View of /trunk/NationalProblemLibrary/Union/setMVderivatives/an14_6_5.pg

    1 ## DESCRIPTION
    2 ##   Calculate Gradient and Directional Derivative
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('Gradient', 'Directional', 'Derivative')
    6 ## Tagged by nhamblet
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Partial Derivatives')
   10 ## DBsection('Directional Derivatives and the Gradient Vector')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Union College')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 ## TitleText2('Calculus: Early Transcendentals')
   20 ## EditionText2('1')
   21 ## AuthorText2('Rogawski')
   22 ## Section2('14.5')
   23 ## Problem2('21 22 23 24 25 26 27 28 29 30')
   24 
   25 DOCUMENT();        # This should be the first executable line in the problem.
   26 
   27 loadMacros(
   28   "PGstandard.pl",
   29   "PGunion.pl",
   30   "Parser.pl",
   31   "parserVectorUtils.pl",
   32   "PGchoicemacros.pl",
   33   "PGcourse.pl"
   34 );
   35 
   36 
   37 TEXT(beginproblem());
   38 BEGIN_PROBLEM();
   39 
   40 ##############################################
   41 #  Setup
   42 
   43 Context("Vector");
   44 
   45 #
   46 #  The function
   47 #
   48 $nx = random(2,5,1);
   49 $ny = random(2,5,1);
   50 $nz = random(2,5,1);
   51 $m = $nx*$ny*$nz;
   52 
   53 $f = Formula("(x^$nx y^$ny z^$nz)/($nx*$ny*$nz)");
   54 
   55 #
   56 #  The point
   57 #
   58 ($x,$y,$z) = (non_zero_random(-3,3,1),
   59               non_zero_random(-3,3,1),
   60               non_zero_random(-3,3,1));
   61 
   62 #
   63 #  The vector
   64 #
   65 $U = Vector((2*non_zero_random(-1,1,1),
   66              2*non_zero_random(-1,1,1),
   67              non_zero_random(-1,1,1))[shuffle(3)]);
   68 
   69 #
   70 #  The derivatives
   71 #
   72 $fx = Formula("x^($nx-1) y^$ny z^$nz/($ny*$nz)");
   73 $fy = Formula("x^$nx y^($ny-1) z^$nz/($nx*$nz)");
   74 $fz = Formula("x^$nx y^$ny z^($nz-1)/($nx*$ny)");
   75 
   76 #
   77 #  Directional derivative
   78 #
   79 
   80 $gradf = Vector($fx,$fy,$fz);
   81 $Duf = $gradf->eval(x=>$x,y=>$y,z=>$z) . unit($U);
   82 
   83 ##############################################
   84 #  Main text
   85 
   86 $u = BoldMath('u');
   87 
   88 Context()->texStrings;
   89 BEGIN_TEXT
   90 
   91 Let \(f(x,y,z) = $DISPLAY $f\).
   92 $PAR
   93 
   94 Then \($GRAD\!f\) = \{ans_rule(60)\}, and \(D_{$u} f($x,$y,$z)\)
   95 in the direction of \($U\) is \{ans_rule(40)\}.
   96 
   97 END_TEXT
   98 Context()->normalStrings;
   99 
  100 ##################################################
  101 #  Answers
  102 
  103 ANS($gradf->cmp);
  104 ANS($Duf->cmp);
  105 
  106 $showPartialCorrectAnswers = 1;
  107 
  108 ##################################################
  109 
  110 END_PROBLEM();
  111 ENDDOCUMENT();        # This should be the last executable line in the problem.