qu.1.topic=x,yIntercepts@

qu.1.1.question=<p>State the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math> intercept(s) using set notation, that is, { }, for the function $F. Enter N (for NONE!) if there aren't any.</p>@
qu.1.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.1.1.allow2d=0@
qu.1.1.maple_answer=$ANS@
qu.1.1.type=maple@
qu.1.1.mode=Maple@
qu.1.1.name=x intercepts@
qu.1.1.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>intercepts, set <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></math>and solve.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(1,10);
$b=rint(1,10);
condition:eq(gcd($a,$b),1);
$as=($a)^2;
$bs=($b)^2;
$f1="($a)*x^4+($b)*x^3";
$f2="($as)*x^4-($bs)*x^2";
$f3="($a)*x^(5/3)-($b)*x^(2/3)";
$f4="x^(2*($a))+($b)";
$z=rint(4);
$f=switch($z,"$f1","$f2","$f3", "$f4");
$F=maple("printf(MathML[ExportPresentation](f(x)=$f))");
$ANS=switch($z, "{0,-($b)/($a)}", "{0,abs(($b)/($a)),-abs(($b)/($a))}", "{0,($b)/($a)}","N");@
qu.1.1.uid=b5d934c5-91f3-4833-89f5-0c33191b0eea@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Intercepts;
@

qu.1.2.question=<p>State the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>intercept for the function $F. Enter N (for NONE!) if there isn't one.</p>@
qu.1.2.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.1.2.allow2d=0@
qu.1.2.maple_answer=$ANS@
qu.1.2.type=maple@
qu.1.2.mode=Maple@
qu.1.2.name=y intercept@
qu.1.2.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>intercepts, set <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.</mn></mrow></math></p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(1,10);
$b=rint(1,10);
$as=($a)^2;
$bs=($b)^2;
$c=rint(-3,3);
condition:eq(gcd($a,$b),1);


$f1="($a)*x^4+($b)*x^3+($c)";
$f2="($as)*x^4-($bs)*x^2+($c)";
$f3="($a)*x^(5/3)-($b)*x^(2/3)+($c)";
$f4="sqrt(x-($a))";
$z=rint(4);
$f=switch($z,"$f1","$f2","$f3","$f4");

$F=maple("printf(MathML[ExportPresentation](f(x)=$f))");

$ANS=switch($z, "$c","$c","$c",N);@
qu.1.2.uid=05e6aa55-ee15-4b27-ab34-513338f2642a@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Intercepts;
@

qu.2.topic=Asymptotes@

qu.2.1.mode=Equation@
qu.2.1.name=Horizontal Asymptotes Some@
qu.2.1.comment=<p>$plot</p>
<p>The horizontal asymptote is the line <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$ANS</mi></mrow></math>.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(1,4);
$z=rint(2);
$b=$a+$z;
$M=maple("
randomize(): 
f1 := 0:
f2 := 0: 
c1 := 0: 
c2 := 0:
while (evalb(simplify(f1-f2) = 0) or evalb(c1 = 0) or evalb(c2 = 0)) do 
f1 := randpoly(x, degree = $a, coeffs = rand(-5 .. 5)): 
f2 := randpoly(x, degree = $b, coeffs = rand(-3 .. 3)):
c1 := coeff(f1, x, $a): 
c2 := coeff(f2, x, $b):
end do:
convert((c1/c2),string),
MathML[ExportPresentation](f(x)=(sort(f1))/(sort(f2))),
convert((f1/f2),string)
");
$c=switch(0,$M);
$F=switch(1,$M);
$f=switch(2,$M);
$ANS=switch($z,"y=$c", "y=0");
$line=switch($z,"$c","0");
$plot=plotmaple("plot([$f,$line],x=-10..10,y=-20..20,thickness=2,color=[red,black], linestyle=[solid,dash],discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.1.uid=14a784c6-22a1-4b45-ac2a-fe29ef0a9ba2@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Asymptotes;
@
qu.2.1.question=<p>Horizontal asymptotes are equations of the form <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi></mrow></math>, which we find by taking the limit as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></math>and/or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow><mrow></mrow></math> .<strong>**</strong></p>
<p>Find the horizontal asymptote(s) for the function $F. Enter your answer as an equation. If there is more than one, separate your answers by a comma. If there isn't one, enter N for none.</p>
<p>&nbsp;</p>
<p>**Vertical asymptotes<strong> </strong>are finite <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>values that make <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>infinite.</p>
<p><strong>&nbsp;&nbsp;&nbsp; </strong>Horizontal asymptotes are finite <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>values that make <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>infinite.</p>@
qu.2.1.answer=$ANS@

