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# View of /trunk/NationalProblemLibrary/Rochester/setVecFunction1Curves/urvc_3_11.pg

Mon May 22 17:36:35 2006 UTC (7 years ago) by jjholt
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Added tags.


    1 ## DESCRIPTION
2 ## Calculus
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS('vector' 'velocity' 'unit' 'tangent')
6 ## Tagged by tda2d
7
8 ## DBsubject('Calculus')
9 ## DBchapter('Vector Functions')
10 ## DBsection('Vector Functions and Space Curves')
11 ## Date('')
12 ## Author('')
13 ## Institution('Rochester')
14 ## TitleText1('')
15 ## EditionText1('')
16 ## AuthorText1('')
17 ## Section1('')
18 ## Problem1('')
19
20 DOCUMENT();        # This should be the first executable line in the problem.
21
23 "PG.pl",
24 "PGbasicmacros.pl",
25 "PGchoicemacros.pl",
27 "PGauxiliaryFunctions.pl"
28 );
29
30 TEXT(beginproblem());
31 $showPartialCorrectAnswers = 1; 32 33$a1 = random(1,5,1);
34 $b1 = random(1,5,1); 35$b2 = $b1*$b1;
36 $c1 = random(-5,5,1); 37$pi = arccos(-1);
38
39
40 BEGIN_TEXT
41 $PAR 42 For the given position vectors $$\mathbf{r}(t)$$ compute the unit tangent 43 vector $$\mathbf{T}(t)$$ for the given value of $$t$$ . 44$PAR
45
46  A.) $BR 47 If 48 $$\mathbf{r}(t)= (\cos a1 t,\sin a1 t )$$$BR
49  Then $$\mathbf{T}(\frac{\pi}{4})$$= ( \{ans_rule(15)\} , \{ans_rule(15)\} ) $BR$BR
50  C.)  $BR 51 If 52 $$\mathbf{r}(t)= (t^2,t^3)$$$BR
53  Then $$\mathbf{T}(b1)$$= ( \{ans_rule(20)\} , \{ans_rule(20)\} ) $BR 54 55 C.)$BR
56 If
57  $$\mathbf{r}(t)= e^{a1 t}\mathbf{i}+ e^{-b1 t}\mathbf{j}+ 58 t\mathbf{k}$$. $BR 59 Then $$\mathbf{T}(c1)$$= 60 \{ans_rule(13)\} $$\mathbf{i}+$$ \{ans_rule(13)\}$$\mathbf{j}+$$ 61 \{ans_rule(13)\} $$\mathbf{k}$$ . 62$BR
63 END_TEXT
64
65
66 $ans0 = -sin($a1*$pi/4) ; 67$ans1 =  cos($a1*$pi/4) ;
68 $ans2 = 2*$b1/((4*$b1*$b1+9*$b1*$b1*$b1*$b1)**.5);
69 $ans3 = 3*$b1*$b1/((4*$b1*$b1+9*$b1*$b1*$b1*$b1)**.5); 70$ans4 = $a1*exp($a1*$c1)/($a1*$a1*exp(2*$a1*$c1)+$b1*$b1*exp(-2*$b1*$c1)+1)**.5 ; 71$ans5 = -$b1*exp(-$b1*$c1)/($a1*$a1*exp(2*$a1*$c1)+$b1*$b1*exp(-2*$b1*$c1)+1)**.5 ; 72$ans6 =  1/($a1*$a1*exp(2*$a1*$c1)+$b1*$b1*exp(-2*$b1*$c1)+1)**.5 ;
73
74 ANS(num_cmp($ans0)); 75 ANS(num_cmp($ans1));
76 ANS(num_cmp($ans2)); 77 ANS(num_cmp($ans3));
78 ANS(num_cmp($ans4)); 79 ANS(num_cmp($ans5));
80 ANS(num_cmp(\$ans6));
81
82 ENDDOCUMENT();        # This should be the last executable line in the problem.