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Fri Oct 7 16:42:13 2011 UTC (20 months, 1 week ago) by glarose
File size: 3258 byte(s)
LoyolaChicago 9.5: 4e tagging, updates.

    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')
21 ## Institution('Loyola University Chicago')
22
23 DOCUMENT();
24
26            "PGbasicmacros.pl",
27            "PGchoicemacros.pl",
29            "PGgraphmacros.pl",
30            "PGauxiliaryFunctions.pl",
32            "MathObjects.pl",
33            "PGcourse.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) \}
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();