qu.2.2.question=<p>Vertical asymptotes are finite <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math> values, that is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>a</mi></mrow></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math> approaches either plus or minus infinity.<strong>**</strong></p>
<p>List in set notation the equation(s) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>a</mi></mrow></math>that give the vertical asymptotes of $F .</p>
<p>eg.,<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='{' close='}' separators=','><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>5</mn></mrow></mfenced></mrow></math></p>
<p><strong>**Vertical Asymptotes are finite <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>values that make <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>infinte.</strong></p>
<p><strong>&nbsp;&nbsp; Horizontal Asymptotes are finite <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>values that make <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>infinite.</strong></p>@
qu.2.2.maple=evalb(($ANSWER)=($RESPONSE));@
qu.2.2.allow2d=0@
qu.2.2.maple_answer={x=$a,x=$b}@
qu.2.2.type=maple@
qu.2.2.mode=Maple@
qu.2.2.name=Vertical Asymptotes@
qu.2.2.comment=<p>$plot</p>
<p>Remember, <strong>vertical asymptotes</strong> are of the form <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>a</mi></mrow></math>.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(-5,5);
$b=rint($a+1,10);
$n=rint(2);
$f="1/(x-($b))^($n+1)/(x-($a))";
$F=maple("printf(MathML[ExportPresentation](y=$f))");
$plot=plotmaple("plot($f,x=-abs($a)-2..abs($b)+2,y=-10..10,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.2.uid=3c2a3df0-52d3-49f6-a20e-8ca45454c2ea@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Graph Sketching;
  Sub-Topic=Asymptotes;
@

qu.2.3.mode=Formula@
qu.2.3.name=Horizontal Asymptote None@
qu.2.3.comment=<p>$plot</p>
<p>There is no horizontal asymptote this time!</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$a=rint(3,6);
$b=rint($a-2,$a);
$M=maple("
randomize(): 
f1 := 0:
f2 := 0: 
c1 := 0: 
c2 := 0:
while (evalb(simplify(f1-f2) = 0) or evalb(c1 = 0) or evalb(c2 = 0)) do 
f1 := randpoly(x, degree = $a, coeffs = rand(-5 .. 5)): 
f2 := randpoly(x, degree = $b, coeffs = rand(-3 .. 3)):
c1 := coeff(f1, x, $a): 
c2 := coeff(f2, x, $b):
end do:
MathML[ExportPresentation](f(x)=(sort(f1))/(sort(f2))),
convert((f1/f2),string)
");
$F=switch(0,$M);
$f=switch(1,$M);
$ANS=N;
$plot=plotmaple("plot($f,x=-10..10,y=-20..20,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.3.uid=0dd9cb67-648d-49a0-9195-370818698782@
qu.2.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Graph Sketching;
  Sub-Topic=Asymptotes;
@
qu.2.3.question=<p>Horizontal asymptotes are equations of the form <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi></mrow></math>which we find by letting <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mi>&infin;</mi></mrow></mrow></math>and/or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></math>.<strong>**</strong></p>
<p>Find the horizontal asymptote(s) for the function $F. Enter your answer as an equation. If there is more than one, separate your answers by a comma. If there isn't one, enter N for NONE!</p>
<p>&nbsp;</p>
<p><strong>**Vertical asymptotes </strong>are finite <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>values that make <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>infinite.</p>
<p><strong>Horizontal asymptotes</strong> are finite <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>values that make <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>infinite.</p>@
qu.2.3.answer=$ANS@

