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

    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 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.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.