EquationImplicitFunction1

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This article has been retained as a historical document. It is not up-to-date and the formatting may be lacking. Use the information herein with caution.

This problem has been replaced with a newer version of this problem


Answer is an Equation that Implicitly Defines a Function

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This PG code shows how to have an answer that is an equation that implicitly defines a function.


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PG problem file Explanation

Problem tagging data

Problem tagging:

DOCUMENT();   

loadMacros('PGstandard.pl','MathObjects.pl',
  'parserImplicitEquation.pl','PGML.pl','PGcourse.pl');
TEXT(beginproblem());

Initialization:

Context('ImplicitEquation');
Context()->{error}{msg}{
"Can't find any solutions to your equation"} = ' ';
Context()->{error}{msg}{
"Can't generate enough valid points for comparison"} = ' ';

Context()->variables->set(
  x=>{limits=>[-6,11]},
  y=>{limits=>[-6,11]},
);

$a = random(1,5,1);
$b = random(1,5,1);
$r = random(2,5,1);
$p = Compute("($a,$b)");

$answer = ImplicitEquation(
  "(x-$a)^2 + (y-$b)^2 = $r^2",
  solutions=>[
     [$a,$b+$r],
     [$a,$b-$r],
     [$a+$r,$b],
     [$a-$r,$b],
     [$a+$r*sqrt(2)/2,$b+$r*sqrt(2)/2],
  ]
);

Setup: We quash some error messages by redefining them to be a blank string " " (notice the space). Since the circle will always be contained in a rectangle with two opposite corners at (-4,-4) and (10,10), we set the limits for the variables x and y to be outside of this rectangle. The ImplicitEquation object allows us to specify as many solutions as we like, and doing so should improve the accuracy of the answer evaluator.

If your equation is linear of the form x=3, 4x+3y=12, or 4x+3y+5z=21, or..., you should probably use the [ImplicitPlane1 implicit plane] context and answer evaluator.

BEGIN_PGML
Enter an equation for a circle in the [`xy`]-plane
of radius [` [$r] `] centered at [` [$p] `].

[________________________]{$answer}

[@ helpLink('equation') @]*
END_PGML

Main Text:

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution:

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