[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.3_Convergence_of_Series_with_Positive_Terms / 10.3.21.pg Repository:
ViewVC logotype

View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.3_Convergence_of_Series_with_Positive_Terms/10.3.21.pg

Parent Directory Parent Directory | Revision Log Revision Log


Revision 2584 - (download) (annotate)
Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2472 byte(s)
Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Infinite Series')
    3 ## DBsection('Convergence of Series with positive terms')
    4 ## KEYWORDS('calculus', 'infinite series', 'series', 'converge', 'convergence', 'comparison test', 'integral test', 'limit')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('10.3')
    9 ## Problem1('21')
   10 ## Author('Christopher Sira')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("PGchoicemacros.pl");
   16 loadMacros("Parser.pl");
   17 loadMacros("freemanMacros.pl");
   18 $context = Context();
   19 
   20 $context->variables->add(n=>'Real');
   21 
   22 
   23 $a = Real(random(4, 9, 1));
   24 $p = Real(random(2, 7, 1));
   25 
   26 $ch = random(0,1,1); ##0 diverge, 1 converge! LAD
   27 
   28 if ($ch >0) {#converge
   29     $answer = "converges";
   30     $wrong = "diverges";
   31     $start = 1;
   32     $func = "\frac{n^{\frac{1}{$a}}}{n^{$p} + n}";
   33     $bn = "\frac{n^{\frac{1}{$a}}}{n^{$p}}";
   34     $b = $a * $p - 1;
   35     $bounds = "\le";
   36     $compare = ">";
   37 
   38 }else{
   39     $answer = "diverges";
   40     $wrong = "converges";
   41     $start = 2;
   42     $c = $a*($p-1)+1;
   43     $func = "\frac{n^{\frac{$c}{$a}}}{n^{$p} - n}";
   44     $bn = "\frac{n^{\frac{$c}{$a}}}{n^{$p}}";
   45     $b = $a * $p - $c;
   46     $bounds = "\ge";
   47     $compare = "<";
   48 }
   49 
   50 $mc = new_multiple_choice();
   51 
   52 $mc->qa("the infinite series \( \displaystyle \sum_{n=$start}^{\infty} $func \) ",
   53     $answer);
   54 $mc->extra($wrong);
   55 $mc->makeLast("diverges");
   56 
   57 Context()->texStrings;
   58 BEGIN_TEXT
   59 \{ beginproblem() \}
   60 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","21") \}
   61 $PAR
   62 Use the Comparison Test to determine whether the infinite series is convergent.
   63 \[ \sum_{n=$start}^{\infty} $func \]
   64 
   65 By the Comparison Test,
   66 
   67 \{ $mc->print_q; \}
   68 \{ $mc->print_a; \}
   69 $BR
   70 $BBOLD Note: $EBOLD You are allowed only one attempt on this problem.
   71 $PAR
   72 END_TEXT
   73 Context()->normalStrings;
   74 
   75 ANS(str_cmp($mc->correct_ans));
   76 
   77 $j1 = 0;
   78 
   79 Context()->texStrings;
   80 SOLUTION(EV3(<<'END_SOLUTION'));
   81 $PAR
   82 $SOL
   83 $PAR
   84 For \( n \ge $start \),
   85 
   86 \[ $func $bounds $bn = \frac{1}{n^{\frac{$b}{$a}}} .\]
   87 
   88 The series \( \displaystyle \sum_{n=$start}^{\infty} \frac{1}{n^{\frac{$b}{$a}}} \) is a \(p\)-series with \( p = \frac{$b}{$a} $compare 1 \), so it $answer.  By the Comparison Test we can therefore conclude that the series \(\sum_{n=$start}^{\infty} $func\) also $answer.
   89 $PAR
   90 END_SOLUTION
   91 
   92 ENDDOCUMENT();
   93 
   94 

aubreyja at gmail dot com
ViewVC Help
Powered by ViewVC 1.0.9