View of /trunk/NationalProblemLibrary/Utah/Calculus_II/set10_Infinite_Series/set10_pr2.pg

    1 #DESCRIPTION
    2 #Taylor_series
    3 #ENDDESCRIPTION
    4 ## Author('Utah ww group')
    5 ## Institution('Univeristy of Utah')
    6 ## DBsubject('Calculus')
    7 ## DBchapter('Infinite Sequences and Series')
    8 ## DBsection('Taylor and MacLaurin Series')
    9 ## AuthorText1('Dale Varberg, Edwin J. Purcell, and Steve E. Rigdon')
   10 ## TitleText1('Calculus')
   11 ## EditionText1('9')
   12 ## Section1('Infinite Series')
   13 ## Problem1('')
   14 ## KEYWORDS('calculus')
   15 
   16 DOCUMENT();        # This should be the first executable line in the problem.
   17 
   18 loadMacros(
   19 "PG.pl",
   20 "PGbasicmacros.pl",
   21 "PGchoicemacros.pl",
   22 "PGanswermacros.pl",
   23 "PGauxiliaryFunctions.pl"
   24 );
   25 
   26 TEXT(beginproblem());
   27 $showPartialCorrectAnswers = 1;
   28 
   29 $b = random(2,3,1);
   30 
   31 if ($b==3) {
   32   $a = random(2,6,1);
   33   $c = 3;
   34 } else {
   35   $a = random(3,7,1);
   36   $c = random(3,5,2);
   37 }
   38 
   39 $p = ($c - 1)/2;
   40 
   41 # We'll ask for the m-th derivative, where m=c*b:
   42 
   43 $m = $c * $b;
   44 
   45 $endstr='th';
   46 
   47 # A quick routine to compute m factorial:
   48 
   49 $j=0;
   50 $mfact=1;
   51 while ($j<$m) {
   52   $j = $j + 1;
   53   $mfact = $j * $mfact;
   54 }
   55 
   56 # Now the answer:
   57 
   58 $ans = (-1)**$p * $mfact/($c * $a**$c);
   59 
   60 
   61 TEXT(EV2(<<EOT));
   62 
   63 Compute the $m$endstr derivative of
   64 \[ f(x) = \arctan\left( \frac{x^{$b}}{$a} \right) \]
   65 at \( x=0 \).
   66 $BR
   67 
   68 \( \displaystyle f^{($m)}(0) = \)
   69 \{ans_rule(30)\}
   70 
   71 $BR
   72 $BR
   73 Hint: Use the MacLaurin series for \( f(x) \).
   74 
   75 EOT
   76 
   77 ANS(num_cmp($ans));
   78 
   79 
   80 ENDDOCUMENT();        # This should be the last executable line in the problem.
   81