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    1 ## DESCRIPTION
    2 ## Calculus: Functions and models
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('calculus', 'functions','models')
    6 ## Tagged by LD
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Functions and Models')
   10 ## DBsection('New Functions from Old Functions')
   11 ## Date('5/26/2005')
   12 ## Author('Jeff Holt')
   13 ## Institution('UVA')
   14 ## TitleText1('Calculus: Early Transcendentals')
   15 ## EditionText1('5')
   16 ## AuthorText1('Stewart')
   17 ## Section1('1.3')
   18 ## Problem1('37')
   19 
   20 ## TitleText2('Calculus: Early Transcendentals')
   21 ## EditionText2('6')
   22 ## AuthorText2('Stewart')
   23 ## Section2('1.3')
   24 ## Problem2('')
   25 
   26 DOCUMENT();        # This should be the first executable line in the problem.
   27 
   28 loadMacros("PG.pl",
   29            "PGbasicmacros.pl",
   30            "PGchoicemacros.pl",
   31            "PGanswermacros.pl",
   32            "PGauxiliaryFunctions.pl",
   33            "extraAnswerEvaluators.pl");
   34 
   35 $a = random(2,8,1);
   36 $b = random(-8,-2,1);
   37 
   38 TEXT(beginproblem());
   39 
   40 TEXT(EV2(<<EOT));
   41 Suppose that
   42 \[
   43     f(x) = \frac{1}{x}  \quad \mathrm{and} \quad
   44     g(x) = {$a} x^3 ? {$b} x.
   45 \]
   46 For each function \(h\) given below, find a formula for \(h(x)\)
   47 and the domain of \(h\).
   48 $BR
   49 $BBOLD Note: $EBOLD When entering interval notation in WeBWorK, use
   50 $BBOLD I $EBOLD for \(\infty\), $BBOLD -I $EBOLD for \(-\infty\),
   51 and $BBOLD U $EBOLD for the union symbol.  If the set is empty,
   52 enter "{}" without the quotation marks.
   53 $BR
   54 (A)\(h(x) = (f \circ g)(x) \).
   55 $PAR
   56 \(h(x)\) = \{ans_rule(25)\}
   57 $PAR
   58 Domain = \{ans_rule(25)\}
   59 $BR
   60 EOT
   61 
   62 $ans1 = "1/($a*x^3+($b)*x)";
   63 $ep1=sqrt(- $b/$a);
   64 $aux1=$ep1+10;
   65 $ans2 = "(-I,- $ep1)U(- $ep1,0)U(0,$ep1)U($ep1,I)";
   66 
   67 ANS(( fun_cmp($ans1, limits=>[$ep1,$aux1], vars=>"x"),interval_cmp($ans2) ) );
   68 
   69 TEXT(EV2(<<EOT));
   70 (B) \(h(x) = (g \circ f)(x) \).
   71 $PAR
   72 \(h(x)\) = \{ans_rule(25)\}
   73 $PAR
   74 Domain = \{ans_rule(25)\}
   75 $BR
   76 EOT
   77 
   78 $ans3 = "$a*(1/x)^3+($b)/x";
   79 $ans4 = "(-I,0)U(0,I)";
   80 
   81 ANS(( fun_cmp($ans3, limits=>[1,10], vars=>"x"),interval_cmp($ans4) ) );
   82 
   83 TEXT(EV2(<<EOT));
   84 (C) \(h(x) = (f \circ f)(x) \).
   85 $PAR
   86 \(h(x)\) = \{ans_rule(25)\}
   87 $PAR
   88 Domain = \{ans_rule(25)\}
   89 $BR
   90 EOT
   91 
   92 $ans5 = "x";
   93 $ans6 = "(-I,0)U(0,I)";
   94 
   95 ANS(( fun_cmp($ans5, limits=>[0.5,10], vars=>"x"),interval_cmp($ans6) ) );
   96 
   97 TEXT(EV2(<<EOT));
   98 (D) \(h(x) = (g \circ g)(x) \).
   99 $PAR
  100 \(h(x)\) = \{ans_rule(25)\}
  101 $PAR
  102 Domain = \{ans_rule(25)\}
  103 $PAR
  104 EOT
  105 
  106 $ans7 = "$a*($a*x^3+($b)*x)^3+($b)*($a*x^3+($b)*x)";
  107 $ans8 = "(-I,I)";
  108 
  109 ANS(( fun_cmp($ans7, vars=>"x"),interval_cmp($ans8) ) );
  110 
  111 ENDDOCUMENT();        # This should be the last executable line in the problem.