View of /trunk/NationalProblemLibrary/Rochester/setIntegrals0Theory/osu_in_0_16.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('integral' 'summation' 'area' 'riemann')
    6 ## Tagged by tda2d
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Integrals')
   10 ## DBsection('Area and Distance')
   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('5.2')
   23 ## Problem2('29')
   24 
   25 DOCUMENT();
   26 
   27 loadMacros(
   28 "PG.pl",
   29 "PGbasicmacros.pl",
   30 "PGchoicemacros.pl",
   31 "PGanswermacros.pl",
   32 "PGauxiliaryFunctions.pl"
   33 );
   34 
   35 
   36 $showPartialCorrectAnswers = 1;
   37 
   38 $a = random(2,7,1);
   39 $b = $a + random(1,4,1);
   40 $c = random(2,6,1);
   41 $c2 = 2*$c;
   42 $c3 = 3*$c;
   43 $answ = 2*($b+$c)**1.5/3 - 2*$b**1.5/3;
   44 
   45 TEXT(beginproblem());
   46 BEGIN_TEXT
   47 The following sum
   48 \[
   49 \sqrt{$b + \frac{$c}{n}} \cdot \left( \frac{$c}{n}\right) +
   50 \sqrt{$b + \frac{$c2}{n}} \cdot \left( \frac{$c}{n}\right) +
   51 \ldots + \sqrt{$b + \frac{$c n}{n}} \cdot \left( \frac{$c}{n}\right)
   52 \]
   53 
   54 is a right Riemann sum for the definite integral
   55 \[\int_{$a}^b f(x)\, dx\]
   56 $BR
   57 
   58 where \(b\) = \{ ans_rule()\}
   59 $BR
   60 
   61 and \(f(x)\) =  \{ ans_rule()\}
   62 $PAR
   63 
   64 It is also a Riemann sum for the definite integral
   65 \[\int_{$b}^c g(x)\, dx\]
   66 $BR
   67 
   68 where \(c\) = \{ ans_rule()\}
   69 $BR
   70 
   71 and \(g(x)\) =  \{ ans_rule()\}
   72 $PAR
   73 
   74 The limit of these Riemann sums as \(n \to \infty\) is
   75 \{ ans_rule(40)\}
   76 END_TEXT
   77 
   78 ##set $PG_environment{'textbook'} in webworkCourse.ph
   79 if (defined($textbook)) {
   80    if ($textbook eq "EllisGulick5") {
   81 BEGIN_TEXT
   82 $PAR
   83 $BBOLD Hint:$EBOLD This is similar to Problem 7 in WeBWorK Problem Set Hmwk \#1.
   84 END_TEXT
   85 }
   86 }
   87 
   88 ANS(num_cmp($a + $c));
   89 ANS(fun_cmp("sqrt($b - $a + x)", vars=>"x"));
   90 ANS(num_cmp($b + $c));
   91 ANS(fun_cmp("sqrt(x)", vars=>"x"));
   92 ANS(num_cmp($answ));
   93 
   94 ENDDOCUMENT();
   95