Vector Parametric Function

Parametric equations: parametric curve in space

Complete Code

Download file: VectorParametricFunction.pg

POD for Macro Files

PG problem file

Explanation

DOCUMENT();
loadMacros(
    'PGstandard.pl',        'PGML.pl',
    'parserVectorUtils.pl', 'parserMultiAnswer.pl',
    'PGcourse.pl'
);

Preamble

Since it is a vector parametric curve, we will want vector utilities from parserVectorUtils.pl. Since we will need to check multiple answer blanks that depend upon each other, we use parserMultiAnswer.pl.

Context('Vector2D');
#Context('Vector'); # for 3D vectors
Context()->variables->are(t => 'Real');
Context()->variables->set(t => { limits => [ 0, 5 ] });
Context()->flags->set(ijk => 0);

$a = random(2, 5);
$Q = Point($a, $a**2);

$multians = MultiAnswer(Vector("<t,t**2>"), 0, $a)->with(
    singleResult => 1,

    checker => sub {

        my ($correct, $student, $self) = @_;           # get the parameters
        my ($f,       $x1,      $x2)   = @{$student};  # extract student answers
        if ((($f . i)**2 == ($f . j))
            && ($f->eval(t => $x1) == Vector("<0,0>"))
            && ($f->eval(t => $x2) == Vector("<$a,$a**2>")))
        {
            return 1;
        } elsif ((($f . i)**2 == ($f . j))
            && ($f->eval(t => $x1) == Vector("<0,0>")))
        {
            $self->setMessage(3, 'Your right endpoint is not correct.');
            return 0;
        } elsif ((($f . i)**2 == ($f . j))
            && ($f->eval(t => $x2) == Vector("<$a,$a**2>")))
        {
            $self->setMessage(2, 'Your left endpoint is not correct.');
            return 0;
        } elsif ((($f . i)**2 == ($f . j))) {
            $self->setMessage(2, 'Your left endpoint is not correct.');
            $self->setMessage(3, 'Your right endpoint is not correct.');
            return 0;
        } else {
            return 0;
        }
    }
);

Setup

The student’s vector-valued function is stored in $f. To get the x- and y-components of the students answer we dot it with the standard basis vectors using $f . i and $f . j. Note: If you want to differentiate the component functions in the student’s answer, you’ll need to use a different method as ($f . i)->D('t') will generate errors since the dot product does not get evaluated. Another problem given in this section describes how to extract formulas from the components of the student’s answer, which can then be differentiated. Notice that we have given the students helpful feedback messages about which endpoints are incorrect.

BEGIN_PGML
Find a vector parametric equation for the parabola
[`y = x^2`] from the origin to the point
[`[$Q]`] using [`t`] as a parameter.

[`\vec{r}(t) =`] [_]{$multians}{10} for [___]{$multians}
[`\leq t \leq`] [___]{$multians}.
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.