qu.3.topic=HorVerTangents@

qu.3.1.mode=Multiple Choice@
qu.3.1.name=VTat$b@
qu.3.1.comment=<p>Vertical tangents occur when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dx</mi></mrow></mfrac></mrow></math>&nbsp;approaches <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&plusmn;</mo></mrow><mrow><mi>&infin;</mi></mrow></mrow></math>.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(3,7,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(x+$a)^$m*(x-$b)^$n/(x-$c)^(1/$p)";
$F=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.3.1.uid=e6fb5a2d-a3fc-4f5c-b3ce-12b067782fde@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Horizontal/Vertical Tangents;
@
qu.3.1.question=<p>A function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>, whose domain is all real numbers, has first derivative$F. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi mathvariant='normal'>$c</mi></mrow></math> , we have . . .<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math></p>
<p>(Note: As you will discover later in Math 1210, because the term <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></math> is raised to the exponent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></math> or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>1</mn><mrow><mn>5</mn></mrow></mfrac></mrow></math> in the denominator, it must have been a factor in the numerator of the original function. Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi mathvariant='normal'>$c</mi></mrow></math> is in the domain of the function.)</p>@
qu.3.1.answer=1@
qu.3.1.choice.1=a vertical tangent.@
qu.3.1.choice.2=an undefined relation since we are dividing by 0.@
qu.3.1.choice.3=a horizontal tangent.@
qu.3.1.choice.4=a horizontal asymptote.@
qu.3.1.choice.5=a vertical asymptote.@
qu.3.1.fixed=@

qu.3.2.mode=Multiple Choice@
qu.3.2.name=HTat-$a,$b@
qu.3.2.comment=<p>Horizontal tangents occur when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>dy</mi><mrow><mi>dx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></math>.</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,7);
$m=rint(1,5,2);
$n=rint(2,6,2);
$p=rint(3,7,2);
condition:lt($b,$c);
$A=rint(2,6);
$f="$A*(x+$a)^$m*(x-$b)^$n/(x-$c)^(1/$p)";
$F=maple("printf(MathML[ExportPresentation](dy/dx=$f))");@
qu.3.2.uid=9b5ccff3-f99b-41ba-8f23-dab94a356961@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Horizontal/Vertical Tangents;
@
qu.3.2.question=<p>A function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>has first derivative $F. At <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></math> , we have</p>@
qu.3.2.answer=1@
qu.3.2.choice.1=horizontal tangents.@
qu.3.2.choice.2=vertical tangents.@
qu.3.2.choice.3=a horizontal tangent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math> and a vertical tangent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math>.@
qu.3.2.choice.4=a vertical tangent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math> and a horizontal tangent <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math>.@
qu.3.2.choice.5=horizontal asymptotes.@
qu.3.2.fixed=@

qu.4.topic=Inc A*B*C@

qu.4.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is increasing. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.4.1.maple=grade("$RESPONSE",$ANS);@
qu.4.1.allow2d=0@
qu.4.1.maple_answer=show($ANS);@
qu.4.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.1.type=maple@
qu.4.1.mode=Maple@
qu.4.1.name=f INC (x+a)^(n/3)@
qu.4.1.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is increasing! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.4.1.editing=useHTML@
qu.4.1.hint.1=Increasing is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow><mrow><mn>0</mn></mrow></math>. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup></mrow></math>is increasing on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> but has positive derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>.@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,3);
$f="surd((x-($a))^$n,3)*(x-($b))*(x-$c)";
$M=maple("
convert(int($f,x) ,string),
MathML[ExportPresentation]($f)
");
$F=switch(1,$M);
$fi=switch(0,$M);
$ANS=switch($n-1, '"[$a,$b] U [$c,infinity)"', '"(-infinity,$b] U [$c,infinity)"', '"[$a,$b] U [$c,infinity)"');
$plot=plotmaple("plot($fi,x=$a-2..$c+2,thickness=2),plotdevice='gif', 
plotoptions='height=250,width=250'");@
qu.4.1.uid=eb2689ec-27a3-486e-8603-0c31b3c9f87a@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Increase/Decrease;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
@

qu.4.2.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is increasing. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.4.2.maple=grade("$RESPONSE",$ANS);@
qu.4.2.allow2d=0@
qu.4.2.maple_answer=show($ANS);@
qu.4.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.2.type=maple@
qu.4.2.mode=Maple@
qu.4.2.name=f INC A<0@
qu.4.2.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is increasing! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.4.2.editing=useHTML@
qu.4.2.hint.1=Increasing is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mn>0</mn></mrow></math>. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup></mrow></math>is increasing on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> but has positive derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>.@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$A=rint(-3,0);
$n=rint(1,5);
$f="$A*(x-($a))^$n*(x-($b))*(x-$c)";
$M=maple("
convert(int($f,x),string),
MathML[ExportPresentation]($f)
");
$fi=switch(0,$M);
$F=switch(1,$M);
$ANS=switch($n-1, '"(-infinity,$a] U [$b,$c]"' , '"[$b,$c]"
' , '"(-infinity,$a] U [$b,$c]"' , '"[$b,$c]"
');
$plot=plotmaple("plot($fi,x=$a-2..$c+2,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.2.uid=5f17ab36-e953-49f9-99e8-ca0635222f83@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Increase/Decrease;
@

