View of /trunk/NationalProblemLibrary/Indiana/Indiana_setIntegrals18Improper/sc5_9_10.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('improper' 'integral')
    6 ## Tagged by tda2d
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Integrals')
   10 ## DBsection('Improper Integrals')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Indiana')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 ## Before doing anything, we must import the macro definitions on the next few lines.
   21 
   22 DOCUMENT();        # This should be the first executable line in the problem.
   23 
   24 loadMacros(
   25 "PG.pl",
   26 "PGbasicmacros.pl",
   27 "PGchoicemacros.pl",
   28 "PGanswermacros.pl",
   29 "PGauxiliaryFunctions.pl"
   30 );
   31 
   32 TEXT(beginproblem());
   33 $showPartialCorrectAnswers = 1;
   34 
   35 $a=non_zero_random(-8, 8);
   36 $b=non_zero_random(-8, 8);
   37 $c=non_zero_random(-8, 8);
   38 
   39 @functions=("$a x^2 + $b x + $c", "$a x^4", "(x-$a)^2+$b");
   40 $choice=random(0, 2);
   41 
   42 $soln = "divergent";
   43 
   44 TEXT(EV2(<<EOT));
   45 Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent."
   46 
   47 $BR \[ \int_{-\infty}^{\infty} ($functions[$choice]) dx \]
   48 $BR $BR \{ans_rule( 30) \}
   49 $BR
   50 EOT
   51 ANS(str_cmp($soln));
   52 
   53 if ($choice == 0) {
   54 &SOLUTION(EV3(<<'EOT'));
   55 $SOL $BR $BR
   56 \[
   57 \begin{align*}
   58 \int_{-\infty}^{\infty} ( $a x^2 + $b x + $c ) \; dx
   59 &=  \left \left( \frac{$a x^3}{3} + \frac{$b x^2}{2} + $c x \right)
   60     \right|_{-\infty}^{\infty}  \\\\
   61 &=  \lim_{x\to\infty} \left( \frac{$a x^3}{3}
   62       + \frac{$b x^2}{2} + $c x \right)
   63     - \lim_{x\to -\infty} \left( \frac{$a x^3}{3}
   64       + \frac{$b x^2}{2} + $c x \right)
   65 \end{align*}
   66 \]
   67 Both of these clearly diverge, because they're polynomials,
   68 so the integral in question diverges.
   69 EOT
   70 }
   71 if ($choice == 1) {
   72 &SOLUTION(EV3(<<'EOT'));
   73 $SOL $BR $BR
   74 \[
   75 \begin{align*}
   76 \int_{-\infty}^{\infty} $a x^4 \; dx
   77 &=  \left \frac{$a x^5}{5} \right|_{-\infty}^{\infty} \\\\
   78 &=  \lim_{x\to\infty} \frac{$a x^5}{5}
   79     - \lim_{x\to -\infty} \frac{$a x^5}{5}
   80 \end{align*}
   81 \]
   82 Both of these clearly diverge, because they're polynomials,
   83 so the integral in question diverges.
   84 EOT
   85 }
   86 if ($choice == 2) {
   87 $soln_2a = 2 * $a;
   88 $soln_bpa2 = $b + $a**2;
   89 &SOLUTION(EV3(<<'EOT'));
   90 $SOL $BR $BR
   91 \[
   92 \begin{align*}
   93 \int_{-\infty}^{\infty} \left( ( x - $a )^2 + $b \right) \; dx
   94 &=  \int_{-\infty}^{\infty} ( x^2 - $soln_2a x + $soln_bpa2 ) \; dx \\\\
   95 &=  \left \left( \frac{x^3}{3} - $a x^2 + $soln_bpa2 x \right)
   96     \right|_{-\infty}^{\infty}  \\\\
   97 &=  \lim_{x\to\infty} \left( \frac{x^3}{3} - $a x^2 + $soln_bpa2 x \right)
   98     - \lim_{x\to\infty} \left( \frac{x^3}{3}
   99       - $a x^2 + $soln_bpa2 x \right)
  100 \end{align*}
  101 \]
  102 Both of these clearly diverge, because they're polynomials,
  103 so the integral in question diverges.
  104 EOT
  105 }
  106 
  107 ENDDOCUMENT();        # This should be the last executable line in the problem.
  108