Vector Parametric Function

Parametric equations: parametric curve in space

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Complete Code

Download file: VectorParametricFunction.pg

POD for Macro Files

See Also

PG problem file

Explanation

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

Preamble

Although not necessary for the code demonstrated in this example, you might want to load parserVectorUtils.pl. It provides methods that are useful for vector problems.

The parserMultiAnswer.pl macro is used since the answers depend upon each other.

Context('Vector2D');
# Context('Vector'); # use for 3D vectors
Context()->variables->are(t => 'Real');

$a = random(2, 5);

Context()->variables->set(t => { limits => [ 0, $a ] });

$Q = Point($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

In the checker the vector-valued function that the student enters is stored in $f. It is dotted with the standard basis vectors i and j to obtain the x and y components.

Note that if you need to differentiate the component functions in the student answer, you will need to use a different method. Attempting to compute ($f . i)->D('t') will generate errors since the dot product does not get evaluated. See Parametric Vector Function with Derivative for an example of how to extract formulas from the components of the student answer, which can then be differentiated.

Notice that feedback messages are provided regarding 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.