Specifications-based Calculus UCalgary

Contrib/UCalgary/249-1-0-0-C/1-1-0-An-C/1-1-0-An-C-graph-interval-jms.pg

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+## DESCRIPTION
+##
+## ENDDESCRIPTION
+
+
+## DBsubject(Algebra)
+## DBchapter(Quadratic equations and functions)
+## DBsection(Graphs)
+## Date(3 February 2025)
+## Institution(University of Calgary)
+## Author(Jerrod Smith)
+## Level(4)
+
+DOCUMENT();
+
+loadMacros("PGstandard.pl", "PGML.pl", "PGtikz.pl", "PGcourse.pl",);
+$a    = non_zero_random(-3, 3, 1);   #nonzero repeated root
+$b    = 2; #tried a random power, too much so go with 2
+if(abs($a)==1)
+{$c = 10;}
+else
+{$c = abs($a)*4}; #dampening factor for graph
+$p = non_zero_random(-1,1,1); #coefficient +/- 1
+# answers depend on $p and are switched based on sign
+if($p==1)
+{
+$inc = List("($a,5)");
+$dec = List("(-5,$a)");
+}
+else
+{
+$inc = List("(-5,$a)");
+$dec = List("($a,5)");
+}
+
+$graph_image = createTikZImage();
+$graph_image->tikzLibraries("arrows.meta");
+
+$graph_image->BEGIN_TIKZ
+\draw[->,thick] (-5.5,0) -- (5.5,0) node[above left,outer sep=2pt]{\(x\)};
+\draw[->,thick] (0,-5.5) -- (0,5.5) node[below right,outer sep=2pt]{\(y\)};
+\draw[very thin,color=gray] (-5.5,-5.5) grid (5.5,5.5);
+\foreach \x in {-4,-2,2,4} \draw[thin] (\x,5pt) -- (\x,-5pt) node[below]{\(\x\)};
+\foreach \y in {-4,-2,2,4} \draw[thin] (5pt,\y) -- (-5pt,\y) node[left]{\(\y\)};
+\draw[-,red, very thick] plot[domain={-4.98}:{4.98}] (\x,{$p*pow((\x-$a),$b)/($c) - $p*abs($a)});
+\draw[red, very thick] (-5,{$p*pow(-5-$a,$b)/$c- $p*abs($a)})  circle (2 pt);
+\draw[red, very thick] (5,{$p*pow(5-$a,$b)/$c- $p*abs($a)})  circle (2 pt);
+\draw[red] (5,{$p*pow(5-$a,$b)/$c- $p*abs($a)})  node[above right, outer sep=2pt]{\(y=f(x)\)};
+END_TIKZ
+
+\Context("Interval");
+BEGIN_PGML
+[@ image($graph_image, width => 600, tex_size => 1000) @]*
+
+Let [`f`] be a function defined on the open interval [`(-5,5)`] and consider the graph of [`y=f(x)`] given above (drawn in red).
+
+Enter your answers below as (open) intervals (use a list separated by commas, if needed).  
+
+If there are no such intervals, type [|None|]*.
+
+a. [`f`] is increasing on the interval(s) [_]{$inc}{20}
+b. [`f`] is decreasing on the interval(s) [_]{$dec}{20}
+
+[@helpLink('intervals')@]*
+
+END_PGML
+
+ENDDOCUMENT();

