View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/11_Parametric_Equations_Polar_Coordinates_and_Conic_Sections/11.1_Parametric_Equations/11.1.9.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Introduction to Differential Equations')
    3 ## DBsection('Parametric Equations')
    4 ## KEYWORDS('calculus', 'parametric', 'parametric equations')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('11.1')
    9 ## Problem1('9')
   10 ## Author('Christopher Sira')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("Parser.pl");
   16 loadMacros("freemanMacros.pl");
   17 
   18 $context = Context("Point");
   19 $context->variables->add(t=>'Real');
   20 #$context->variables->add(x=>'Real');
   21 Context()->flags->set(reduceConstants=>0);
   22 Context()->flags->set(reduceConstantFunctions=>0);
   23 
   24 $a = Real(random(1, 3, 1));
   25 $b = Real(random(2, 7, 1));
   26 
   27 $xform = Formula("$a * t")->reduce;
   28 $txform = Formula("x/$a")->reduce;
   29 
   30 $yform = Formula("arctan(t**$b + e**t)")->reduce;
   31 
   32 $ans = $yform->substitute(t=>$txform)->reduce();
   33 
   34 #$txform = Formula("1 / $a * x")->reduce;
   35 
   36 #$yform = Formula("arctan(t**$b + e**t)")->reduce;
   37 
   38 #$ans = $yform->substitute(t=>$txform)->reduce();
   39 
   40 Context()->texStrings;
   41 BEGIN_TEXT
   42 \{ beginproblem() \}
   43 \{ textbook_ref_exact("Rogawski ET 2e", "11.1","9") \}
   44 $PAR
   45 Express \( x = $xform \), \( y = $yform \) in the form \( y = f(x) \) by eliminating the parameter.
   46 $PAR
   47 y = \{ans_rule()\}
   48 END_TEXT
   49 Context()->normalStrings;
   50 
   51 ANS($ans->cmp);
   52 
   53 Context()->texStrings;
   54 SOLUTION(EV3(<<'END_SOLUTION'));
   55 $PAR
   56 $SOL
   57 We eliminate the parameter.  Since \( x = $xform \), we have \( t = $txform \).  Substituting into \( y = $yform \) we obtain
   58 $PAR
   59 \( y = $ans \)
   60 END_SOLUTION
   61 
   62 ENDDOCUMENT();