View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.7_Taylor_Series/10.7.58.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('58')
   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 
   27 
   28 
   29 $a = random(2, 9);
   30 $f = Formula("sin(x^($a)) cos(x^($a))");
   31 
   32 $firstfoursin = Formula("x^($a) - x^(3*$a)/3! + x^(5*$a)/5! - x^(7*$a)/7!")->reduce;
   33 $firstfourcos = Formula("1 - x^(2*$a)/2! + x^(4*$a)/4! - x^(6*$a)/6!")->reduce;
   34 
   35 $expanded = Formula("x^($a)-x^(3*$a)/6-x^(3*$a)/2+x^(5*$a)/24+x^(5*$a)/12+x^(5*$a)/120-x^(7*$a)/720-x^(7*$a)/144-x^(7*$a)/240-x^(7*$a)/5040");
   36 
   37 $answer = Formula("x^($a) - 2*x^(3*$a)/3 + 2*x^(5*$a)/15 - 4*x^(7*$a)/315")->reduce;
   38 
   39 
   40 Context()->texStrings;
   41 
   42 BEGIN_TEXT
   43 \{ beginproblem() \}
   44 \{ textbook_ref_exact("Rogawski ET 2e", "10.7", "58") \}
   45 $PAR
   46 Find the first four non-zero terms of the Maclaurin series for \( f(x) = $f \).
   47 $PAR
   48 \( f(x) = \) \{ ans_rule() \} \( + \cdots \)
   49 END_TEXT
   50 
   51 Context()->normalStrings;
   52 
   53 #Answer Check Time!
   54 ANS($answer->cmp);
   55 
   56 Context()->texStrings;
   57 SOLUTION(EV3(<<'END_SOLUTION'));
   58 $PAR
   59 $SOL
   60 Substituting \(x^{$a}\) in the Maclaurin series for \(\sin x\) and \(\cos x\), we find
   61 
   62 \[$f = ($firstfoursin+\cdots) ($firstfourcos+\cdots)\]
   63 \[ = $expanded+\cdots\]
   64 \[ = $answer + \cdots \]
   65 END_SOLUTION
   66 
   67 ENDDOCUMENT()