BCIT_ehqw.pg

## DESCRIPTION
## Parametric equations: graphing a polar curve
## ENDDESCRIPTION

## DBsubject(Calculus - multivariable)
## DBchapter(Differentiation of multivariable functions)
## DBsection(Differentiability, linearization and tangent planes)
## Date(11/23/2021)
## Institution(British Columbia Institute of Technology)
## Author(Stefan Lukits)
## KEYWORDS('newtons method')
## supported by the BCcampus Open Homework Systems Project Grant

###########################
#  Initialization

DOCUMENT();

loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "PGgraphmacros.pl",

  "unionTables.pl",
  "PGcourse.pl"
);

TEXT(beginproblem());

$refreshCachedImages = 1;
$discriminant=0;
while ($xs<=0)
{
while ($discriminant<=0)
{
$a=random(2,5,1);
$b=random(8,11,1);
$c=random(16,19,1);
$d=random(12,15,1);
$e=random(41,49,1);
$discriminant=(1-((2*$a*$d*$e)/($c**2)))**2-(4*(($a*$d**2)/($c**2))*((($a*$e**2)/($c**2))-$b));
}
$xs=((((2*$a*$d*$e)/($c**2))-1)+sqrt($discriminant))/((2*$a*$d**2)/($c**2));
}
$x=sqrt($xs);
$y=(($d*$x**2)-$e)/$c;
$xr=(int(($x*100)+0.5))/100;
$yr=(int(($y*100)+0.5))/100;
$fabx=$xr**2+$a*$yr**2-$b;
$faby=$c*$yr-$d*$xr**2+$e;
$J11=2*$xr;
$J12=2*$a*$yr;
$J21=(-2)*$d*$xr;
$J22=$c;
$con1=$J11*$xr+$J12*$yr-$fabx;
$con2=$J21*$xr+$J22*$yr-$faby;
$det=($J11*$J22)-($J12*$J21);
$detC1=($con1*$J22)-($J12*$con2);
$detC2=($J11*$con2)-($con1*$J21);
$x1=$detC1/$det;
$y1=$detC2/$det;
$x1r=(int(($x1*100000)+0.5))/100000;
$y1r=(int(($y1*100000)+0.5))/100000;

##########################
#  Setup

Context("Numeric")->variables->are(t=>"Real");

$gr = init_graph(-4,-3,4,3,axes=>[0,0],grid=>[4,4],size=>[300,300]);

$gr->new_color("lightblue", 198,217,253); # RGB
$gr->new_color("darkblue",   77,137,249);
$gr->new_color("lightred",  255,127,127);
$gr->new_color("darkred",   255, 55, 55);
$gr->new_color("lightorange",  255,204,127);
$gr->new_color("darkorange",   255, 153, 0);
$gr->new_color("lightgreen", 187, 255, 153);
$gr->new_color("darkgreen",    0, 208, 0);

$x = Formula("sqrt($b)*cos(t)");
$y = Formula("sqrt($b/$a)*sin(t)");

$f = new Fun( $x->perlFunction, $y->perlFunction, $gr );
$f->domain(0,2*pi);
$f->steps(90);
$f->color('darkgreen');
$f->weight('2');

$x = Formula("t");
$y = Formula("(1/$c)*($d*(t*t)-$e)");

$w = new Fun( $x->perlFunction, $y->perlFunction, $gr );
$w->domain(-4,4);
$w->steps(90);
$w->color('darkorange');
$w->weight('2');

##########################
#  Main Text

Context()->texStrings;
BEGIN_TEXT
Use Newton's method to solve the system
\[
\begin{array}{l}
        x^{2}+$a y^{2} = $b \\
        $c y-$d x^{2} = -$e \\
\end{array}
\]
$BR
\(($xr,$yr)\) is close to the solution. Consider it your first estimate \((x_{0},y_{0})\). What are \(x_{1}\) and \(y_{1}\) after one iteration of using Newton's method? Provide at least six significant digits.
$BR
$BR
\(x_{1}\)=\{ans_rule(20)\}
$BR
\(y_{1}\)=\{ans_rule(20)\}
$BR
$BCENTER \{image(insertGraph($gr),width=>300,height=>300 )\}
$BR
Graph of \( x^{2}+$a y^{2} = $b \) and \( $c y-$d x^{2} = -$e\)$ECENTER
END_TEXT
Context()->normalStrings;

########################
#  Answer Evaluation

$showPartialCorrectAnswers = 1;
ANS(num_cmp($x1,
  tolType => 'absolute',
    tolerance => .0001,
    ));
ANS(num_cmp($y1,
  tolType => 'absolute',
    tolerance => .0001,
    ));

############################
#  Solution

Context()->texStrings;
BEGIN_SOLUTION

Solution explanation goes here.
END_SOLUTION
Context()->normalStrings;

;

ENDDOCUMENT();