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

Wed Jun 27 19:34:12 2007 UTC (5 years, 11 months ago) by jjholt
File size: 2712 byte(s)
Updated/Corrected tags.  --JJH

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
25 "PG.pl",
26 "PGbasicmacros.pl",
27 "PGchoicemacros.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