View of /trunk/NationalProblemLibrary/Rochester/setDerivatives0Theory/s2_R_1-16.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS ('limit')
    6 ## Tagged by cmd6a 4/20/06
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Limits and Derivatives')
   10 ## DBsection('The Limit of a Function')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.pl",
   27 "PGauxiliaryFunctions.pl"
   28 );
   29 
   30 TEXT(beginproblem());
   31 $showPartialCorrectAnswers = 0;
   32 
   33 TEXT(EV2(<<EOT));
   34 Enter a T or an F in each answer space below
   35 to indicate whether the corresponding statement is true or
   36 false.  A statement is true only if it is true for all possibilities.
   37 You must get all of the answers correct to receive credit.
   38 $BR
   39 EOT
   40 $x1 = random(1,3,1);
   41 $a1 = random(2,7,1);
   42 $b1 = $x1**2 + $a1*$x1;
   43 $c1 = random(3,9,1);
   44 $d1 = $x1**2 + $c1*$x1;
   45 $questStr1 = EV2(" \(\displaystyle \lim_{x\rightarrow $x1}\frac
   46 {x^2+$a1 x- $b1}{x^2+$c1 x- $d1} = \frac
   47 {\displaystyle \lim_{x\rightarrow $x1} x^2+$a1 x- $b1}{\lim_{x\rightarrow $x1} x^2+$c1 x- $d1}\)");
   48 $ansStr1 = "F";
   49 
   50 $e1 = $x1**2+$a1*$x1-1;
   51 $f1 = $x1**2+$c1*$x1-2;
   52 $questStr2 = EV2(" \(\displaystyle \lim_{x\rightarrow $x1}\frac
   53 {x^2+$a1 x- $e1}{x^2+$c1 x- $f1} = \frac
   54 {\displaystyle \lim_{x\rightarrow $x1} x^2+$a1 x- $e1}{\displaystyle \lim_{x\rightarrow $x1} x^2+$c1
   55 x- $f1}\)");
   56 $ansStr2 = "T";
   57 
   58 $x3 = random(1,7,1);
   59 $questStr3 = EV2(" If \(\displaystyle \lim_{x\rightarrow $x3} f(x) = $a1\)
   60 and \(\displaystyle \lim_{x\rightarrow $x3} g(x) = 0\), then \(\displaystyle \lim_{x\rightarrow $x3}
   61 [f(x)/g(x)] \)
   62 does not exist");
   63 $ansStr3 = "T";
   64 
   65 $questStr4 = EV2(" If \(\displaystyle \lim_{x\rightarrow $x3} f(x) = 0\)
   66 and \(\displaystyle \lim_{x\rightarrow $x3} g(x) = 0\), then \(\displaystyle \lim_{x\rightarrow $x3}
   67 [f(x)/g(x)] \)
   68 does not exist");
   69 $ansStr4 = "F";
   70 
   71 $questStr5 = EV2(" If \(\displaystyle \lim_{x\rightarrow $x3} f(x) = 0\)
   72 and \(\displaystyle \lim_{x\rightarrow $x3} g(x) = $a1\), then \(\displaystyle \lim_{x\rightarrow
   73 $x3} [f(x)/g(x)] \)
   74 does not exist");
   75 $ansStr5 = "F";
   76 
   77 $x4 = random(1,7,1);
   78 $questStr6 = EV2(" If \(\displaystyle \lim_{x\rightarrow $x4} [f(x)g(x)]\) exists,
   79 then the limit is  \( f($x4)g($x4)\) ");
   80 $ansStr6 = "F";
   81 
   82 $questStr7 = EV2(" If \(p(x)\) is a polynomial, then
   83 then the limit \(\displaystyle \lim_{x\rightarrow $x4} p(x)\) is  \( p($x4)\)");
   84 $ansStr7 = "T";
   85 
   86 $questStr8 = EV2(" If \(\displaystyle \lim_{x\rightarrow $x3} f(x) = \infty \)
   87 and \(\displaystyle \lim_{x\rightarrow $x3} g(x) = \infty \), then \(\displaystyle \lim_{x\rightarrow
   88 $x3} [f(x)-g(x)] =0 \)
   89 ");
   90 $ansStr8 = "F";
   91 
   92 $questStr9 = EV2(" If \(f'($x4)\) exists, then
   93 then the limit \(\displaystyle \lim_{x\rightarrow $x4} f(x)\) is  \( f($x4)\)");
   94 $ansStr9 = "T";
   95 
   96 $questStr10 = EV2(" If \(f(x)\) is continuous at \(a\), then
   97  \(f(x)\) is differentiable at \(a\)");
   98 $ansStr10 = "F";
   99 
  100 $questStr11 = EV2(" If \(f(x)\) is differentiable at \(a\), then
  101  \(f(x)\) is continuous at \(a\)");
  102 $ansStr11 = "T";
  103 
  104 @questions =($questStr1,$questStr2,$questStr3,$questStr4,$questStr5,$questStr6,$questStr7,$questStr8,$questStr9,$questStr10,$questStr11);
  105 @answers =($ansStr1,$ansStr2,$ansStr3,$ansStr4,$ansStr5,$ansStr6,$ansStr7,$ansStr8,$ansStr9,$ansStr10,$ansStr11);
  106 
  107 ## Now choose radomly 5  questions out of the 11 question strings above.
  108 
  109 @slice = NchooseK(scalar(@questions),5);
  110 
  111 ## Next we output the 5 chosen questions.
  112 TEXT(
  113 &match_questions_list(@questions[@slice])
  114 );
  115 
  116 ANS(str_cmp([@answers[@slice]]));
  117 ENDDOCUMENT();        # This should be the last executable line in the problem.