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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 3486 byte(s)
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('5')
10 ## Author('Christopher Sira')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
18 $context = Context(); 19 20$context->variables->add(n=>'Real');
21
22 $start = Real(random(10, 25, 1)); 23$pc = Real(random(3, 9, 1));
24 $a = Real(random(3, 9, 2)); 25$am2 = $a-2; 26$ap2 = $a+2; 27$pc4 = 4*$pc; 28$a3m4 = 3*$a-4; 29$a3m2 = 3*$am2; 30$spc = $start**3+$pc;
31
32
33 $int_lim = Real(2/(3*$am2*($spc)**($am2/2)));
34
35 ($func,$f, $ans_lim,$func2, $deriv,$lim, $answer) = @{ list_random( 36 ["\frac{n^2}{\left(n^3 +$pc\right)^{\frac{$a}{2}}}", 37 Formula("x^2/(x^3 +$pc)^($a/2)"), 38$int_lim ,
39 "\frac{x^2}{\left(x^3 + $pc\right)^{\frac{$a}{2}}}",
40 "\frac{2x\left(x^3+$pc \right)^{\frac{$a}{2}}-x^2\cdot \frac{$a}{2}\left( x^3+$pc \right)^{\frac{$am2}{2}}\cdot 3x^2}{\left(x^3+$pc \right)^{$a}} = \frac{x($pc4-$a3m4 x^3)}{2(x^3+$pc)^{\frac{$ap2}{2}}}", 41 "-\frac{2}{$a3m2} \lim_{R\to\infty} \left(\frac{1}{(R^3 + $pc)^{\frac{$am2}{2}}} - \frac{1}{$spc^{\frac{$am2}{2}}}\right) = \frac{2}{$a3m2 \cdot$spc^{\frac{$am2}{2}}} ", 42 "converges"] 43 ) }; 44 45$extra = "To evaluate the improper integral, we use the substitution $$u = x^3 + pc$$, $$du = 3x^2 \, dx$$.  We then find$PAR"; 46 47$wrong = "converges";
48
49 if ($answer eq "converges") { 50$wrong = "diverges";
51 }
52
53 $mc = new_multiple_choice(); 54 55$mc->qa("the infinite series $$\displaystyle \sum_{n=start}^{\infty} func$$ ",
56     $answer); 57$mc->extra($wrong); 58$mc->makeLast("diverges");
59
60 Context()->texStrings;
61 BEGIN_TEXT
62 \{ beginproblem() \}
63 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","5") \}
64 $PAR 65 Use the Integral Test to determine whether the infinite series is convergent. 66 $\sum_{n=start}^{\infty} func$ 67 Fill in the corresponding integrand and 68 the value of the improper integral. 69$BR
70 Enter $BBOLD inf$EBOLD for $$\infty$$, $BBOLD -inf$EBOLD for $$-\infty$$,
71 and $BBOLD DNE$EBOLD if the limit does not exist.
72 $PAR 73 Compare with 74 $$\int_{start}^{\infty}$$ \{ ans_rule() \} $$dx$$ = \{ ans_rule() \} 75$PAR
76 By the Integral Test,
77 $BR 78 \{$mc->print_q; \}
79 \{ $mc->print_a; \} 80$PAR
81 END_TEXT
82 Context()->normalStrings;
83
84 ANS($f->cmp); 85 ANS(num_cmp($ans_lim,strings=>["inf","INF", "-inf","-INF","DNE","dne"]));
86 ANS(str_cmp($mc->correct_ans)); 87 88 Context()->texStrings; 89 SOLUTION(EV3(<<'END_SOLUTION')); 90$PAR
91 $SOL 92$PAR
93 Let $$f(x) = func2$$.  This function is continuous and positive on the interval $$x \ge start$$.  Moreover, because
94
95 $f'(x) = deriv,$
96
97 we see that $$f'(x) < 0$$ for $$x \ge start$$, so f is decreasing on the interval $$x \ge start$$.  The Integral Test therefore applies.  $extra 98$PAR
99 $\int_{start}^{\infty} func2 \, dx = \lim_{R\to\infty} \int_{start}^{R} func2 \, dx = \frac{1}{3}\lim_{R\to\infty}\int_{start^3 + pc}^{R^3 +pc} \frac{du}{u^{a/2}}$
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