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('8/23/07')
   12 ## Author('')
   13 ## Institution('Union College')
   14 ## TitleText1('Calculus')
   15 ## EditionText1('7')
   16 ## AuthorText1('Anton')
   17 ## Section1('14.6')
   18 ## Problem1('5')
   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   "MathObjects.pl",
   31   "parserVectorUtils.pl",
   32   "PGchoicemacros.pl",
   33   "PGcourse.pl",
   34 );
   35 
   36 
   37 TEXT(beginproblem);
   38 
   39 ##############################################
   40 #  Setup
   41 
   42 Context("Vector");
   43 
   44 #
   45 #  The function
   46 #
   47 $nx = random(2,5,1);
   48 $ny = random(2,5,1);
   49 $nz = random(2,5,1);
   50 $m = $nx*$ny*$nz;
   51 
   52 $f = Formula("(x^$nx y^$ny z^$nz)/($nx*$ny*$nz)");
   53 
   54 #
   55 #  The point
   56 #
   57 ($x,$y,$z) = (non_zero_random(-3,3,1),
   58               non_zero_random(-3,3,1),
   59               non_zero_random(-3,3,1));
   60 
   61 #
   62 #  The vector
   63 #
   64 $U = Vector((2*non_zero_random(-1,1,1),
   65              2*non_zero_random(-1,1,1),
   66              non_zero_random(-1,1,1))[shuffle(3)]);
   67 
   68 #
   69 #  The derivatives
   70 #
   71 $fx = Formula("x^($nx-1) y^$ny z^$nz/($ny*$nz)");
   72 $fy = Formula("x^$nx y^($ny-1) z^$nz/($nx*$nz)");
   73 $fz = Formula("x^$nx y^$ny z^($nz-1)/($nx*$ny)");
   74 
   75 #
   76 #  Directional derivative
   77 #
   78 
   79 $gradf = Vector($fx,$fy,$fz);
   80 $Duf = $gradf->eval(x=>$x,y=>$y,z=>$z) . unit($U);
   81 
   82 ##############################################
   83 #  Main text
   84 
   85 $u = BoldMath('u');
   86 
   87 Context()->texStrings;
   88 BEGIN_TEXT
   89 
   90 Let \(f(x,y,z) = $DISPLAY $f\).
   91 $PAR
   92 
   93 Then \($GRAD\!f\) = \{ans_rule(60)\}, and \(D_{$u} f($x,$y,$z)\)
   94 in the direction of \($U\) is \{ans_rule(40)\}.
   95 
   96 END_TEXT
   97 Context()->normalStrings;
   98 
   99 ##################################################
  100 #  Answers
  101 
  102 ANS($gradf->cmp);
  103 ANS($Duf->cmp);
  104 
  105 $showPartialCorrectAnswers = 1;
  106 
  107 ##################################################
  108 
  109 ENDDOCUMENT();        # This should be the last executable line in the problem.