View of /trunk/NationalProblemLibrary/maCalcDB/setParametric1Curves/ur_pa_1_15.pg

    1 ##DESCRIPTION
    2 #  First Created: 3/2/02
    3 #  Last Modified: 10/22/02
    4 #  Authors: Minock/Mueller
    5 #  Math 162 Problems - Assignment 8 - Problem 12
    6 #  Stewart 4th ed. Section 10.2
    7 #  Find area of loop
    8 ##ENDDESCRIPTION
    9 
   10 ##KEYWORDS('parametric')
   11 ## tcao tagged and PAID on 12-12-2003
   12 
   13 ## DBsubject('Calculus')
   14 ## DBchapter('Parametric Equations and Polar Coordinates')
   15 ## DBsection('Tangents and Areas')
   16 ## Date('6/3/2002')
   17 ## Author('')
   18 ## Institution('')
   19 ## TitleText1('Calculus Early Transcendentals')
   20 ## EditionText1('4')
   21 ## AuthorText1('Stewart')
   22 ## Section1('10.2')
   23 ## Problem1('19')
   24 DOCUMENT();
   25 
   26 loadMacros(
   27 "PG.pl",
   28 "PGbasicmacros.pl",
   29 "PGchoicemacros.pl",
   30 "PGanswermacros.pl",
   31 "PGauxiliaryFunctions.pl"
   32 );
   33 
   34 TEXT(&beginproblem);
   35 $showPartialCorrectAnswers = 1;
   36 
   37 $a = random(1,5);
   38 $b = random(1,9);
   39 $c = random(1,9);
   40 $b2=$b*$b;
   41 $a2=$a*$a;
   42 
   43 $ans1 = $b**2-($a**2)/3;
   44 $ans2 = 0;
   45 $ans3 = (8/15)*($a**5);
   46 BEGIN_TEXT
   47 
   48 Notice that the curve given by the parametric equations
   49 \[ x=$b2 - t^2      \]
   50 \[    y= t^3 - $a2 t     \]
   51 is symetric about the x-axis.  (If t gives us the point (x,y), then -t will give
   52 (x,-y) ).
   53 $BR
   54 $BR
   55 At which x value is the tangent to this curve horizontal?
   56 $BR
   57 At x =  \{ ans_rule(10) \}
   58 END_TEXT
   59 ANS(std_num_cmp($ans1));
   60 
   61 BEGIN_TEXT
   62 $BR
   63 At which t value is the tangent to this curve vertical?
   64 $BR
   65 At t =  \{ ans_rule(10) \}
   66 END_TEXT
   67 ANS(std_num_cmp($ans2));
   68 
   69 BEGIN_TEXT
   70 $BR The curve makes a loop which lies along the x-axis.  What is
   71 the total area inside the loop?
   72 $BR
   73 Area =  \{ ans_rule(40) \}
   74 END_TEXT
   75 ANS(std_num_cmp($ans3));
   76 
   77 ENDDOCUMENT();
   78 
   79 
   80