View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.7_Taylor_Series/10.7.3.pg

    1 # DBsubject('Calculus')
    2 # DBchapter('')
    3 # DBsection('')
    4 # KEYWORDS('')
    5 # TitleText1('Calculus: Early Transcendentals')
    6 # EditionText1('2')
    7 # AuthorText1('Rogawski')
    8 # Section1('10.7')
    9 # Problem1('3')
   10 # Author('Emily Price')
   11 # Institution('W.H.Freeman')
   12 DOCUMENT();
   13 
   14 
   15 
   16 #Load Necessary Macros
   17 
   18 loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", );
   19 loadMacros("Parser.pl");
   20 loadMacros("freemanMacros.pl");
   21 
   22 
   23 Context()->variables->add(n=>'Real');
   24 
   25 #Book Values
   26 #$constant = -2
   27 
   28 $constant = random(2, 9);
   29 $sign = list_random(-1, 1);
   30 $coef = $sign*$constant;
   31 
   32 $denominator = Formula("1 - $coef*x")->reduce;
   33 #$mcterms = Formula("($coef*x)^n")->reduce;
   34 $mcterms = Formula("($coef*x)^n");
   35 $mcterms->{test_points}=[[1,0.1],[2,0.2],[3,0.3]];
   36 
   37 $interval = Interval("(-1/$constant,1/$constant)");
   38 
   39 Context()->texStrings;
   40 
   41 BEGIN_TEXT
   42 \{ beginproblem() \}
   43 \{ textbook_ref_exact("Rogawski ET 2e", "10.7", "3") \}
   44 $PAR
   45 Find the Maclaurin series and corresponding interval of convergence of the following function.
   46 $PAR
   47 \[f(x) = \frac{1}{$denominator} \]
   48 $PAR
   49 \( f(x) = \sum\limits_{n=0}^{\infty} \) \{ ans_rule() \}
   50 $PAR
   51 The interval of convergence is: \{ans_rule() \}
   52 END_TEXT
   53 
   54 Context()->normalStrings;
   55 
   56 #Answer Check Time!
   57 ANS($mcterms->cmp);
   58 ANS($interval->cmp);
   59 
   60 Context()->texStrings;
   61 SOLUTION(EV3(<<'END_SOLUTION'));
   62 $PAR
   63 $SOL
   64 
   65 Substituting \($coef x\) for \(x\) in the Maclaurin series for \(\frac{1}{1-x}\) gives
   66 \[ \frac{1}{$denominator} = \sum_{n=0}^{\infty} $mcterms \]
   67 This series is valid for \( |$constant x| < 1 \), or \( |x| < \frac{1}{$constant} \).  Thus, the interval of convergence is \((-\frac{1}{$constant},\frac{1}{$constant}) \).
   68 
   69 
   70 END_SOLUTION
   71 
   72 ENDDOCUMENT()