SeriesTest1

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Requiring Students to Justify Series Tests

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This PG code shows how to require students to justify series tests.


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PG problem file Explanation

Problem tagging data

Problem tagging:

DOCUMENT();

loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"unionTables.pl",
"parserPopUp.pl",
"PGgraders.pl",
"parserMultiAnswer.pl",
);

TEXT(beginproblem());

Initialization: We load unionTables.pl to create a table in which answer blanks are stacked on top of each other to form a fraction. We use PGgraders.pl to give partial credit incrementally. We use parserMultiAnswer.pl for the fraction answer so that we can accept two correct answers, depending on how much a student has simplified their answer.

Context("Numeric")->variables->are(n=>"Real");

$a = random(2,9,1);
$b = random(2,9,1);
$c = random(5,20,1);
$d = random(3,9,1);
$e = random(2,9,1);

$dm1 = $d - 1;
$dm2 = $d - 2;


# TeX
$series = "\sum_{n=$c}^{\infty} \frac{$a n + $b}{$c n^{$d} + $e}";
$fraction = "\lim_{n\to\infty} \frac{a_n}{b_n} = \lim_{n\to\infty}";

$num1 = Formula("$a n^$d + $b n^$dm1");
$den1 = Formula("$c n^$d + $e");

$num2 = Formula("$a + $b/n");
$den2 = Formula("$c + $e/(n^$d)"); 

$multians = MultiAnswer($num1, $den1)->with(
  singleResult => 0,
  checker => sub {
      my ( $correct, $student, $ansHash ) = @_;
      my ( $stu1, $stu2 ) = @{$student};    
      
      if (($num1 == $stu1 && $den1 == $stu2) ||
          ($num2 == $stu1 && $den2 == $stu2) ) {
          return [1,1];
      } elsif (($num1 == $stu1 && $den2 == $stu2) || 
               ($num2 == $stu1 && $den1 == $stu2)) {
          $ansHash->setMessage(1,"Check your algebra");
          $ansHash->setMessage(2,"Check your algebra");
          return [0,0];
      } elsif ($num1 == $stu1 || $num2 == $stu1) {
          return [1,0];
      } elsif ($den1 == $stu2 || $den2 == $stu2) {
          return [0,1];
      } else {
          return [0,0];
      }
  }
);


$limit = Formula("$a/$c");
$popup = PopUp(["Choose","Converges","Diverges","Inconclusive"],"Converges");


# 
#  Display the fraction and answer blanks nicely
#
Context()->texStrings;
if ($displayMode eq 'TeX') {
  $showfraction =
  "\[ $fraction ".$multians->ans_rule(10).$multians->ans_rule(10)." \]";
} else {
  $showfraction =
  ColumnTable(
  "\( \displaystyle $fraction \)",
  $multians->ans_rule(20).$BR.$HR.$multians->ans_rule(20),
  indent => 0, separation => 10, valign => "MIDDLE"
  );
}
Context()->normalStrings;

Setup: We use the MultiAnswer object $multians to allow students to enter one of two correct answers. We could have also accomplished this using two custom answer checkers.

We display the answerblanks nicely as a fraction in HTML and TeX modes by how we constructed $showfraction.

Context()->texStrings;
BEGIN_TEXT
Use the limit comparison test to determine whether 
\( \displaystyle \sum_{n=$c}^{\infty} a_n =  $series \) 
converges or diverges.
$BR
$BR
(a) Choose a series \( \displaystyle \sum_{n=$c}^\infty b_n \) 
with terms of the form \( \displaystyle b_n = \frac{1}{n^p} \) 
and apply the limit comparison test.
Write your answer as a fully reduced fraction.
For \( n \geq $c \), 
$showfraction
$BR
(b) Evaluate the limit in the previous part.  Enter \( \infty \) 
as ${BITALIC}infinity${EITALIC} and \( -\infty \) 
as ${BITALIC}-infinity.${EITALIC}  If the limit does not exist, 
enter ${BITALIC}DNE.${EITALIC}
$BR$SPACE
\( \displaystyle \lim_{n\to\infty} \frac{a_{n}}{b_{n}} \, \) = 
\{ ans_rule(20) \}
$BR
$BR
(c) By the limit comparison test, does the series
converge, diverge, or is the test inconclusive?
\{ $popup->menu() \}
END_TEXT
Context()->normalStrings;

Main Text:

$showPartialCorrectAnswers=1;

install_problem_grader(~~&custom_problem_grader_fluid);

$ENV{'grader_numright'} = [2,4];
$ENV{'grader_scores'} = [0.4,1];
$ENV{'grader_message'} = "You can earn " .
"40% partial credit for 2 - 3 correct answers.";

ANS( $multians->cmp() );
ANS( $limit->cmp() );
ANS( $popup->cmp() );

Answer Evaluation: We use the problem grader fluid to give partial credit incrementally: 0% for 0-1 correct answers, 40% for 2-3 correct answers, and full credit for 4 correct answers.

Context()->texStrings;
BEGIN_SOLUTION
Solution explanation goes here.
END_SOLUTION
Context()->normalStrings;

COMMENT('MathObject version.');

ENDDOCUMENT();

Solution:

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