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merged 2 commits into from
Nov 6, 2025
Merged

Update graph problems #1308

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d05f86b
Convert problem to PGML and add solution
dlglin Oct 15, 2025
18bb4f2
Convert dynamic graph problems to use plots and PGML
dlglin Oct 15, 2025
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184 changes: 82 additions & 102 deletions OpenProblemLibrary/ASU-topics/setLimitConcepts/3-2-36.pg
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Expand Up @@ -4,7 +4,6 @@

## Tagged by LD


## DBsubject(Calculus - single variable)
## DBchapter(Limits and continuity)
## DBsection(Finding limits using graphs)
Expand Down Expand Up @@ -36,107 +35,88 @@
## Problem4('45')
## KEYWORDS('calculus','limits', 'derivatives','limit')

DOCUMENT(); # This should be the first executable line in the problem.

loadMacros(
"PGstandard.pl",
"PGchoicemacros.pl",
"PGgraphmacros.pl",
"PGcourse.pl"
DOCUMENT(); # This should be the first executable line in the problem.

loadMacros('PGstandard.pl', 'PGML.pl', 'plots.pl', 'PGcourse.pl');

$a = random(-3, 3, 1);
$b = random(-2, 3, 1);
$c = random(-3, 2, 1);
$ap1 = 1 + $a;
$bp1 = 1 + $b;
$m1 = random(-1, 1, 0.5);
$m2 = ($b - $a) / 2;
$m3 = ($c - $b - 1) / 2;
$m4 = random(-1, 1, 0.5);
$ystart = $a - $m1;
$yend = $c + $m4;

$color = list_random("blue", "red", "green");

$plot = Plot(
xmin => -3,
xmax => 5,
ymin => -6,
ymax => 6,
xtick_delta => 1,
xminor => 0,
ytick_delta => 1,
yminor => 0,
xlabel => '\(x\)',
ylabel => '\(y\)',
aria_label => 'The graph of the function y=F(x)',
axes_on_top => 1,
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(-3,3,1);
$b=random(-2,3,1);
$c=random(-3,2,1);
$m1=random(-1,1,0.5);
$m2=($b - $a)/2;
$m3=($c - $b - 1)/2;
$m4=random(-1,1,0.5);
@slice = NchooseK(3,3);

@colors = ("blue", "red", "green");
@sc = @colors[@slice]; #scrambled colors
@sa = ('A','B','C')[@slice];

$f1 = FEQ("${m1}*(x+1)+$a for x in [-2,-1) using color:$sc[0] and weight:2");
$f2 = FEQ("${m2}*(x-1)+$b for x in (-1,1) using color=$sc[0] and weight:2");
$f3 = FEQ("${m3}*(x-3)+$c for x in [1,3) using color=$sc[0] and weight=2");
$f4 = FEQ("1+$a for x in [-1,-1] using color=$sc[0] and weight=2");
$f5 = FEQ("${m4}*(x-3)+$c for x in (3,4] using color=$sc[0] and weight=2");

$graph = init_graph(-3,-6,5,6,'axes'=>[0,0],'grid'=>[8,12]);

($f1Ref,$f2Ref,$f3Ref,$f4Ref,$f5Ref) = plot_functions($graph,$f1,$f2,$f3,$f4,$f5);

TEXT(EV2(<<EOT));
Let F be the function below.$PAR
If you are having a hard time seeing the picture clearly, click on the picture. It will expand to a larger picture on its own page so that you can inspect it more clearly.$PAR
EOT

TEXT(image( insertGraph($graph) ));