qu.4.3.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is increasing. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.4.3.maple=grade("$RESPONSE",$ANS);@
qu.4.3.allow2d=0@
qu.4.3.maple_answer=show($ANS);@
qu.4.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.3.type=maple@
qu.4.3.mode=Maple@
qu.4.3.name=f INC A*B*C@
qu.4.3.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is increasing! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.4.3.editing=useHTML@
qu.4.3.hint.1=Increasing is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo></mrow></math>0. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup></mrow></math>is increasing on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> but has positive derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>.@
qu.4.3.solution=@
qu.4.3.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,4);
$f="(x-($a))^$n*(x-($b))*(x-$c)";
$M=maple("
convert(int($f,x),string),
MathML[ExportPresentation]($f)
");
$fi=switch(0,$M);
$F=switch(1,$M);
$ANS=switch($n-1, '"[$a,$b] U [$c,infinity)"
' , '"(-infinity,$b] U [$c,infinity)"' ,'"[$a,$b] U [$c,infinity)"
');
$plot=plotmaple("plot($fi,x=$a-2..$c+2,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.3.uid=c4340411-b1dc-4374-ad4f-88c4299e0bb7@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Increase/Decrease;
  Course=Introduction to Calculus I;
@

qu.5.topic=Dec A*B*C@

qu.5.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is decreasing. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.5.1.maple=grade("$RESPONSE",$ANS);@
qu.5.1.allow2d=0@
qu.5.1.maple_answer=show($ANS);@
qu.5.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.5.1.type=maple@
qu.5.1.mode=Maple@
qu.5.1.name=f DEC (x+a)^(n/3)@
qu.5.1.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is decreasing! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.5.1.editing=useHTML@
qu.5.1.hint.1=Decreasing is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>0. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup></mrow></math>is decreasing on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></math> but has negative derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mn>0</mn></mrow></mfenced></mrow></math>.@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,3);
$M=maple("
f:=surd((x-($a))^$n,3)*(x-($b))*(x-$c):
f1:=int(f,x):
convert(f,string),
convert(f1,string),
MathML[ExportPresentation]($f)
");
$f=switch(0,$M);
$F=switch(2,$M);
$fi=switch(1,$M);
$ANS=switch($n-1,'"(-infinity,$a] U [$b,$c]"','"[$b,$c]"','"(-infinity,$a] U [$b,$c]"');
$plot=plotmaple("plot($fi,x=$a-2..$c+2,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.1.uid=c557ea53-b2c0-49b3-851f-98e1b2a939c4@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Increase/Decrease;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
@

qu.5.2.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is decreasing. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.5.2.maple=grade("$RESPONSE",$ANS);@
qu.5.2.allow2d=0@
qu.5.2.maple_answer=show($ANS);@
qu.5.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.5.2.type=maple@
qu.5.2.mode=Maple@
qu.5.2.name=f DEC A*B*C@
qu.5.2.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is decreasing! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.5.2.editing=useHTML@
qu.5.2.hint.1=Decreasing is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mn>0</mn></mrow></math>. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup></mrow></math>is decreasing on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></math> but has negative derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></math>.@
qu.5.2.solution=@
qu.5.2.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,5);
$f="(x-($a))^$n*(x-($b))*(x-$c)";
$M=maple("
convert(int($f,x),string),
MathML[ExportPresentation]($f)
");
$fi=switch(0,$M);
$F=switch(1,$M);
$ANS=switch($n-1, '"(-infinity,$a] U [$b,$c]"','"[$b,$c]"','"(-infinity,$a] U [$b,$c]"','"[$b,$c]"');
$plot=plotmaple("plot($fi,x=$a-2..$c+2,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.2.uid=cbb4caeb-7b26-450f-ab2a-95fe1b674352@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Increase/Decrease;
  Difficulty=Easy;
@

