View of /trunk/rochester_problib/setIntegrals0Theory/osu_in_0_18.pg

    1 ##DESCRIPTION
    2 ##KEYWORDS('integrals', 'inverse functions', 'area')
    3 ##Find integral of inverse function
    4 ##by relating it to integral of original function
    5 ##via relation between areas
    6 ##Authored by Zig Fiedorowicz 2/4/2000
    7 ##ENDDESCRIPTION
    8 
    9 DOCUMENT();
   10 
   11 loadMacros(
   12 "PG.pl",
   13 "PGbasicmacros.pl",
   14 "PGchoicemacros.pl",
   15 "PGanswermacros.pl",
   16 "PGauxiliaryFunctions.pl"
   17 );
   18 $showPartialCorrectAnswers = 1;
   19 
   20 $a= random(1,5,1);
   21 $b= random(2,15,1);
   22 $c= random(2,10,1);
   23 $quad = 3*$a;
   24 $lin =  3*($a**2+$b);
   25 $y1 = 1 - $quad + $lin + $c;
   26 $y2 = 8 - $quad*4 + $lin*2 +$c;
   27 $a1 = 1/4 - $quad/3 + $lin/2 + $c;
   28 $a2 = 16/4 - $quad*8/3 + $lin*4/2 + $c*2;
   29 $ans = 2*$y2 - $y1 - $a2 + $a1;
   30 
   31 TEXT(beginproblem());
   32 BEGIN_TEXT
   33 Consider the function
   34 \[f(x) = x^3 - $quad x^2 + $lin x + $c\]
   35 By drawing a suitable picture, find a relation between the definite
   36 integrals \( \displaystyle \int_1 ^2 f(x)\,dx\) and \( \displaystyle \int_{$y1}^{$y2}f^{-1}(x)\,dx\).
   37 Use this relation to find the second of these two integrals
   38 $PAR
   39  \( \displaystyle \int_{$y1}^{$y2}f^{-1}(x)\,dx\) =  \{ ans_rule(30)\}
   40 $PAR
   41 
   42 END_TEXT
   43 
   44 ANS(num_cmp($ans));
   45 
   46 ENDDOCUMENT();