View of /trunk/NationalProblemLibrary/ma123DB/set4/s7_8_11.pg

    1 ##KEYWORDS('integrals', 'improper')
    2 ##DESCRIPTION
    3 ## Evaluate an improper integral
    4 ##ENDDESCRIPTION
    5 
    6 ## Shotwell cleaned
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Techniques of Integration')
   10 ## DBsection('Improper Integrals')
   11 ## Date('6/3/2002')
   12 ## Author('')
   13 ## Institution('')
   14 ## TitleText1('Calculus Early Transcendentals')
   15 ## EditionText1('4')
   16 ## AuthorText1('Stewart')
   17 ## Section1('7.8')
   18 ## Problem1('11')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PGbasicmacros.pl",
   24 "PGanswermacros.pl",
   25 "PGauxiliaryFunctions.pl"
   26 );
   27 
   28 TEXT(beginproblem());
   29 $showPartialCorrectAnswers = 1;
   30 
   31 $a=non_zero_random(-8, 8);
   32 $b=non_zero_random(-8, 8);
   33 $c=non_zero_random(-8, 8);
   34 $pol = nicestring([$a, $b, $c], ['x^2', 'x', '']);
   35 
   36 @functions=($pol, "$a x^4", "(x-$a)^2+$b");
   37 $choice=random(0, 2);
   38 
   39 BEGIN_TEXT
   40 Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as $BITALIC div $EITALIC .
   41 
   42 $BR \[ \int_{-\infty}^{\infty} ($functions[$choice]) dx \]
   43 $BR Answer:  \{ans_rule( 30) \}
   44 $BR
   45 END_TEXT
   46 ANS(num_cmp('div', strings=>['div']));
   47 
   48 if ($choice == 0) {
   49 &SOLUTION(EV3(<<'EOT'));
   50 $SOL $BR $BR
   51 \[
   52 \begin{align*}
   53 \int_{-\infty}^{\infty} ( $a x^2 + $b x + $c ) \; dx
   54 &=  \left \left( \frac{$a x^3}{3} + \frac{$b x^2}{2} + $c x \right)
   55     \right|_{-\infty}^{\infty}  \\\\
   56 &=  \lim_{x\to\infty} \left( \frac{$a x^3}{3}
   57       + \frac{$b x^2}{2} + $c x \right)
   58     - \lim_{x\to -\infty} \left( \frac{$a x^3}{3}
   59       + \frac{$b x^2}{2} + $c x \right)
   60 \end{align*}
   61 \]
   62 Both of these clearly diverge, because they're polynomials,
   63 so the integral in question diverges.
   64 EOT
   65 }
   66 if ($choice == 1) {
   67 &SOLUTION(EV3(<<'EOT'));
   68 $SOL $BR $BR
   69 \[
   70 \begin{align*}
   71 \int_{-\infty}^{\infty} $a x^4 \; dx
   72 &=  \left \frac{$a x^5}{5} \right|_{-\infty}^{\infty} \\\\
   73 &=  \lim_{x\to\infty} \frac{$a x^5}{5}
   74     - \lim_{x\to -\infty} \frac{$a x^5}{5}
   75 \end{align*}
   76 \]
   77 Both of these clearly diverge, because they're polynomials,
   78 so the integral in question diverges.
   79 EOT
   80 }
   81 if ($choice == 2) {
   82 $soln_2a = 2 * $a;
   83 $soln_bpa2 = $b + $a**2;
   84 &SOLUTION(EV3(<<'EOT'));
   85 $SOL $BR $BR
   86 \[
   87 \begin{align*}
   88 \int_{-\infty}^{\infty} \left( ( x - $a )^2 + $b \right) \; dx
   89 &=  \int_{-\infty}^{\infty} ( x^2 - $soln_2a x + $soln_bpa2 ) \; dx \\\\
   90 &=  \left \left( \frac{x^3}{3} - $a x^2 + $soln_bpa2 x \right)
   91     \right|_{-\infty}^{\infty}  \\\\
   92 &=  \lim_{x\to\infty} \left( \frac{x^3}{3} - $a x^2 + $soln_bpa2 x \right)
   93     - \lim_{x\to\infty} \left( \frac{x^3}{3}
   94       - $a x^2 + $soln_bpa2 x \right)
   95 \end{align*}
   96 \]
   97 Both of these clearly diverge, because they're polynomials,
   98 so the integral in question diverges.
   99 EOT
  100 }
  101 
  102 ENDDOCUMENT();        # This should be the last executable line in the problem.
  103