+$a)^$m*(t-$b)^$n/(t-$c)^$p";
$displayf=maple("printf(MathML[ExportPresentation]($f))");@
qu.9.6.uid=7cfc3128-ac05-4a24-abdd-62c7ffe699ce@
qu.9.6.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Parametric equations;
  Sub-Topic=Concave up/down;
@
qu.9.6.question=<p>A pair of parametric equations <strong>is defined for all real numbers</strong> <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    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>t</mi></mrow></mstyle></math>. The second derivative is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>d</mi><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><msup><mi>dx</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$displayf. For what <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi></mrow></mstyle></math> intervals is the graph concave up?</p>
<p>&nbsp;</p>
<p><strong>Note/Hint</strong>: A function can still be concave up on an interval even if its second derivative is 0 or undefined somewhere in the interval!</p>@
qu.9.6.answer=1@
qu.9.6.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.9.6.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.6.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=']' separators=','><mrow><mo separator='true' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&infin;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.9.6.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mfenced><mrow><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.9.6.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&InvisibleTimes;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$c</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow></mrow></mrow></math>@
qu.9.6.fixed=@