Contrib/UCalgary/249-1-0-0-C/1-1-0-An-C/1-1-0-An-C-interpret-graph1.pg

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+## DESCRIPTION
+## 
+## ENDDESCRIPTION
+
+
+## DBsubject(Algebra)
+## DBchapter(Functions)
+## DBsection(Graphs)
+## Date(30 January 2025)
+## Institution(University of Calgary)
+## Author(Aiden Taylor)
+## Level(4)
+
+DOCUMENT();
+
+loadMacros(
+    "PGstandard.pl",
+    "PGML.pl",
+    "PGcourse.pl",
+    "PGtikz.pl"
+);
+
+$showPartialCorrectAnswers = 1;
+
+$graph_image = createTikZImage();
+$graph_image->tikzLibraries("arrows.meta");
+
+$graph_image->BEGIN_TIKZ
+\draw[<->,thick] (-7,0) -- (7,0) node[above left,outer sep=2pt]{\(x\)};
+\draw[<->,thick] (0,-3) -- (0,11) node[below right,outer sep=2pt]{\(y\)};
+\foreach \x in {-6,-4,...,-2,2,4,...,6} \draw[thin] (\x,5pt) -- (\x,-5pt) node[below]{\(\x\)};
+\foreach \y in {-2,2,4,...,10} \draw[thin] (5pt,\y) -- (-5pt,\y) node[left]{\(\y\)};
+\draw[-,red, ultra thick] plot[domain={-3}:{2.99}] (\x,{pow(\x,2)});
+\filldraw[red, ultra thick] (-3,9) circle (3.5 pt);
+\draw[red, ultra thick] (3,9) circle (3.5 pt);
+\draw[very thin,color=gray]  (-7,-3) grid (7,11);
+\draw[red] (1,9) node[outer sep=2pt]{\(y=f(x)\)};
+END_TIKZ
+
+\Context("Interval");
+BEGIN_PGML
+[@ image($graph_image, width => 600, tex_size => 1000) @]*
+
+Refer to the graph of the function, [`f`], drawn above in red.
+
+(a) The value of [`f(1)`] is: [_]{"1"}{2}
+
+(b) For what values of [`x`] is [`f(x) = 4`]? [____]{"-2,2"}{5}
+(Enter the values of [`x`] as a list, separated by commas.)
+
+(c) State the domain of [`f`]. [______]{"[-3,3)"}{20}
+(Enter your answer as an interval. [@helpLink('intervals')@]*) 
+
+(d) State the range of [`f`]. [______]{"[0,9]"}{20}
+(Enter your answer as an interval.)
+END_PGML
+
+ENDDOCUMENT();

Contrib/UCalgary/249-1-0-0-C/1-1-0-An-C/1-1-0-An-C-quad-linear-jms.pg

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+## DESCRIPTION
+##
+## ENDDESCRIPTION
+
+
+## DBsubject(Algebra)
+## DBchapter(Quadratic equations and functions)
+## DBsection(Graphs)
+## Date(3 February 2025)
+## Institution(University of Calgary)
+## Author(Jerrod Smith)
+## Level(4)
+
+DOCUMENT();
+
+loadMacros("PGstandard.pl", "PGML.pl", "PGtikz.pl", "PGcourse.pl","parserPopUp.pl",);
+$a    = non_zero_random(-1, 1, 1);   #nonzero repeated root
+$b    = random(1,2,1); #linear or quadratic
+if($b==2)
+{$c = 10*$b;}
+else
+{$c = 2}; #dampening factor for graph
+$p = non_zero_random(-1,1,1); #coefficient +/- 1
+# popup_type answers depend on $b
+if($b==1)
+{$type = 'linear function';}
+else
+{$type = 'quadtratic function';}
+$popup_type = DropDown(
+    [
+        'linear function',
+        'quadratic function',
+    ],
+    $type
+);
+# popup_slope answers depend on $b and $p
+if($p==1 and $b==1)
+{$slope = 'positive';}
+elsif($p==-1 and $b==1)
+{$slope = 'negative';}
+else
+{$slope = 'neither';}
+$popup_slope = DropDown(
+    [
+        'negative',
+        'positive',
+        'neither',
+    ],
+    $slope
+);
+
+$graph_image = createTikZImage();
+$graph_image->tikzLibraries("arrows.meta");
+
+$graph_image->BEGIN_TIKZ
+\draw[->,thick] (-5.5,0) -- (5.5,0) node[above left,outer sep=2pt]{\(x\)};
+\draw[->,thick] (0,-5.5) -- (0,5.5) node[below right,outer sep=2pt]{\(y\)};
+\draw[very thin,color=gray] (-5.5,-5.5) grid (5.5,5.5);
+\foreach \x in {-4,-2,2,4} \draw[thin] (\x,5pt) -- (\x,-5pt) node[below]{\(\x\)};
+\foreach \y in {-4,-2,2,4} \draw[thin] (5pt,\y) -- (-5pt,\y) node[left]{\(\y\)};
+\draw[<->,red, very thick] plot[domain={-5}:{5}] (\x,{$p*pow($b*(\x-$a),$b)/($c) - $p*abs($a*$b)});
+\draw[red] (5,{$p*pow($b*(5-$a),$b)/($c) - $p*abs($a*$b)})  node[above right, outer sep=2pt]{\(y=f(x)\)};
+END_TIKZ
+
+\Context("Interval");
+BEGIN_PGML
+[@ image($graph_image, width => 600, tex_size => 1000) @]*
+
+Consider the graph of the of the function [`y=f(x)`] given above in red.
+
+Based on the visible portion of the graph [`f`] is a [_]{$popup_type}.
+
+If [`f`] is a linear function, then the slope of [`y=f(x)`] is [_]{$popup_slope}. (If [`f`] is a quadratic function, choose "neither").
+
+END_PGML
+
+ENDDOCUMENT();

