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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('1')
   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(-10, 10, 1));
   22 $b = Real(random(-10, 10, 1));
   23 $c = Real(random(1, 5, 1));
   24 $d = Real(random(-10, 10, 1));
   25 $e = Real(random(-10, 10, 1));
   26 $f = Real(random(1, 5, 1));
   27 
   28 
   29 $xform = Formula("$a + $b * t ** $c")->reduce();
   30 $yform = Formula("$d + $e * t ** $f")->reduce();
   31 
   32 $t1 = 0;
   33 $t2 = $t1 + Real(random(1, 5, 1));
   34 $t3 = $t2 + Real(random(1, 5, 1));
   35 
   36 $ans1 = Point( $xform->eval(t=>$t1), $yform->eval(t=>$t1) );
   37 $ans2 = Point( $xform->eval(t=>$t2), $yform->eval(t=>$t2) );
   38 $ans3 = Point( $xform->eval(t=>$t3), $yform->eval(t=>$t3) );
   39 
   40 
   41 
   42 Context()->texStrings;
   43 BEGIN_TEXT
   44 \{ beginproblem() \}
   45 \{ textbook_ref_exact("Rogawski ET 2e", "11.1","1") \}
   46 $PAR
   47 Find the coordinates at times t = $t1, $t2, $t3 of a particle following the path \( x = $xform \), \( y = $yform \).
   48 $PAR
   49 t = $t1: \{ans_rule()\}
   50 $PAR
   51 t = $t2: \{ans_rule()\}
   52 $PAR
   53 t = $t3: \{ans_rule()\}
   54 END_TEXT
   55 Context()->normalStrings;
   56 
   57 ANS($ans1->cmp, $ans2->cmp, $ans3->cmp);
   58 
   59 Context()->texStrings;
   60 SOLUTION(EV3(<<'END_SOLUTION'));
   61 $PAR
   62 $SOL
   63 Substituting t = $t1, t = $t2, and t = $t3 into \( x = $xform \), \( y = $yform \) gives the coordinates of the particle at these times respectively.  that is,
   64 $PAR
   65 (t = $t1): \( x = \{$xform->eval(t=>$t1) \},\, y = \{$yform->eval(t=>$t1) \} \, \to \, $ans1 \)
   66 $PAR
   67 (t = $t2): \( x = \{$xform->eval(t=>$t2) \},\, y = \{$yform->eval(t=>$t2) \} \, \to \, $ans2 \)
   68 $PAR
   69 (t = $t3): \( x = \{$xform->eval(t=>$t3) \},\,y = \{$yform->eval(t=>$t3) \} \, \to \, $ans3 \)
   70 $PAR
   71 
   72 END_SOLUTION
   73 
   74 ENDDOCUMENT();
   75