qu.6.topic=Inc A*B/C@

qu.6.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow></mrow><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><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is increasing. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>&nbsp;</p>@
qu.6.1.maple=grade("$RESPONSE",$ANS);@
qu.6.1.allow2d=0@
qu.6.1.maple_answer=show($ANS);@
qu.6.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.6.1.type=maple@
qu.6.1.mode=Maple@
qu.6.1.name=Inc (x+a)^n/(x-b)*(x-c)@
qu.6.1.comment=<p>Here is the plot of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is increasing. Does this match your choice of intervals?</p>
<p>$plot</p>@
qu.6.1.editing=useHTML@
qu.6.1.hint.1=Increasing is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mn>0</mn></mrow></math>. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup></mrow></math>is increasing on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> but has positive derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>.@
qu.6.1.hint.2=The derivative is undefined at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>, but because the exponent on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math>is between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>1</mn></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math> <strong>IS</strong> in the domain of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math>.@
qu.6.1.solution=@
qu.6.1.algorithm=$a=rint(-8,-1);
$b=rint(1,5);
$c=rint(6,10);
$n=rint(1,3);
$f="(x-($a))/surd((x-($b))^($n),3)*(x-($c))";
$M=maple("
convert(int($f,x),string),
MathML[ExportPresentation]($f)
");
$F=switch(1,$M);
$fi=switch(0,$M);
$ANS=switch($n-1, '"[$a,$b] U [$c,infinity)"' , '"(-infinity,$a] U [$c,infinity)"' );
$plot=plotmaple("plot($fi,x=$a-4..$c+4,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.6.1.uid=6d227a93-b609-4511-94cd-09590f462149@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Increase/Decrease;
  Difficulty=Easy;
@

qu.7.topic=Dec A*B/C@

qu.7.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow></mrow><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><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is decreasing. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>&nbsp;</p>@
qu.7.1.maple=grade("$RESPONSE",$ANS);@
qu.7.1.allow2d=0@
qu.7.1.maple_answer=show($ANS);@
qu.7.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.7.1.type=maple@
qu.7.1.mode=Maple@
qu.7.1.name=Dec (x+a)^n/(x-b)*(x-c)@
qu.7.1.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is decreasing. Does this match your choice of intervals?</p>
<p>$plot</p>@
qu.7.1.editing=useHTML@
qu.7.1.hint.1=Decreasing is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow><mrow><mn>0</mn></mrow></math>. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup></mrow></math>is decreasing on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></math> but has negative derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced></mrow></math>.@
qu.7.1.hint.2=The derivative is undefined at <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>, but because of the exponent on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math> <strong>IS</strong> in the domain of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math>.@
qu.7.1.solution=@
qu.7.1.algorithm=$a=rint(-8,-1);
$b=rint(1,5);
$c=rint(6,10);
$n=rint(1,3);
$M=maple("
f:=(x-($a))/surd((x-($b))^($n),3)*(x-($c)):
f1:=int(f,x):
convert(f,string),
convert(f1,string),
MathML[ExportPresentation]($f)
");
$f=switch(0,$M);
$F=switch(2,$M);
$fi=switch(1,$M);
$ANS=switch($n-1,'"(-infinity,$a]U[$b,$c]"','"[$a,$c]"');
$plot=plotmaple("plot($fi,x=$a-4..$c+4,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.7.1.uid=afc7193a-7596-4bf0-ac69-d9319373dc64@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Increase/Decrease;
@

qu.8.topic=ConcaveUp@

qu.8.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has second derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is concave up. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.8.1.maple=grade("$RESPONSE",$ANS);@
qu.8.1.allow2d=0@
qu.8.1.maple_answer=show($ANS);@
qu.8.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.8.1.type=maple@
qu.8.1.mode=Maple@
qu.8.1.name=f ConUp A*B*C A<0@
qu.8.1.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is concave up! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.8.1.editing=useHTML@
qu.8.1.hint.1=Concave up is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mn>0</mn></mrow></math>. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>5</mn></mrow></msup></mrow></math>is concave up on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> while it has positive second derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>.@
qu.8.1.solution=@
qu.8.1.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$A=rint(-3,0);
$n=rint(1,3);
$f="($A)*(x-($a))^$n*(x-($b))*(x-$c)";
$M=maple("
f1:=int($f,x):
f2:=int(f1,x):
convert(f2,string), MathML[ExportPresentation]($f)
");
$fi2=switch(0,$M);
$F=switch(1,$M);
$ANS=switch($n-1, '"(-infinity,$a] U [$b,$c]"', '"[$b,$c]"');
$plot=plotmaple("plot($fi2,x=$a-4..$c+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.8.1.uid=3f9df4fc-4aad-4d7d-b334-17d6ed3aa5c2@
qu.8.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Concavity;
@

