#DESCRIPTION
#KEYWORDS('limits', 'sequences')
# Find limits (intuitively or by experimental calculations).
#ENDDESCRIPTION

##KEYWORDS('Calculus')
##Tagged by ynw2d

##DBsubject('Calculus')
## DBchapter('Limits and Derivatives')
##DBsection('Continuity')


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

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

$showPartialCorrectAnswers=1;
# define a broken graph f and a broken graph g:

do {
# first define the parameters (value and first derivatives)
foreach my $i (0..5) {
	$f1y[$i] = random(0, 3, 1);
	$f1yp[$i] = random(-1,1, 2);
	$f2y[$i] = random(0, 3, 1);
	$f2yp[$i] = random(-1,1, 2);
	$g1y[$i] = random(0, 3, 1);
	$g1yp[$i] = random(-1,1, 2);
	$g2y[$i] = random(0, 3, 1);
	$g2yp[$i] = random(-1,1, 2);
}
  
# define the break points where we switch from one hemite function to the other
# and the value of the function at that point   

               
$xf = random(0,3);
$yf = random(-1,4);
$xg = random(0,3);
$yg = random(-1,4);

# Make sure the two functions break at different places (or we can get some
# questions repeated), and make sure they don't both vanish at the same time,
# regardless of which pieces are compared, to avoid L'hopital situations.

} until ( $xg != $xf and 
          ($f1y[$xf+1]**2 + $g1y[$xf+1]**2>0) and
          ($f1y[$xf+1]**2 + $g2y[$xf+1]**2>0) and
          ($f2y[$xf+1]**2 + $g1y[$xf+1]**2>0) and
          ($f2y[$xf+1]**2 + $g2y[$xf+1]**2>0) and
          ($f1y[$xg+1]**2 + $g1y[$xg+1]**2>0) and
          ($f2y[$xg+1]**2 + $g1y[$xg+1]**2>0) and
          ($f1y[$xg+1]**2 + $g2y[$xg+1]**2>0) and
          ($f2y[$xg+1]**2 + $g2y[$xg+1]**2>0)
);


# define the functions
$hermite_f1 = new Hermite( [ -1  , 0,  1, 2 , 3, 4 ],
                   [ @f1y ],
                   [ @f1yp  ]);
$hermite_f2 = new Hermite( [ -1  , 0,  1, 2 , 3, 4 ],
                   [ @f2y ],
                   [ @f2yp  ]);
$hermite_g1 = new Hermite( [ -1  , 0,  1, 2 , 3, 4 ],
                   [ @g1y ],
                   [ @g1yp  ]);
$hermite_g2 = new Hermite( [ -1  , 0,  1, 2 , 3, 4 ],
                   [ @g2y ],
                   [ @g2yp  ]);

# define the broken functions themselves -- we'll need them to answer questions.
sub f_rule {
	 my $x = shift;
     my $out;
     if ($x < $xf ) {
        $out = $hermite_f1->rf_f->($x);
     } elsif ($x == $xf) {
     	$out = $yf;
     } else {
        $out = $hermite_f2->rf_f->($x);
     }
      $out;
};
sub g_rule {
	my $x = shift;
    my $out;
    if ($x < $xg ) {
    	$out = $hermite_g1->rf_f->($x);
     } elsif ($x == $xg) {
        $out = $yg;
     } else {
        $out = $hermite_g2->rf_f->($x);
     }
     $out;
};


# plot the f graph
 
$graphf = init_graph(-2,-2,5,5,grid =>[7,7], axes => [0,0]);
$graphg = init_graph(-2,-2,5,5,,grid =>[7,7], axes => [0,0]);
$f1 = new Fun($hermite_f1->rf_f,$graphf);
$f2 = new Fun($hermite_f2->rf_f,$graphf);
$graphf->stamps(open_circle($xf, $hermite_f1->rf_f->($xf), 'red')   );
$graphf->stamps(open_circle($xf, $hermite_f2->rf_f->($xf), 'red')   );
$graphf ->stamps( closed_circle($xf, $yf, 'red') ); # value at $xf;
$f1->color('red');
$f2->color('red');
# restrict the two partial graph domains so that only the 'active' one is drawn
$f1->domain(-1, $xf);
$f2->domain($xf, 4);

#plot the g graph 
$g1 = new Fun($hermite_g1->rf_f,$graphg);
$g2 = new Fun($hermite_g2->rf_f,$graphg);
$graphg->stamps(open_circle($xg, $hermite_g1->rf_f->($xg), 'blue')  );
$graphg->stamps(open_circle($xg, $hermite_g2->rf_f->($xg), 'blue')  );
$graphg ->stamps(closed_circle($xg, $yg, 'blue') ); # value at $xg;
$g2->color('blue');
$g2->color('blue'); 
# restrict the two partial graph domains so that only the 'active' one is drawn 
$g1->domain(-1, $xg);
$g2->domain($xg, 4);

TEXT(beginproblem());


