##KEYWORDS('Sequences')
##DESCRIPTION
## Find a formula for the given sequences
##ENDDESCRIPTION

## Shotwell cleaned


## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('9')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 0;

$showHints = 1;

qa(~~@questions, ~~@answers,

"\( { \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, . . . } \)"	,	"1/(2^n)"	,
"\( { \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, . . . } \)"	,	"1/(2*n)"	,
"\( { \frac{3}{16}, \frac{4}{25}, \frac{5}{36}, \frac{6}{49}, . . . } \)",	"(n+2)/((n+3)^2)",
"\( { \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5} , . . . } \)"	,	"n/(n+1)"	);

# Now randomize the questions:
@slice = &NchooseK(4,2);
@shuffle = &shuffle(scalar(@slice));
($ans1, $ans2) = @answers[@slice] ;

BEGIN_TEXT
For each sequence, find a formula for the general term, \( a_n \).
For example, answer \(n ^2 \) if given the sequence 
\( \lbrace 1, 4, 9, 16, 25, 36, . . . \rbrace \)$BR
END_TEXT

TEXT(EV2(match_questions_list(@questions[@slice])));
ANS(fun_cmp($ans1, vars=> ['n']), fun_cmp($ans2, vars=> ['n']));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('algebra','geometric sequences')
##
## tsch tagged and PAID on 3-22-2004 

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('6')


##ENDDESCRIPTION

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGgraphmacros.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(-10,10,1);
$r = non_zero_random(2,6,.5);
$n = random(4,7,1);
$a2 = $a*$r;
$a3 = $a*$r**2;
$an = $a*$r**($n-1);


BEGIN_TEXT

Find the \( $n^{th} \) term of the geometric sequence 
$PAR
\( $a , $a2 , $a3 , ...\)


Answer: \{ans_rule(10)\}

END_TEXT

ANS(num_cmp($an, mode=>"arith"));

ENDDOCUMENT();       # This should be the last executable line in the problem.
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('sequences', 'limits')
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('43')


##ENDDESCRIPTION

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=non_zero_random(-8,8,2);
$b=random(2,4,1);

if ($a>0) {
	$soln=0;
} else {
	$soln='INF';
}
	
TEXT(EV2(<<EOT));

Determine whether the sequence is divergent or convergent.
If it is convergent, evaluate its limit.
If it diverges to infinity, state your answer as "INF" (without the
quotation marks).
If it diverges to negative infinity, state your answer as "MINF".
If it diverges without being infinity or negative infinity, state your
answer as "DIV".

$BR
\[ \lim_{n\rightarrow\infty} \frac{n^{$b}}{e^{$a n}} \]
$BR \{ans_rule( 30) \}
EOT
ANS(num_cmp($soln, strings=>['INF', 'MINF', 'DIV']));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

#DESCRIPTION
#Sequences. Find the limit of Convergence
#ENDDESCRIPTION

#Keywords('Sequences', 'convergence')
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('23')




DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,6,.1);



TEXT(EV3(<<'EOT'));

Find the limit of the sequence whose terms are given by
 $BR

\( a_n = (n^2)(1- \cos (\frac{$a}{n})  ) .  \)
$BR
\{ans_rule(40)\}
EOT

$answer = $a*$a/2 ;
ANS(num_cmp($answer));

&ENDDOCUMENT;
________________________________________________________________________________

##KEYWORDS('Sequences', 'convergence')
##DESCRIPTION
## Determine the limit of a sequence.
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('17')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,9,1);
$b = random(1,9,1);
$c = random(1,9,1);
$d = random(1,9,1);
$e = random(1,9,1);
$f  = random(1,9,1);
$num = nicestring([$a, $b, $c], ['n^2', 'n', '']);
$den= nicestring([$d, $e, $f], ['n^2', 'n', '']);

BEGIN_TEXT
Find the limit of the sequence: 
\( a_n = \frac{$num }{$den } \) 
$BR$BR
Answer: \{ans_rule(20)\}
END_TEXT

ANS(num_cmp("$a/$d"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('sequences', 'convergent', 'divergent')
##DESCRIPTION
## Determine if a sequence converges and compute its limit
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('19')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(2,5,1);
$b=random(2,5,1);
$c=random(4,19,1);
$d=random(3,18,1);
$f=random(3,18,1);

if ($a>$b) {
	$soln='inf';
} else {
	if ($a==$b) {
		$soln="$d/$f";
	} else {
		$soln=0;
	}
}
	
BEGIN_TEXT

Determine whether the sequence is divergent or convergent.
If it is convergent, evaluate its limit.$BR$BR

(If it diverges to infinity, state your answer as $BITALIC inf $EITALIC .
If it diverges to negative infinity, state your answer as $BITALIC -inf $EITALIC . If it diverges without being infinity or negative infinity, state your answer as $BITALIC div $EITALIC .)

