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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
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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('7')
   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 
   24 $a = Real(random(1, 9, 1));
   25 $start = $a;
   26 $asqr = $a**2;
   27 $a4 = 4*$a;
   28 $spasqr = $start +$asqr;
   29 
   30 $int_lim = Real(pi/(4*$a));
   31 
   32 ($func, $f, $ans_lim, $func2, $lim, $answer) = @{ list_random(
   33 ["\frac{1}{n^2 + $asqr}",
   34 Formula("1/(x^2+$asqr)"),
   35 $int_lim,
   36 "\frac{1}{x^2 + $asqr}",
   37 "\frac{1}{$a}\lim_{R\to\infty}\left(\tan^{-1}\left(\frac{R}{$a}\right) - \tan^{-1}\left(\frac{$a}{$a}\right)\right) = \frac{1}{$a}\left(\frac{\pi}{2} - \frac{\pi}{4}\right) = \frac{\pi}{$a4}",
   38 "converges"],
   39 ["\frac{1}{n + $asqr}",
   40 Formula("1/(x+$asqr)"),
   41 "inf",
   42 "\frac{1}{x + $asqr}",
   43 "\lim_{R\to\infty}\left(\ln(R+$asqr) - \ln($spasqr)\right) = \infty",
   44 "diverges"]
   45 ) };
   46 
   47 
   48 
   49 $wrong = "converges";
   50 
   51 if ($answer eq "converges") {
   52     $wrong = "diverges";
   53 }
   54 
   55 $mc = new_multiple_choice();
   56 
   57 $mc->qa("the infinite series \( \displaystyle \sum_{n=$start}^{\infty} $func \) ",
   58     $answer);
   59 $mc->extra($wrong);
   60 $mc->makeLast("diverges");
   61 
   62 Context()->texStrings;
   63 BEGIN_TEXT
   64 \{ beginproblem() \}
   65 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","7") \}
   66 $PAR
   67 Use the Integral Test to determine whether the infinite series is convergent.
   68 \[ \sum_{n=$start}^{\infty} $func \]
   69 Fill in the corresponding integrand and
   70 the value of the improper integral.
   71 $BR
   72 Enter $BBOLD inf $EBOLD for \(\infty\), $BBOLD -inf $EBOLD for \(-\infty\),
   73 and $BBOLD DNE $EBOLD if the limit does not exist.
   74 $PAR
   75 Compare with
   76 \(\int_{$start}^{\infty} \) \{ ans_rule() \} \(dx\) = \{ ans_rule() \}
   77 $PAR
   78 By the Integral Test,
   79 $BR
   80 \{ $mc->print_q; \}
   81 \{ $mc->print_a; \}
   82 $PAR
   83 END_TEXT
   84 Context()->normalStrings;
   85 
   86 ANS($f->cmp);
   87 ANS( num_cmp($ans_lim,strings=>["inf","INF", "-inf","-INF","DNE","dne"]));
   88 ANS(str_cmp($mc->correct_ans));
   89 
   90 $j1 = 0;
   91 
   92 Context()->texStrings;
   93 SOLUTION(EV3(<<'END_SOLUTION'));
   94 $PAR
   95 $SOL
   96 $PAR
   97 Let \( f(x) = $func2 \).  This function is continuous, positive and decreasing on the interval \( x \ge $start \), so the Integral Test applies.  Moreover,
   98 
   99 \[ \int_{$start}^{\infty} $func2 \, dx = \lim_{R\to\infty} \int_{$start}^{R} $func2 \, dx \]
  100 \[=  $lim\].
  101 $PAR
  102 The integral $answer; hence the series \( \displaystyle \sum_{n=$start}^{\infty} $func \) also $answer.
  103 $PAR
  104 END_SOLUTION
  105 
  106 ENDDOCUMENT();
  107 
  108 

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