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```Update to current Union versions (using MathObjects)
```

```    1 ## DESCRIPTION
2 ##   Tangent Plane to a Multivariate Function
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS('Tangent', 'Plane', 'Multivariable', 'Implicit', 'Parametric')
6 ## Tagged by nhamblet
7
8 ## DBsubject('Calculus')
9 ## DBchapter('Partial Derivatives')
10 ## DBsection('Tangent Planes')
11 ## Date('8/23/07')
12 ## Author('')
13 ## Institution('Union College')
14 ## TitleText1('Calculus')
15 ## EditionText1('7')
16 ## AuthorText1('Anton')
17 ## Section1('14.7')
18 ## Problem1('15')
19
20 DOCUMENT();        # This should be the first executable line in the problem.
21
23   "PGstandard.pl",
24   "PGunion.pl",
25   "parserParametricLine.pl",
26   "parserImplicitPlane.pl",
27   "parserVectorUtils.pl",
28   "PGchoicemacros.pl",
29   "PGcourse.pl",
30 );
31
32
33 TEXT(beginproblem);
34
35 ##############################################
36 #  Setup
37
38 Context("Vector");
39
40 #
41 #  Ceofficients for function
42 #
43 (\$a,\$b,\$c) = (random(3,5,1),random(1,2,1),1)[shuffle(3)];
44
45 #
46 #  The point
47 #
48 (\$x,\$y,\$z) = (non_zero_random(-1,1,1),
49         2*non_zero_random(-1,1,1),
50               non_zero_random(-3,3,1));
51
52 \$P = Point(\$x,\$y,\$z);
53
54 \$d = \$a + 4*\$b + \$c*\$z**2;
55
56 #
57 #  The function
58 #
59 \$f = Formula("\$a x^2 + \$b y^2 + \$c z^2")->reduce;
60
61 #
62 #  The Normal Vector
63 #
64 \$N = Vector(\$a*\$x,\$b*\$y,\$c*\$z);
65
66 #
67 #  The tangent plane
68 #
69 Context("ImplicitPlane");
70 \$T = ImplicitPlane(\$P,\$N);
71
72 #
73 #  The Normal Line
74 #
75 Context("ParametricLine");
76 \$L = ParametricLine(\$P,\$N);
77
78 ##############################################
79 #  Main text
80
81 Context()->texStrings;
82 BEGIN_TEXT
83
84 Consider the ellipsoid \(\$f = \$d\).
85 \$PAR
86
87 The implicit form of the tangent plane to this ellipsoid at
88 \(\$P\) is \{ans_rule(40)\}.
89 \$PAR
90
91 The parametric form of the line through this point that is
92 perpendicular to that tangent plane is \(L(t)\) = \{ans_rule(30)\}.
93
94 END_TEXT
95 Context()->normalStrings;
96
97 ##################################################