TEXT(EV2(<<EOT));
$BR
$BR
Evaluate each of the following expressions. $PAR
Note: Enter 'DNE' if the limit does not exist or is not defined. $PAR

a) \( \displaystyle \lim_{x \to -1^-} F(x) \) = \<ans_rule(7)\>
$SPACE

\( \displaystyle \lim_{x \to -1^+} F(x) \) = \<ans_rule(7)\>
$SPACE

\( \displaystyle \lim_{x \to -1} F(x) \) = \<ans_rule(7)\>
$SPACE

\( F(-1) \) = \<ans_rule(7)\>
$PAR

b) \( \displaystyle \lim_{x \to 1^-} F(x) \) = \<ans_rule(8)\>
$SPACE

\( \displaystyle \lim_{x \to 1^+} F(x) \) = \<ans_rule(8)\>
$SPACE

\( \displaystyle \lim_{x \to 1} F(x) \) = \<ans_rule(8)\>
$SPACE

\( F(1) \) = \<ans_rule(8)\>
$PAR

c) \( \displaystyle \lim_{x \to 3^-} F(x) \) = \<ans_rule(8)\>
$SPACE

\( \displaystyle \lim_{x \to 3^+} F(x) \) = \<ans_rule(8)\>
$SPACE

\( \displaystyle \lim_{x \to 3} F(x) \) = \<ans_rule(8)\>
$SPACE

\( F(3) \) = \<ans_rule(8)\>
$PAR

EOT

$ap1 = 1 + $a;
$bp1 = 1 + $b;

ANS(num_cmp($a, strings=>['DNE'])) ;
ANS(num_cmp($a, strings=>['DNE'])) ;
ANS(num_cmp($a, strings=>['DNE'])) ;
ANS(num_cmp($ap1, strings=>['DNE']));
ANS(num_cmp($b, strings=>['DNE'])) ;
ANS(num_cmp($bp1, strings=>['DNE']));
ANS(num_cmp('DNE', strings=>['DNE'])) ;
ANS(num_cmp($bp1, strings=>['DNE']));
ANS(num_cmp($c, strings=>['DNE'])) ;
ANS(num_cmp($c, strings=>['DNE'])) ;
ANS(num_cmp($c, strings=>['DNE'])) ;
ANS(num_cmp('DNE', strings=>['DNE'])) ;

ENDDOCUMENT(); # This should be the last executable line in the problem.
$plot->add_function(
["${m1}*(x+1)+$a", 'x', -2, -1, start_mark=>'circle', end_mark=>'open_circle', color=>$color, weight=>2],
["${m2}*(x-1)+$b", 'x', -1, 1, start_mark=>'open_circle', end_mark=>'open_circle', color=>$color, weight=>2],
["${m3}*(x-3)+$c", 'x', 1, 3, start_mark=>'circle', end_mark=>'open_circle', color=>$color, weight=>2],
["${m4}*(x-3)+$c", 'x', 3, 4, start_mark=>'open_circle', end_mark=>'circle', color=>$color, weight=>2]
);

$plot->add_dataset([ -1, $ap1 ], color => $color, marks => 'circle');

$alttext =
"A graph showing a line segment starting with a solid circle at the point (-2,$ystart) and ending with an open circle at the point (-1,$a). There is a solid circle at the point (-1,$ap1). There is a line segment starting with an open circle at the point (-1,$a) and ending with an open circle at the point (1,$b). There is a line segment starting with a solid circle at the point (1,$bp1) and ending with an open circle at the point (3,$c). There is a line segment starting with an open circle at the point (3,$c) and ending with a solid circle at the point (4,$yend).";

$ans1 = Compute($a);
$ans2 = Compute($a);
$ans3 = Compute($a);
$ans4 = Compute($ap1);
$ans5 = Compute($b);
$ans6 = Compute($bp1);
$ans7 = Compute('DNE');
$ans8 = Compute($bp1);
$ans9 = Compute($c);
$ans10 = Compute($c);
$ans11 = Compute($c);
$ans12 = Compute('DNE');

BEGIN_PGML
The graph of [`y=F(x)`] is given below.

If you are having a hard time seeing the picture clearly, click on the picture. It will expand to a larger picture on its own page so that you can inspect it more clearly.

[! $alttext !]{$plot}

Evaluate each of the following expressions.

