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Tue Nov 8 15:17:41 2011 UTC (18 months, 2 weeks ago) by aubreyja
File size: 1362 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('')
3 ## DBsection('')
4 ## KEYWORDS('calculus', 'integrals', 'integration', 'volume')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('6.2')
9 ## Problem1('55')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
15            "Parser.pl",
16            "freemanMacros.pl",
17            "PGgraphmacros.pl"
18            );
19 TEXT(beginproblem());
20 $showPartialCorrectAnswers = 0; 21 Context()->variables->are(y=>'Real'); 22 23$n = random(2,10,1);
24 $a = random(1,10,1); 25$m=$n+1; 26$ans = $a/($m**(1/$n)); 27 28 BEGIN_TEXT 29 \{ textbook_ref_exact("Rogawski ET 2e", "6.2","55") \}$BR
30 Let M be the average value of $$f(x) = x^{n}$$ on $$[0,a]$$. Find a value of $$c$$ in $$[0,a]$$
31 such that $$f(c) = M$$.
32
33 $PAR 34 $$c=$$\{ans_rule()\} 35 END_TEXT 36 37 ANS(num_cmp($ans));
38 SOLUTION(EV3(<<'END_SOLUTION'));
39 $PAR 40$SOL
41 \$PAR
42
43 We have
44 $45 \begin{array}{ll} 46 M &= \frac1{a - 0}\int_0^{a} x^{n} dx \cr 47 &= \frac1{a}\left .\frac{x^{m}}{m}\right|_0^{a}\cr 48 &= \frac1{a}\frac{a^{m}}{m}\cr 49 &= \frac{a^{n}}{m} 50 \end{array} 51$
52
53 Then $$M = f(c) = c^{n}$$ implies $$\frac{a^{n}}{m} = c^{n}$$ so that $$c =\frac{a}{\sqrt[n]{m}} = ans$$.
54
55
56 END_SOLUTION
57
58
59
60
61
62 ENDDOCUMENT();
63