qu.8.2.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has second derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is concave up. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.8.2.maple=grade("$RESPONSE",$ANS);@
qu.8.2.allow2d=0@
qu.8.2.maple_answer=show($ANS);@
qu.8.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.8.2.type=maple@
qu.8.2.mode=Maple@
qu.8.2.name=f ConUp A*B*C@
qu.8.2.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is concave up! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.8.2.editing=useHTML@
qu.8.2.hint.1=Concave up is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mn>0</mn></mrow></math>. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>5</mn></mrow></msup></mrow></math>is concave up on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> while it has positive second derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>.@
qu.8.2.solution=@
qu.8.2.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,4);
$f="(x-($a))^$n*(x-($b))*(x-$c)";
$M=maple("
f1:=int($f,x):
f2:=int(f1,x):
convert(f2,string), MathML[ExportPresentation]($f)
");
$F=switch(1,$M);
$fi2=switch(0,$M);
$ANS=switch($n-1, '"[$a,$b] U [$c,infinity)"' , '"(-infinity,$b] U [$c,infinity)"' , '"[$a,$b] U [$c,infinity)"');
$plot=plotmaple("plot($fi2,x=$a-4..$c+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.8.2.uid=1c52a3d7-8843-4d62-888c-0b098089db1f@
qu.8.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Concavity;
  Course=Introduction to Calculus I;
@

qu.8.3.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has second derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is concave up. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.8.3.maple=grade("$RESPONSE",$ANS);@
qu.8.3.allow2d=0@
qu.8.3.maple_answer=show($ANS);@
qu.8.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.8.3.type=maple@
qu.8.3.mode=Maple@
qu.8.3.name=f ConUp (x+a)^(n/3)@
qu.8.3.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is concave up! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.8.3.editing=useHTML@
qu.8.3.hint.1=Concave up is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>0. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>5</mn></mrow></msup></mrow></math>is concave up on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> but has positive second derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>.@
qu.8.3.solution=@
qu.8.3.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,3);
$M=maple("
f:=surd((x-($a))^$n,3)*(x-($b))*(x-$c):
f1:=int(f,x):
f2:=int(f1,x):

convert(f,string),
convert(f2, string),
MathML[ExportPresentation]($f)
");
$f=switch(0,$M);
$F=switch(2,$M);
$fi2=switch(1,$M);
$ANS=switch($n-1, '"[$a,$b] U [$c,infinity)"' , '"(-infinity,$b] U [$c,infinity)"' , '"[$a,$b] U [$c,infinity)"');
$plot=plotmaple("plot($fi2,x=$a-4..$c+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.8.3.uid=0d9d0c16-dc35-4e48-9208-39dc58faca26@
qu.8.3.info=  Author=Jack Weiner, Gord Clement;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Concavity;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
@

qu.9.topic=ConcaveDown@

qu.9.1.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has second derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is concave down. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.9.1.maple=grade("$RESPONSE",$ANS);@
qu.9.1.allow2d=0@
qu.9.1.maple_answer=show($ANS);@
qu.9.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.9.1.type=maple@
qu.9.1.mode=Maple@
qu.9.1.name=f ConDown (x+a)^(n/3)@
qu.9.1.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is concave down! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.9.1.editing=useHTML@
qu.9.1.hint.1=Concave down is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>0. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>5</mn></mrow></msup></mrow></math>is concave up on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> but has negative second derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>.@
qu.9.1.solution=@
qu.9.1.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,3);
$M=maple("
f:=surd((x-($a))^$n,3)*(x-($b))*(x-$c):
f1:=int(f,x):
f2:=int(f1,x):
convert(f,string),
convert(f1,string),
convert(f2,string),
MathML[ExportPresentation](f)
");
$f=switch(0,$M);
$F=switch(3,$M);
$fi=switch(1,$M);
$fi2=switch(2,$M);
$ANS=switch($n-1,'"(-infinity,$a] U [$b,$c]"','"[$b,$c]"','"(-infinity,$a] U [$b,$c]"');
$plot=plotmaple("plot($fi2,x=$a-4..$c+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.1.uid=4c8d89fb-dec9-4b8d-9c75-27bd2361fae6@
qu.9.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Concavity;
@