Note: Enter 'DNE' if the limit does not exist or is not defined.
[#
[. a) .] [. [`` \lim_{x \to -1^-} F(x) ``] = [_]{$ans1}{5} .]
[. [`` \lim_{x \to -1^+} F(x) ``] = [_]{$ans2}{5} .]
[. [`` \lim_{x \to -1} F(x) ``] = [_]{$ans3}{5} .]
[. [` F(-1) `] = [_]{$ans4}{5} .]*
[. b) .] [. [`` \lim_{x \to 1^-} F(x) ``] = [_]{$ans5}{5} .]
[. [`` \lim_{x \to 1^+} F(x) ``] = [_]{$ans6}{5} .]
[. [`` \lim_{x \to 1} F(x) ``] = [_]{$ans7}{5} .]
[. [` F(1) `] = [_]{$ans8}{5} .]*
[. c) .] [. [`` \lim_{x \to 3^-} F(x) ``] = [_]{$ans9}{5} .]
[. [`` \lim_{x \to 3^+} F(x) ``] = [_]{$ans10}{5} .]
[. [`` \lim_{x \to 3} F(x) ``] = [_]{$ans11}{5} .]
[. [` F(3) `] = [_]{$ans12}{5} .]*
#]*
END_PGML

ENDDOCUMENT();
56 changes: 20 additions & 36 deletions OpenProblemLibrary/AlfredUniv/anton8e/chapter2/2.2/prob18.pg
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Original file line number Diff line number Diff line change
Expand Up @@ -27,49 +27,33 @@
DOCUMENT();

# Load whatever macros you need for the problem
loadMacros(
"PGstandard.pl",
"PGchoicemacros.pl",
"PGgraphmacros.pl",
"MathObjects.pl",
"PGcourse.pl"
);

## Show partial correct answers
$showPartialCorrectAnswers = 1;
## Display the problem information
TEXT(beginproblem());

## Declare Variables
Context()->variables->are(x=>'Real',y=>'Real');
loadMacros("PGstandard.pl", "PGML.pl", "PGcourse.pl");

## Function Definition
$a = random(1,9);
$a = random(1, 9);

$f = Formula("(sqrt(x+$a**2)-$a)/x")->reduce;

$ans1 = Formula("1/(2*$a)")->reduce;

$twoa = 2 * $a;
$a2 = $a**2;
$ans1 = Compute("1/($twoa)");
$conj = Formula("sqrt(x+$a**2)+$a");
BEGIN_PGML
Let [`` f(x)= [$f]``].

Context()->texStrings;
BEGIN_TEXT
Let \(\> f(x)= $f\).
$BR
$BR
Find \(\> \lim_{x \rightarrow 0} f(x) \>\) by first rationalizing the numerator.
$BR
$BR
Enter $BBOLD I $EBOLD for \(\infty\), $BBOLD -I $EBOLD for \(-\infty\),
and $BBOLD DNE $EBOLD if the limit does not exist.
$BR
$BR
Answer: \{ans_rule(20)\}

END_TEXT
Context()->normalStrings;
ANS(num_cmp($ans1, strings=>["I", "-I", "DNE"]));
Find [`` \lim_{x \rightarrow 0} f(x) ``] by first rationalizing the numerator (multiplying and dividing by the conjugate).

[`` \lim_{x \rightarrow 0} f(x) =``][_]{$ans1}{5}[@helpLink('limits')@]*
END_PGML

BEGIN_PGML_SOLUTION
[``\begin{array}{rl}\displaystyle \lim_{x \rightarrow 0} [$f] & = \displaystyle\lim_{x \rightarrow 0} [$f]\cdot\frac{[$conj]}{[$conj]}\\
& =\displaystyle\lim_{x \rightarrow 0}\frac{(x+[$a2])-[$a2]}{x\left([$conj]\right)}\\
& =\displaystyle\lim_{x \rightarrow 0}\frac{x}{x\left([$conj]\right)}\\
& =\displaystyle\lim_{x \rightarrow 0}\frac{1}{[$conj]}\\
& =\displaystyle\frac{1}{\sqrt{0+[$a2]}+[$a]}\\
& =\displaystyle\frac{1}{[$twoa]}
\end{array}``]
END_PGML_SOLUTION

ENDDOCUMENT();

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