View of /trunk/NationalProblemLibrary/ASU-topics/setCalculus/stef/stef4_3p12.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('calculus', 'derivatives')
    6 ## Tagged by YL
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Applications of Differentiation')
   10 ## DBsection('How Derivatives Affect the Shape of a Graph')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('ASU')
   14 ## TitleText1('Calculus: Early Transcendentals')
   15 ## EditionText1('5')
   16 ## AuthorText1('Stewart')
   17 ## Section1('4.3')
   18 ## Problem1('')
   19 
   20 ## TitleText2('Calculus: Early Transcendentals')
   21 ## EditionText2('6')
   22 ## AuthorText2('Stewart')
   23 ## Section2('4.3')
   24 ## Problem2('')
   25 ## TitleText3('Calculus: Early Transcendentals')
   26 ## EditionText3('1')
   27 ## AuthorText3('Rogawski')
   28 ## Section3('4.5')
   29 ## Problem3('21')
   30 
   31 DOCUMENT();        # This should be the first executable line in the problem.
   32 
   33 loadMacros("PG.pl",
   34            "PGbasicmacros.pl",
   35            "PGchoicemacros.pl",
   36            "PGanswermacros.pl",
   37            "PGauxiliaryFunctions.pl",
   38     "extraAnswerEvaluators.pl");
   39 
   40 
   41 $a = random(1,6,1);
   42 $b = random(1,6,1);
   43 
   44 TEXT(beginproblem());
   45 
   46 $showPartialCorrectAnswers = 1;
   47 
   48 TEXT(EV2(<<EOT));
   49 Suppose that
   50 \[ f(x) = ($a-x)(x+$b)^2.  \]
   51 $BR
   52 (A) Find all critical numbers of \(f\).
   53 If there are no critical values, enter 'NONE'.
   54 $BR
   55 Critical values = \{ans_rule(20)\}
   56 $PAR
   57 EOT
   58 
   59 ANS(number_list_cmp( "-$b,(2*$a-$b)/3" , strings=>["none"] ));
   60 
   61 TEXT(EV2(<<EOT));
   62 (B) Use interval notation to indicate where \( f(x) \) is increasing.
   63 $BR
   64 $BBOLD Note: $EBOLD  Use 'INF' for \(\infty\), '-INF' for \(-\infty\),
   65 and use 'U' for the union symbol.
   66 $BR
   67 Increasing: \{ans_rule(25)\}
   68 $PAR
   69 EOT
   70 
   71 @answers = (interval_cmp("(-$b,-($b-2*$a)/3)"));
   72 ANS(@answers );
   73 
   74 TEXT(EV2(<<EOT));
   75 (C) Use interval notation to indicate where \( f(x) \) is decreasing.
   76 $BR
   77 Decreasing: \{ans_rule(25)\}
   78 $PAR
   79 EOT
   80 
   81 @answers = (interval_cmp("(-Inf,-$b)U(-($b-2*$a)/3,Inf)"));
   82 ANS(@answers );
   83 
   84 TEXT(EV2(<<EOT));
   85 (D) List the \(x\)-coordinates of all local maxima of \(f\).
   86 If there are no local maxima, enter 'NONE'.
   87 $BR
   88  \(x\) values of local maxima = \{ans_rule(10)\}
   89 $PAR
   90 EOT
   91 
   92 ANS(number_list_cmp( "(2*$a-$b)/3" , strings=>["none"] ));
   93 
   94 TEXT(EV2(<<EOT));
   95 (E) Find the \(x\)-coordinates of all local minima of \(f\).
   96  If there are no local minima, enter 'NONE'.
   97 $BR
   98 \(x\) values  of local minima = \{ans_rule(10)\}
   99 $PAR
  100 EOT
  101 
  102 ANS(number_list_cmp( "-$b" , strings=>["none"] ));
  103 
  104 TEXT(EV2(<<EOT));
  105 (F) Use interval notation to indicate where \( f(x) \) is concave up.
  106 $BR
  107 Concave up: \{ans_rule(25)\}
  108 $PAR
  109 EOT
  110 
  111 @answers = (interval_cmp("(-Inf,-(2*$b-$a)/3)"));
  112 ANS(@answers );
  113 
  114 TEXT(EV2(<<EOT));
  115 (G) Use interval notation to indicate where \( f(x) \) is concave down.
  116 $BR
  117 Concave down: \{ans_rule(25)\}
  118 $PAR
  119 EOT
  120 
  121 @answers = (interval_cmp("(-(2*$b-$a)/3,Inf)"));
  122 ANS(@answers );
  123 
  124 TEXT(EV2(<<EOT));
  125 (H) List the \(x\) values of  all inflection points of \(f\).
  126  If there are no inflection points, enter 'NONE'.
  127 $BR
  128 \(x\) values of inflection points = \{ans_rule(20)\}
  129 $PAR
  130 $PAR
  131 EOT
  132 
  133 ANS(number_list_cmp( "-(2*$b-$a)/3" , strings=>["none"] ));
  134 
  135 TEXT(EV2(<<EOT));
  136 (I) Use all of the preceding information to sketch a
  137 graph of \(f\).  When you're finished, enter a "1" in the box
  138 below.
  139 $BR
  140 Graph Complete: \{ans_rule(12)\}
  141 $PAR
  142 EOT
  143 
  144 @answers = (num_cmp(1) );
  145 ANS(@answers );
  146 
  147 ENDDOCUMENT();        # This should be the last executable line in the problem.