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    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('49')
   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 
   21 $a = Real(random(2, 5, 1));
   22 $b = Real(random(2, 5, 1));
   23 $c = Real(random(1, 10, 1));
   24 
   25 $t0 = Real(random(1, 10, 1));
   26 
   27 $xt = Formula("t ** $a");
   28 $yt = Formula("t ** $b - $c");
   29 
   30 $anst = Formula("( ($b * t**($b - 1)) / ($a * t**($a - 1) ))")->reduce();
   31 
   32 $fp = $anst->eval(t=>$t0);
   33 
   34 
   35 Context()->texStrings;
   36 BEGIN_TEXT
   37 \{ beginproblem() \}
   38 \{ textbook_ref_exact("Rogawski ET 2e", "11.1","49") \}
   39 $PAR
   40 Find \( \frac{dy}{dx} \) at the point t = $t0.
   41 $PAR
   42 \( c(t) = ($xt, $yt) \)
   43 $PAR
   44 \( \frac{dy}{dx} \) (at t = $t0) = \{ans_rule()\}
   45 END_TEXT
   46 Context()->normalStrings;
   47 
   48 ANS($fp->cmp);
   49 
   50 Context()->texStrings;
   51 SOLUTION(EV3(<<'END_SOLUTION'));
   52 $PAR
   53 $SOL
   54 \( \frac{dy}{dx} = \frac{y'(t)}{x'(t)} = \frac{({$yt})'}{({$xt})'} = $anst \)
   55 $PAR
   56 Substituting t = $t0 we get
   57 $PAR
   58 \( \frac{dy}{dx} = $fp \).
   59 END_SOLUTION
   60 
   61 ENDDOCUMENT();
   62