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    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')
   10 ## Author('Justask')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl",
   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