## DESCRIPTION ## Max/min problems ## ENDDESCRIPTION ## KEYWORDS('Critical', 'Point', 'Partial', 'Multivariable') ## Tagged by nhamblet ## DBsubject('Calculus') ## DBchapter('Partial Derivatives') ## DBsection('Maximum and Minimum Values') ## Date('October 29, 2009') ## Author('Shifrin') ## Institution('UGA') ## TitleText1('') ## EditionText1('') ## AuthorText1('') ## Section1('') ## Problem1('') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGstandard.pl", "PGunion.pl", "Parser.pl", "parserVectorUtils.pl", "PGcourse.pl" ); TEXT(beginproblem()); BEGIN_PROBLEM(); ############################################## # Setup Context("Vector"); $a = random(1,4); do{$b = non_zero_random(-3,3)} until ($b!=1); do{$c = non_zero_random(-4,4)} until ($c**2<4*$a); $f = Formula("x^2 +$b y^2 + $c x")->reduce;$arg = '\left(\begin{array}{c} x\\y \end{array}\right)'; $max = max($c**2/(4*($b-1))+$a*$b,$a+sqrt($a)*$c, $a-sqrt($a)*$c, -$c**2/4); $min = min($c**2/(4*($b-1))+$a*$b,$a+sqrt($a)*$c, $a-sqrt($a)*$c, -$c**2/4); ############################################## # Main text Context()->texStrings; BEGIN_TEXT 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 $$D$$. $PAR maximum temperature = \{ans_rule(10)\}$PAR minimum temperature = \{ans_rule(10)\} $PAR END_TEXT Context()->normalStrings; ################################################## # Answers ANS(num_cmp($max)); ANS(num_cmp($min));$showPartialCorrectAnswers = 1; ################################################## END_PROBLEM(); ENDDOCUMENT(); # This should be the last executable line in the problem.