qu.9.2.question=<p>The function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> has second derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><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><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$F. Find the intervals on which <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> is concave down. Give your answer using interval notation. Use infinity for<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>@
qu.9.2.maple=grade("$RESPONSE",$ANS);@
qu.9.2.allow2d=0@
qu.9.2.maple_answer=show($ANS);@
qu.9.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.9.2.type=maple@
qu.9.2.mode=Maple@
qu.9.2.name=f ConDown A*B*C@
qu.9.2.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Look at where the graph is concave down! Does it match your choice of intervals?</p>
<p>$plot</p>@
qu.9.2.editing=useHTML@
qu.9.2.hint.1=Concave down is <strong>NOT</strong> the same as <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><mfrac><mrow><msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mn>2</mn></mrow></msup><mi>y</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow><mrow><mn>0</mn></mrow></math>. For example, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>5</mn></mrow></msup></mrow></math>is concave down on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> while it has negative second derivative on <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lpar;</mo><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow></math>.@
qu.9.2.solution=@
qu.9.2.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,4);
$f="(x-($a))^$n*(x-($b))*(x-$c)";
$M=maple("
f1:=int($f,x):
f2:=int(f1,x):
convert(f2,string),
MathML[ExportPresentation]($f)
");
$fi2=switch(0,$M);
$F=switch(1,$M);
$ANS=switch($n-1,'"(-infinity,$a] U [$b,$c]"','"[$b,$c]"' , '"(-infinity,$a] U [$b,$c]"');
$plot=plotmaple("plot($fi2,x=$a-4..$c+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.2.uid=640dfd25-3ad3-45bd-8ff0-566d8072e72d@
qu.9.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Graph Sketching;
  Sub-Topic=Intervals of Concavity;
@

qu.10.topic=Slope of inflection tangent@

qu.10.1.mode=Inline@
qu.10.1.name=InfTang@
qu.10.1.comment=<p>$plot</p>
<p align="left">The polynomial $F is black and the tangent line at the single inflection point is blue. The slope of this tangent line may not look like <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$m</mi></mrow></math>. If this is the case, the scale on the axes will not be one to one.</p>@
qu.10.1.editing=useHTML@
qu.10.1.solution=@
qu.10.1.algorithm=$b=rint(1,5);
$f="x^3+($b)*x^2";
$fp="3*x^2+2*($b)*x";
$fp2="6*x+2*($b)";
$M=maple("
c:= -$b/3:
m:= 3*(c)^2+2*($b)*(c):
y:= (c)^3+($b)*(c)^2:
convert(c,string),
convert(m,string),
convert(y,string),
MathML[ExportPresentation]('y'=$f)
");
$C=switch(0,$M);
$m=switch(1,$M);
$y=switch(2,$M);
$F=switch(3,$M);
$plot=plotmaple("plot([$f,($m)*(x-($C))+($y)],x=-8..8,thickness=2,
color=[black,blue]),plotdevice='gif', plotoptions='height=250,width=250'");
$plot1=plotmaple("plot($f,x=-8..8,thickness=2,
color=[black,blue]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.10.1.uid=545a8700-e5fa-4b79-b266-a1e2c0aa48d0@
qu.10.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Graph Sketching;
  Sub-Topic=Point of Inflection;
@
qu.10.1.weighting=1,1@
qu.10.1.numbering=alpha@
qu.10.1.part.1.editing=useHTML@
qu.10.1.part.1.question=(Unset)@
qu.10.1.part.1.name=sro_id_1@
qu.10.1.part.1.answer=($C,$y)@
qu.10.1.part.1.mode=Ntuple@
qu.10.1.part.2.name=sro_id_2@
qu.10.1.part.2.maple_answer=$m@
qu.10.1.part.2.editing=useHTML@
qu.10.1.part.2.question=(Unset)@
qu.10.1.part.2.libname=@
qu.10.1.part.2.mode=Maple@
qu.10.1.part.2.allow2d=1@
qu.10.1.part.2.plot=@
qu.10.1.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.10.1.part.2.type=formula@
qu.10.1.question=<p align="center">$plot1</p><p>Above is the graph of the cubic polynomial $F, which has a single point of inflection. Find the following.</p><p>Give <strong>EXACT </strong>answers.</p><p><br />(a) The point of inflection <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mi>y</mi></mrow></mfenced></mrow></math></p><p><1></p><p>&nbsp;</p><p><span> </span></p><p>(b) The slope of the tangent line at the point of inflection</p><p><2><span> </span></p><p>Give <strong>EXACT </strong>answers.</p>@

