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Updated/Consolidated tags.


    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
29            "PGbasicmacros.pl",
30            "PGchoicemacros.pl",
32            "PGauxiliaryFunctions.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.