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| 1 : | ted shifri | 1457 | ## 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. |
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