Contrib/UCalgary/249-1-0-0-C/1-1-1-U-C/1-1-1-U-C-slope-int-1-jms.pg

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+## DESCRIPTION
+##
+## ENDDESCRIPTION
+
+
+## DBsubject(Algebra)
+## DBchapter(Linear equations and functions)
+## DBsection(Equations of lines: slope-intercept form)
+## Level(4)
+
+DOCUMENT();
+
+loadMacros("PGstandard.pl", "PGML.pl", "PGtikz.pl", "PGcourse.pl",);
+$a    = random(1, 3, 1);              #random positive slope numerator
+$m    = $a / 2;                       #slope is m = a/2
+$b    = non_zero_random(-1, 1, 1);    #random nonzero y-intercept
+$l    = $m * (-4) + $b;               #lower limit, min of L
+$u    = $m * (4) + $b;                #upper limit, max of L
+$um   = $u - 1;                       # for graph grid
+$lm   = $l + 1;                       # for graph grid
+$xint = -$b / $m;                     #calculate x-intercept
+
+$graph_image = createTikZImage();
+$graph_image->tikzLibraries("arrows.meta");
+
+$graph_image->BEGIN_TIKZ
+\draw[->,thick] (-5,0) -- (5,0) node[above left,outer sep=2pt]{\(x\)};
+\draw[->,thick] (0,$l) -- (0,$u) node[below right,outer sep=2pt]{\(y\)};
+\draw[very thin,color=gray] (-5,$l) grid (5,$u);
+\foreach \x in {-4,-2,2,4} \draw[thin] (\x,5pt) -- (\x,-5pt) node[below]{\(\x\)};
+\foreach \y in {$lm,...,$um} \draw[thin] (5pt,\y) -- (-5pt,\y) node[left]{\(\y\)};
+\draw[<->,red, very thick] plot[domain={-4}:{4}] (\x,$m*\x+$b);
+\draw[red] (4,$u) node[below right, outer sep=2pt]{\(y=L(x)\)};
+END_TIKZ
+
+\Context("Interval");
+BEGIN_PGML
+[@ image($graph_image, width => 600, tex_size => 1000) @]*
+
+Consider the graph of the line [`L`] given above.
+
+(a) The slope of [`L`] is [`m = `] [_]{"$m"}
+
+(b) The [`y`]-intercept of [`L`] is [`b = `] [_]{"$b"}
+
+(c) The [`x`]-intercept of [`L`] is [`x = `] [_]{"$xint"} 
+
+Note: You may need to calculate the [`x`]-intercept by using your answers to parts (a) and (b).
+
+END_PGML
+
+ENDDOCUMENT();

