View of /trunk/NationalProblemLibrary/LoyolaChicago/Precalc/Chap9Sec5/Q13.pg

    1 # DESCRIPTION
    2 # Problem from Functions Modeling Change, Connally et al., 3rd ed.
    3 # WeBWorK problem written by Adam Spiegler, <aspiegler@luc.edu>
    4 # ENDDESCRIPTION
    5 
    6 ## DBsubject('Precalculus')
    7 ## DBchapter('Polynomial And Rational Functions')
    8 ## DBsection('The Short-Run Behavior Of Rational Functions')
    9 ## KEYWORDS('rational','fraction','polynomial,'asymptote','intercept')
   10 ## TitleText1('Functions Modeling Change')
   11 ## EditionText1('3')
   12 ## AuthorText1('Connally')
   13 ## Section1('9.5)
   14 ## Problem1('13')
   15 ## TitleText2('Functions Modeling Change');
   16 ## EditionText2('4')
   17 ## AuthorText2('Connally')
   18 ## Section2('11.5')
   19 ## Problem2('13')
   20 ## Author('Adam Spiegler')
   21 ## Institution('Loyola University Chicago')
   22 
   23 DOCUMENT();
   24 
   25 loadMacros("PG.pl",
   26            "PGbasicmacros.pl",
   27            "PGchoicemacros.pl",
   28            "PGanswermacros.pl",
   29            "PGgraphmacros.pl",
   30            "PGauxiliaryFunctions.pl",
   31            "extraAnswerEvaluators.pl",
   32            "MathObjects.pl",
   33            "PGcourse.pl",
   34      "AnswerFormatHelp.pl",
   35 );
   36 
   37 TEXT(beginproblem());
   38 
   39 Context("Numeric");
   40 
   41 
   42 $showPartialCorrectAnswers = 1;
   43 
   44 $pick = random(0,1,1);
   45 $b = non_zero_random(-9,9,1);
   46 
   47 if ($pick == 0) {
   48    $eqn = "\frac{x}{(x+$b)^3}";
   49    $right = "INFINITY";
   50    $right_sym = "\infty";
   51    $left = "-INFINITY";
   52    $left_sym = "- \infty"}
   53 else {
   54    $eqn = "\frac{-x}{(x+$b)^3}";
   55    $right = "-INFINITY";
   56    $left = "INFINITY";
   57    $right_sym = "- \infty";
   58    $left_sym = "\infty"};
   59 
   60 $q = -$b;
   61 
   62 $d[0] = .001;
   63 $d[1] = .01;
   64 $d[2] = .1;
   65 $d[3] = 1;
   66 
   67 for ($i = 0; $i <= 3; $i++){
   68   $x[$i] = $q+$d[$i];
   69   $r[$i] = round((-1)**{$pick}*$x[$i]/($x[$i]+$b)**3)};
   70 
   71 for ($i = 0; $i <= 3; $i++){
   72   $z[$i] = $q-$d[3-$i];
   73   $l[$i] = round((-1)**{$pick}*$z[$i]/($z[$i]+$b)**3)};
   74 
   75 Context()->texStrings;
   76 BEGIN_TEXT
   77 
   78 Let \( \displaystyle f(x) = $eqn \) and estimate the one-sided limits below.  If you need to enter \( \infty \) or \( - \infty \), enter INFINITY or -INFINITY.
   79 $PAR
   80 (a) \( \displaystyle \lim_{x \to $q^+} f(x) = \) \{ ans_rule(12) \}
   81 \{ AnswerFormatHelp("limits") \}
   82 $PAR
   83 (b) \( \displaystyle \lim_{x \to $q^-} f(x) = \) \{ ans_rule(12) \}
   84 \{ AnswerFormatHelp("limits") \}
   85 
   86 END_TEXT
   87 Context()->normalStrings;
   88 
   89 ANS( Compute($right)->cmp() );
   90 ANS( Compute($left)->cmp() );
   91 
   92 Context()->texStrings;
   93 SOLUTION(EV3(<<'END_SOLUTION'));
   94 $PAR
   95 $BBOLD  SOLUTION $EBOLD
   96 $PAR
   97 To find \( \displaystyle \lim_{x \to $q^+} f(x) \) we consider what happens to the function when \( x \) is slightly larger than \( $q \).  One way to investigate this behavior is to use a table such as the one below:
   98 $BR
   99 $BCENTER
  100 \{ begintable(6) \}
  101 \{ row ( "\( x \)", "$q", @x ) \}
  102 \{ row ( "\( f(x) \)", "Undefined", @r ) \}
  103 \{ endtable() \}
  104 $ECENTER
  105 $BR
  106 From the table we can see that \( \displaystyle \lim_{x \to $q^+} f(x) = $right_sym \).
  107 $PAR
  108 
  109 To find \( \displaystyle \lim_{x \to $q^-} f(x) \) we consider what happens to the function when \( x \) is slightly less than \( $q \).  Again we can use a table to investigate this behavior:
  110 $BR
  111 $BCENTER
  112 \{ begintable(6) \}
  113 \{ row ( "\( x \)", @z , "$q" ) \}
  114 \{ row ( "\( f(x) \)", @l, "Undefined" ) \}
  115 \{ endtable() \}
  116 $ECENTER
  117 $BR
  118 From the table we can see that \( \displaystyle \lim_{x \to $q^-} f(x) = $left_sym \).
  119 
  120 END_SOLUTION
  121 Context()->normalStrings;
  122 
  123 
  124 ENDDOCUMENT();