View of /trunk/NationalProblemLibrary/Rochester/setSeries9Taylor/S11.10.TaylorPolynomials.PTP02.pg

    1 ## DESCRIPTION
    2 ##   Approximating a Function with a Taylor Series
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('Series', 'Taylor', 'Approximate', 'Error')
    6 ## Tagged by nhamblet
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Infinite Sequences and Series')
   10 ## DBsection('Application of Taylor Polynomials')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 ## TitleText2('Calculus: Early Transcendentals')
   20 ## EditionText2('1')
   21 ## AuthorText2('Rogawski')
   22 ## Section2('8.4')
   23 ## Problem2('1')
   24 
   25 DOCUMENT();        # This should be the first executable line in the problem.
   26 
   27 loadMacros(
   28 "PG.pl",
   29 "PGbasicmacros.pl",
   30 "PGchoicemacros.pl",
   31 "PGanswermacros.pl",
   32 "PGauxiliaryFunctions.pl"
   33 );
   34 
   35 TEXT(beginproblem());
   36 $showPartialCorrectAnswers = 1;
   37 
   38 
   39 TEXT(EV2(<<EOT));
   40 
   41 (A) Find the fifth degree Taylor polynomial approximation \( T_5(x) \) centered at \(a = 0  \) to the function \( f(x) = \cos(x). \) $BR
   42 \(T_{5}(x) =  \) \{ans_rule(60)\}  $BR$BR
   43 
   44 (B) Find the fifth degree Taylor polynomial approximation \( T_5(x) \) centered at \(a = 0  \) to the function \( f(x) = \sin(x). \) $BR
   45 \(T_{5}(x) =  \) \{ans_rule(60)\}  $BR$BR
   46 
   47 
   48 (C) Find the fifth degree Taylor polynomial approximation \( T_5(x) \) centered at \(a = 0  \) to the function \( f(x) = e^x. \) $BR
   49 \(T_{5}(x) =  \) \{ans_rule(60)\}  $BR$BR
   50 
   51 
   52 EOT
   53 
   54 $taylorcos = "1-x^2/2+x^4/24" ;
   55 $taylorsin = "x - x^3/6 + x^5/120";
   56 $taylore = "1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120";
   57 
   58 ANS(fun_cmp($taylorcos, limits=>[5,6]));
   59 ANS(fun_cmp($taylorsin, limits=>[5,6]));
   60 ANS(fun_cmp($taylore, limits=>[5,6]));
   61 
   62 ENDDOCUMENT();