Parametric Curve Graph

Graphing a parametric curve.

Complete Code

POD for Macro Files

PGtikz.pl

PG problem file

Explanation

DOCUMENT();

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

Preamble

We use PGtikz.pl to generate the graph.

Context()->variables->add(t => 'Real');

$graph = createTikZImage();
$graph->tikzLibraries('arrows.meta');
$graph->BEGIN_TIKZ
\tikzset{>={Stealth[scale=1.5]}}
\filldraw[
    draw=LightBlue,
    fill=white,
    rounded corners=10pt,
    thick,use as bounding box
] (-3.5,-3.5) rectangle (3.5,3.5);
\draw[->] (-3.5,0) -- (3.5,0) node[above left,outer sep=3pt] {\(x\)};
\foreach \x in {-3,-2,-1,1,2,3} \draw (\x,0.15) -- (\x,-0.15) node [below] {\x};
\draw[->] (0,-3.5) -- (0,3.5) node[below right,outer sep=3pt] {\(y\)};
\foreach \y in {-3,-2,-1,1,2,3} \draw (0.15,\y) -- (-0.15,\y) node [left] {\y};
\draw[DarkBlue,very thick]
    plot [samples=250,domain=0:{2*pi},variable=\t]
    ({2*sin(2*\t r)},{2*sin(3*\t r)});
END_TIKZ

$x = Compute('2sin(2t)');
$x0 = $x->eval(t => 'pi/3');
$y = Compute('2sin(3t)');
$y0 = $y->eval(t => 'pi/3');

$m = $y->D('t')->eval(t => 'pi/3')/$x->D('t')->eval(t => 'pi/3');
$line = Compute("$m*(x-$x0)+$y0");

Setup

The package PGtikz.pl is used to produce the curve. The plotting routine used in this way plots parametrically, which is shown in the last statement in the TIKZ section with \draw[DarkBlue, very thick] plot [....].

The first two statements of the TIKZ section scales the plot and then gives a nice background that will contrast the rest of the problem page.

BEGIN_PGML
Find the tangent line to the parametric curve:
>> [``x(t) = [$x], \qquad y(t) = [$y]``] <<

when [`t=\pi/3`].  The graph of the curve is

>>[@ image($graph, width => 300) @]*<<

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.