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Revision 1457 - (download) (annotate)
Sat Jul 24 17:09:50 2010 UTC (4 years, 4 months ago) by ted shifrin
File size: 1785 byte(s)
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    1 ## DESCRIPTION
    2 ##   Max/min problems
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('Critical', 'Point', 'Partial', 'Multivariable')
    6 ## Tagged by nhamblet
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Partial Derivatives')
   10 ## DBsection('Maximum and Minimum Values')
   11 ## Date('October 29, 2009')
   12 ## Author('Shifrin')
   13 ## Institution('UGA')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23   "PGstandard.pl",
   24   "PGunion.pl",
   25   "Parser.pl",
   26   "parserVectorUtils.pl",
   27   "PGcourse.pl"
   28 );
   29 
   30 
   31 TEXT(beginproblem());
   32 BEGIN_PROBLEM();
   33 
   34 ##############################################
   35 #  Setup
   36 
   37 Context("Vector");
   38 
   39 $a = random(1,4);
   40 do{$b = non_zero_random(-3,3)} until ($b!=1);
   41 do{$c = non_zero_random(-4,4)} until ($c**2<4*$a);
   42 
   43 $f = Formula("x^2 + $b y^2 + $c x")->reduce;
   44 $arg = '\left(\begin{array}{c} x\\y \end{array}\right)';
   45 
   46 
   47 $max = max($c**2/(4*($b-1))+ $a*$b, $a+sqrt($a)*$c, $a-sqrt($a)*$c, -$c**2/4);
   48 $min = min($c**2/(4*($b-1))+ $a*$b, $a+sqrt($a)*$c, $a-sqrt($a)*$c, -$c**2/4);
   49 
   50 
   51 ##############################################
   52 #  Main text
   53 
   54 Context()->texStrings;
   55 BEGIN_TEXT
   56 
   57 The temperature of the circular plate \(D = \left\{"\{"\} $arg: x^2+y^2 \le $a \right\}\) is given by the function \(f $arg=$f\). Find the maximum and minimum temperatures of \(D\).
   58 
   59 $PAR
   60 maximum temperature = \{ans_rule(10)\}
   61 $PAR
   62 minimum temperature = \{ans_rule(10)\}
   63 
   64 $PAR
   65 
   66 END_TEXT
   67 Context()->normalStrings;
   68 
   69 ##################################################
   70 #  Answers
   71 
   72 ANS(num_cmp($max));
   73 ANS(num_cmp($min));
   74 
   75 $showPartialCorrectAnswers = 1;
   76 
   77 ##################################################
   78 
   79 END_PROBLEM();
   80 ENDDOCUMENT();        # This should be the last executable line in the problem.

aubreyja at gmail dot com
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