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    1 ## DESCRIPTION
    2 ##  Calculus: Newton's Method
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('derivatives', 'Newtons method')
    6 ## Tagged by YL
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Applications of Differentiation')
   10 ## DBsection('Newton's Method')
   11 ## Date('5/29/2005')
   12 ## Author('Jeff Holt')
   13 ## Institution('UVA')
   14 ## TitleText1('Calculus: Early Transcendentals')
   15 ## EditionText1('5')
   16 ## AuthorText1('Stewart')
   17 ## Section1('4.9')
   18 ## Problem1('10')
   19 
   20 ## TitleText2('Calculus: Early Transcendentals')
   21 ## EditionText2('6')
   22 ## AuthorText2('Stewart')
   23 ## Section2('4.8')
   24 ## Problem2('')
   25 ## TitleText3('Calculus: Early Transcendentals')
   26 ## EditionText3('1')
   27 ## AuthorText3('Rogawski')
   28 ## Section3('4.8')
   29 ## Problem3('11')
   30 
   31 DOCUMENT();        # This should be the first executable line in the problem.
   32 
   33 loadMacros(
   34 "PG.pl",
   35 "PGbasicmacros.pl",
   36 "PGchoicemacros.pl",
   37 "PGanswermacros.pl",
   38 "PGauxiliaryFunctions.pl"
   39 );
   40 
   41 TEXT(beginproblem());
   42 $showPartialCorrectAnswers = 1;
   43 
   44 $a = random(400,600,1);
   45 $r = random(4,7,1);
   46 $x1 = 2;
   47 $y1 =  $x1**$r - $a;
   48 $der1 = $r*$x1**($r-1);
   49 $x2 = $x1 - $y1/$der1;
   50 $y2 =  $x2**$r - $a;
   51 $der2 = $r*$x2**($r-1);
   52 $x3 = $x2 - $y2/$der2;
   53 
   54 TEXT(EV2(<<EOT));
   55 Use Newton's method to approximate the value of
   56  \[  \sqrt[$r]{$a} \] as follows: $BR
   57 Let \( x_1 = $x1 \)
   58 be the initial approximation.
   59 $BR The second approximation \( x_2 \)
   60 is \{ans_rule(20) \}
   61 EOT
   62 
   63 $ans = $x2;
   64 ANS(num_cmp($ans));
   65 TEXT(EV2(<<EOT));
   66 $BR
   67 and the third approximation \( x_3 \)
   68 is \{ans_rule(20) \}
   69 EOT
   70 
   71 $ans = $x3;
   72 ANS(num_cmp($ans));
   73 ENDDOCUMENT();        # This should be the last executable line in the problem.