Contrib/UCalgary/249-1-0-0-C/1-1-1-U-C/1-1-1-U-C-slope-int-2-jms.pg

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+## DESCRIPTION
+##
+## ENDDESCRIPTION
+
+
+## DBsubject(Algebra)
+## DBchapter(Linear equations and functions)
+## DBsection(Equations of lines: slope-intercept form)
+## Date(31 January 2025)
+## Institution(University of Calgary)
+## Author(Jerrod Smith)
+## Level(2)
+
+DOCUMENT();
+
+loadMacros("PGstandard.pl", "PGML.pl", "PGtikz.pl", "PGcourse.pl",);
+$a    = random(-4, -2, 2);              #random even negative slope numerator
+$b    = non_zero_random(-2, 2, 1);    #random nonzero
+$c    = random(3,5,2); #positive odd slope denominator relatively prime to $a*$b
+$m    = $a / $c;                       #slope is m = a/c
+# L = $m(x-$b);
+$l    = $m * (4 - $b);               #lower limit, min of L at x=4 since decreasing
+$u    = $m * (-4 - $b) ;                #upper limit, max of L at x=-4 since decreasing
+$um   = $u + 1;   # for graph grid
+$umc  = int($um); #floor of $um
+$lm   = $l - 1;     # graph grid
+$lmf  = int($lm); # floor of $lm
+# floor for y-axis grid marks
+$yint = -$b * $m;                     #calculate y-intercept
+
+
+
+$graph_image = createTikZImage();
+$graph_image->tikzLibraries("arrows.meta");
+
+$graph_image->BEGIN_TIKZ
+\draw[->,thick] (-5,0) -- (5,0) node[above left,outer sep=2pt]{\(x\)};
+\draw[->,thick] (0,$lm) -- (0,$um) node[below right,outer sep=2pt]{\(y\)};
+\draw[very thin,color=gray] (-5,$lm) grid (5,$um);
+\foreach \x in {-4,-2,2,4} \draw[thin] (\x,5pt) -- (\x,-5pt) node[below]{\(\x\)};
+\foreach \y in {$lmf,...,$umc} \draw[thin] (5pt,\y) -- (-5pt,\y) node[left]{\(\y\)};
+\draw[<->,red, very thick] plot[domain={-4.5}:{4.5}] (\x,$m*(\x-$b));
+\draw[red] (4,$l) node[above right, outer sep=2pt]{\(y=L(x)\)};
+END_TIKZ
+
+\Context("Interval");
+BEGIN_PGML
+[@ image($graph_image, width => 600, tex_size => 1000) @]*
+
+Consider the graph of the line [`L`] given above.
+
+(a) The slope of [`L`] is [`m = `] [_]{"$m"}
+
+(b) The [`x`]-intercept of [`L`] is [`x = `] [_]{"$b"}
+
+(c) The [`y`]-intercept of [`L`] is [`b = `] [_]{"$yint"} 
+
+Note: You may need to calculate the [`y`]-intercept by using your answers to parts (a) and (b).
+
+END_PGML
+
+ENDDOCUMENT();

Contrib/UCalgary/249-1-0-0-C/1-1-3-U-C/1-1-3-U-C-delta-slope-jms.pg

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+## DESCRIPTION
+##
+## ENDDESCRIPTION
+
+
+## DBsubject(Algebra)
+## DBchapter(Linear equations and functions)
+## DBsection(Linear equations)
+## Date(31 January 2025)
+## Institution(University of Calgary)
+## Author(Jerrod Smith)
+## Level(2)
+
+DOCUMENT();
+
+loadMacros(
+    "PGstandard.pl",    # Standard macros for PG language
+    "PGML.pl",          # PGML markup and Math Objects
+    "PGcourse.pl",      # Customization file for the course
+);
+
+# Uncomment the following if you don't want to show which
+# answers are correct and which are incorrect
+#$showPartialCorrectAnswers = 0;
+
+# Uncomment the following to override the default numerical
+# tolerances
+# Context()->->flags->set(tolerance => 0.0001, tolType => 'absolute');
+
+$m = non_zero_random(-7,7,1); #random nonzero slope
+if(abs($m)==1)# avoid slope of +/- 1
+{
+$m = $m + 3;
+}
+$b = non_zero_random(-20,20,1); #random y-int
+$delx = random(2,9,1); #given change in x, pos
+$dely = random(-9,-2,1); #given change in y, neg
+$magy = abs($dely);
+$f = Formula("($m*x + $b)")->reduce();
+
+$yans = Compute("$m*$delx");
+$xans = Compute("$dely/$m");
+
+BEGIN_PGML
+Suppose that [`x`] and [`y`] are related by the linear equation [`y = [$f]`].
+
++ If [`x`] is increased by [`[$delx]`] units, then the corresponding change in [`y`] is [`\Delta y = \, `][_]{$yans}{5}.
++ If [`y`] is decreased by [`[$magy]`] units, then the corresponding change in [`x`] is [`\Delta x = \, `][_]{$xans}{5}.
+END_PGML
+
+BEGIN_PGML_SOLUTION
+Recall that the slope of a line is [`m = \dfrac{\Delta y}{\Delta x}`], where [`\Delta y`] is the change in [`y`] and [`\Delta x`] is the change in [`x`].
+
+For the line [`y = [$f]`] we have [`m = [$m]`].
+
++ Given [`\Delta x = [$delx]`], we get [`\Delta y = m \Delta x = [$yans]`].
++ Given [`\Delta y = [$dely]`], we get [`\Delta x = \dfrac{\Delta y}{m} = \dfrac{[$dely]}{[$m]}`].
+END_PGML_SOLUTION
+
+ENDDOCUMENT();