View of /branches/UGA/3.2.3a.pg

    1 ##DESCRIPTION
    2 #
    3 
    4 #
    5 # Asks for the tangent plane to a surface.
    6 #
    7 ##ENDDESCRIPTION
    8 
    9 ##KEYWORDS('Multivariable','Tangent Plane')
   10 
   11 ## DBsubject('Calculus')
   12 ## DBchapter('Partial Derivatives')
   13 ## DBsection('Tangent Planes')
   14 ## Date('9/7/2009')
   15 ## Author('Shifrin')
   16 ## Institution('UGA')
   17 
   18 
   19 DOCUMENT();        # This should be the first executable line in the problem.
   20 
   21 loadMacros(
   22   "PGstandard.pl",
   23   "PGunion.pl",
   24   "Parser.pl",
   25   "alignedChoice.pl",
   26   "PGcourse.pl",
   27   "PGanswermacros.pl",
   28   "PGbasicmacros.pl",
   29  );
   30 
   31 
   32 
   33 # loadMacros("PGstandard.pl",
   34 #           "PGbasicmacros.pl",
   35 #           "PGchoicemacros.pl",
   36 #           "PGanswermacros.pl",
   37 #           "PGauxiliaryFunctions.pl");
   38 
   39 
   40 TEXT(beginproblem());
   41 BEGIN_PROBLEM();
   42 
   43 
   44 Context("Numeric")->variables->are(x=>'Real',y=>'Real');
   45 $showPartialCorrectAnswers = 1;
   46 
   47 $a = non_zero_random( -4, 4, 1 );
   48 $aa = random(1,5,1);
   49 do{$b = non_zero_random( -4, 4, 1 )} until ($b!=$a);
   50 do{$bb = random(1,5,1)} until ($bb!=$aa);
   51 do{$c = random( 1, 10, 1 )} until ($c**2 > $a**2+$b**2);
   52 $d = non_zero_random( -4, 4, 1 );
   53 
   54 @func = ("e^($d x y)", "sqrt($c^2 - x^2 - y^2)", "sqrt($aa x^2+$bb y^2)",
   55 "$a x^2+$b y^2");
   56 
   57 @choice=NchooseK($#func,2);
   58 @sub_func=@func[@choice];
   59 
   60 for ($k=0; $k<2; $k++){
   61 $f[$k] = Formula("$sub_func[$k]")->reduce;
   62 $fx[$k] = $f[$k]->D('x');
   63 $fy[$k] = $f[$k]->D('y');
   64 $evalf[$k] = $f[$k]->eval(x=>$a,y=>$b);
   65 $evalfx[$k] = $fx[$k]->eval(x=>$a,y=>$b);
   66 $evalfy[$k] = $fy[$k]->eval(x=>$a,y=>$b);}
   67 
   68 
   69 $ans1x = $evalfx[0]->cmp; $ans1y = $evalfy[0]->cmp; $ans1z=$evalf[0]->cmp;
   70 $ans2x = $evalfx[1]->cmp; $ans2y = $evalfy[1]->cmp; $ans2z=$evalf[1]->cmp;
   71 
   72 
   73 Context()->texStrings;
   74 BEGIN_TEXT
   75 
   76 Let \( f(x,y) = $f[0] \).
   77 Find the equation of the tangent plane of the graph \( z = f(x,y) \) at the point \( \left( $a, $b, f($a,$b) \right) \).
   78  $PAR
   79 \( z = \)\{ans_rule(10)\}\(x + \) \{ans_rule(10)\} \( y + \) \{ans_rule(15)\}.
   80  $PAR
   81 $PAR
   82 Now let \( f(x,y) = $f[1] \).
   83 Find the equation of the tangent plane of the graph \( z = f(x,y) \) at the point \( \left( $a, $b, f($a,$b) \right) \).
   84  $PAR
   85 \( z = \)\{ans_rule(10)\}\(x + \) \{ans_rule(10)\} \( y + \) \{ans_rule(15)\}.
   86  $PAR
   87 $PAR
   88 
   89 END_TEXT
   90 
   91 ANS($ans1x); ANS($ans1y); ANS($ans1z);
   92 ANS($ans2x); ANS($ans2y); ANS($ans2z);
   93 
   94 END_PROBLEM();
   95 ENDDOCUMENT();        # This should be the last executable line in the problem.