View of /trunk/NationalProblemLibrary/Utah/Calculus_II/set10_Infinite_Series/set10_pr13.pg

    1 #DESCRIPTION
    2 #  Taylor_Polynomials
    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 = 0;
   28 
   29 $a = 1 ;
   30 $b = random(3,7,2) ;
   31 $d = 1 ;
   32 
   33 # a simple code to perform a calculation.
   34   $i = 1;
   35         $f = 1;
   36 
   37   while ($i <= $b ) {
   38               $f = $f*$i ;
   39         $i = $i + 1;
   40     }
   41 
   42 TEXT(EV2(<<EOT));
   43 
   44 Let \( T_{k}(x) \): be the Taylor polynomial of degree k of the function
   45   \( f(x) = \sin(x) \)
   46 at \(a = 0 \). $BR
   47 
   48 Suppose you approximate \( f(x) \) by \( T_{k}(x) \), and if \( |x| \leq 1 \), how many terms
   49 do you need (that is, what is k) for you to have your error to be less than
   50 \( \frac{1}{$f} \) ?
   51 (Hint: use the alternating series approximation.)
   52 
   53 $BR
   54 
   55  \{ans_rule(20)\}
   56 
   57 EOT
   58 
   59 $answer = $b - 2;
   60 ANS(num_cmp($answer));
   61 
   62 ENDDOCUMENT();        # This should be the last executable line in the problem.
   63