$BR$BR
\[ \lim_{n\rightarrow\infty} \frac{$d ($a^n) + $c}{$f ($b^n)} \]
$BR$BR 
Answer: \{ans_rule( 30) \}
END_TEXT

ANS(num_cmp($soln, strings=>['inf', '-inf', 'div']));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('sequences', 'limits')
##DESCRIPTION
## Determine if a sequence converges and compute its limit
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('24')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(2,20,1);
$b=random(2,30,1);
$c=random(4,35,1);
$d=random(2,7,1);
$soln = "$b*pi/2";
	
BEGIN_TEXT

Determine whether the sequence is divergent or convergent.
If it is convergent, evaluate its limit. $BR$BR
(If it diverges to infinity, state your answer as $BITALIC inf $EITALIC . If it diverges to negative infinity, state your answer as $BITALIC -inf $EITALIC .
If it diverges without being infinity or negative infinity, state your
answer as $BITALIC div $EITALIC .

$BR$BR
\[ \lim_{n\rightarrow\infty} \frac{$c}{$a^n} + $b \arctan (n^{$d}) \]
$BR$BR
Answer: \{ans_rule( 30) \}

END_TEXT

ANS(num_cmp($soln, strings=>['inf', '-inf', 'div'] ) );

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('sequences', 'convergent', 'divergent')
##DESCRIPTION
## Determine if a sequence is convergent and compute its limit
##ENDDESCRIPTION

##Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('25')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a=random(4,19,1);

	
BEGIN_TEXT
Determine whether the sequence is divergent or convergent.
If it is convergent, evaluate its limit. $BR$BR

(If it diverges to infinity, state your answer as $BITALIC inf $EITALIC . If it diverges to negative infinity, state your answer as $BITALIC -inf $EITALIC .
If it diverges without being infinity or negative infinity, state your answer as $BITALIC div $EITALIC .)

$BR$BR
\[ \lim_{n\rightarrow\infty} \frac{$a + (-1)^n}{n^2} \]
$BR$BR
Answer:  \{ans_rule( 30) \}

END_TEXT

ANS(num_cmp(0, strings=>['inf', '-inf', 'div'] ) );


ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

#DESCRIPTION
#  Polar Coordinates
#  Arc Length.
#ENDDESCRIPTION
#KEYWORDS('Integration', 'Arc length', 'parametric')
## tsch tagged and PAID on 3-22-2004 

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('68')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",          
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$d = random(1,9,1);
$k = random(2,7,1);
$p = random(-1,1,2);
$e = $p + $k +$k*$k;
$w = $e**.5;
TEXT(EV3(<<'EOT'));
  
Let 
\(  \displaystyle f(x) = \frac{x}{x^2 +$d x +$e}. \)
$BR
A.  Find the smallest real number \(  r \) such that
\( f(x) \) is decreasing for all \( x \) greater than \( r. \)$BR


              \(r = \) \{ans_rule(20)\}
$BR 
    
B. Find the smallest integer \( s \) such that \( f(n) \) is decreasing for all integers \( n \) greater than or equal to
\( s \).

$BR

          \( s = \) \{ans_rule(20)\}

$BR 
EOT

$ans0 = $w;
$ans1 = $k + .5 +.5*$p;
#$ans2 = ($a + $b*cos( ($b/(2*$a) +1)*$pi) )*sin( ($b/(2*$a) +1)*$pi );
#$ans3 = 0 ;
ANS(num_cmp($ans0));
ANS(num_cmp($ans1));
#ANS(num_cmp($ans2));
#ANS(num_cmp($ans3));
ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('Sequences', 'limits')
##DESCRIPTION
##Sequences. Increasing or Decreasing?
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Sequences')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.1')
## Problem1('52 53 54 58')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

# No partial credit on this problem, so we say:

install_problem_grader(~~&std_problem_grader);


TEXT(beginproblem());
$showPartialCorrectAnswers = 0;

$a = random(2,5,1);
$b = random(6,9,1);


qa(~~@questions, ~~@answers,
"\( a_n = \frac{\cos n}{$a^n} \)", "N" ,
"\( a_n = \frac{n - $a}{n + $a} \)", "I" ,
"\( a_n = \frac{1}{$a n + $b} \)", "D" ,
"\( a_n = \frac{\sqrt{n + $a}}{$b n + $a} \)", "D"
);


# Now randomize the questions:
@slice = &NchooseK(4,4);
@shuffle = &shuffle(scalar(@slice));
($ans1, $ans2, $ans3, $ans4) = @answers[@slice] ;

BEGIN_TEXT
Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing,
enter $BITALIC I $EITALIC as your answer. If decreasing, enter $BITALIC D $EITALIC as your answer. If not monotonic, enter $BITALIC N $EITALIC as your answer. $BR
END_TEXT

TEXT(EV2(match_questions_list(@questions[@slice])));

ANS(str_cmp($ans1), str_cmp($ans2),
str_cmp($ans3), str_cmp($ans4));

ENDDOCUMENT();        # This should be the last executable line in the problem.