Parametric Curve Graph (plots)

Graphing a parametric curve.

Complete Code

POD for Macro Files

plots.pl

PG problem file

Explanation

DOCUMENT();

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

Preamble

The plots.pl macro is used to generate the graph.

Context()->variables->add(t => 'Real');
@tvals     = ('pi/12',   'pi/6',   '5pi/12',   'pi/3',   '2pi/3',   '7pi/12');
@tvals_tex = ('\pi/12', '\pi/6', '5\pi/12', '\pi/3', '2\pi/3', '7\pi/12');
$n         = random(1, $#tvals);
$x         = Compute('2sin(2t)');
$x0        = $x->eval(t => $tvals[$n]);
$y         = Compute('2sin(3t)');
$y0        = $y->eval(t => $tvals[$n]);

$m    = $y->D('t')->eval(t => $tvals[$n]) / $x->D('t')->eval(t => $tvals[$n]);
$line = Compute("$m(x - $x0) + $y0");

$plot = Plot(
    xmin        => -2.5,
    xmax        =>  2.5,
    ymin        => -2.5,
    ymax        =>  2.5,
    xtick_delta =>  0.5,
    ytick_delta => 0.5
);

$plot->add_function([ $x, $y ], 't', 0, '2pi', color => 'blue');

Setup

This problem asks the student to find the equation of a tangent line to a parametric curve at a random point. There is an array of t-values as just a string and a tex string and a random index $n. The parametric function is defined as the x and y parts as well as the derivatives which results in the slope.

For the plot, the plotting domain is listed, then the parametric function is added with the add_function and it is important to add the components as an array reference.

BEGIN_PGML
Find the tangent line to the parametric curve

>> [``x(t) = [$x], \quad y(t) = [$y]``] <<

when [`t = [$tvals_tex[$n]]`].  The graph of the curve is

>>[!a Lissajous curve!]{$plot}{500}<<

Tangent line in slope-intercept form: [`y =`] [_____]{$line}
END_PGML

Statement

This is the problem statement in PGML.

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

A solution should be provided here.