# draw the graphs
BEGIN_TEXT
\{ begintable(2) \}
\{ row( image( insertGraph($graphf), tex_size =>400), image( insertGraph($graphg) , tex_size =>400 )   )\}
\{ row( '\(f(x)\)', '\(g(x)\)') \}
\{ endtable() \}
END_TEXT

sub rnd {
	my $x = shift;
	my $places = shift;
	$places = 0 unless defined($places);
	my $sign = ($x <0 ) ? -1 : 1;
	$x = abs($x);
	
	$x = int( $x*10 **(-$places) +.5) *10 **($places);
	$sign*$x;
}
# now construct the questions
@questions = ();
@answers = ();

qa( ~~@questions, ~~@answers,
'\( \displaystyle \lim_{x\to $xf^-} [f(x) + g(x) ] \) ',
num_cmp(rnd( f_rule($xf-.00001) +g_rule($xf-.00001)  ), strings=>['DNE'] ),  # we are expecting integer answers
'\( \displaystyle \lim_{x\to $xf^+} [f(x) + g(x) ] \) ',
num_cmp(rnd( f_rule($xf+.00001) +g_rule($xf+.00001)  ), strings=>['DNE'] ),  # we are expecting integer answers
'\( f($xf) + g($xf) \)',
num_cmp(f_rule($xf) + g_rule($xf), strings=>['DNE'] ),

'\( \displaystyle \lim_{x\to $xf^-} [f(x)g(x) ] \) ',
num_cmp(rnd( f_rule($xf-.00001)*g_rule($xf-.00001) ), strings=>['DNE'] ),  # we are expecting integer answers
'\( \displaystyle \lim_{x\to $xf^+} [f(x)g(x) ] \) ',
num_cmp(rnd( f_rule($xf+.00001)*g_rule($xf+.00001)  ), strings=>['DNE'] ),  # we are expecting integer answers
'\( f($xf)g($xf) \)',
num_cmp(f_rule($xf)*g_rule($xf), strings=>['DNE'] ),

'\( \displaystyle \lim_{x\to $xf^-} [f( g(x) ) ] \) ',
num_cmp(rnd( f_rule(g_rule($xf-.00001)  ) ), strings=>['DNE'] ),  # we are expecting integer answers
'\( \displaystyle \lim_{x\to $xf^+} [f( g(x) ) ] \) ',
num_cmp(rnd( f_rule(g_rule($xf+.00001) ) ), strings=>['DNE'] ),  # we are expecting integer answers
'\( f( g($xf) ) \)',
num_cmp(f_rule( g_rule($xf) ), strings=>['DNE'] ),

#check for zeros in answering the division limits
'\( \displaystyle \lim_{x\to $xf^-} [f(x)/g(x) ] \) ',
  ( not rnd(g_rule($xf-.00001) ) and  rnd(f_rule($xf-.00001) ) ) ? num_cmp('DNE', strings=>['DNE'])        # if g is zero, but f is not
           : num_cmp(rnd( f_rule($xf-.00001)/g_rule($xf-.00001) , -3 ), tol =>.002, strings=>['DNE'] ),  # the chances of g being identically zero are small since the derivatives at defined points are never zero.
# we are expecting integer answers
'\( \displaystyle \lim_{x\to $xf^+} [f(x)/g(x) ] \) ',
( not rnd(g_rule($xf+.00001) ) and  rnd(f_rule($xf+.00001) ) ) ? num_cmp('DNE', strings=>['DNE'])
           : num_cmp(rnd( f_rule($xf+.00001)/g_rule($xf+.00001) , -3 ), tol =>.002, strings=>['DNE'] ) ,  
# we are expecting integer answers
'\( f($xf)/g($xf) \)',
(rnd(g_rule($xf)) ) ? num_cmp( rnd(f_rule($xf)/g_rule($xf), -3), tol =>.002, strings=>['DNE'] ) : num_cmp('DNE', strings=>['DNE']),

'\( \displaystyle \lim_{x\to $xg^-} [f(x) + g(x) ] \) ',
num_cmp(rnd( f_rule($xg-.00001) +g_rule($xg-.00001) ), strings=>['DNE'] ),   # we are expecting integer answers
'\( \displaystyle \lim_{x\to $xg^+} [f(x) + g(x) ] \) ',
num_cmp(rnd( f_rule($xg+.00001) +g_rule($xg+.00001)  ), strings=>['DNE'] ),  # we are expecting integer answers
'\( f($xg) + g($xg) \)',
num_cmp(f_rule($xg) + g_rule($xg), strings=>['DNE'] ),

'\( \displaystyle \lim_{x\to $xg^-} [f(x)g(x) ] \) ',
num_cmp(rnd( f_rule($xg-.00001)*g_rule($xg-.00001) ), strings=>['DNE'] ),  # we are expecting integer answers
'\( \displaystyle \lim_{x\to $xg^+} [f(x)g(x) ] \) ',
num_cmp(rnd( f_rule($xg+.00001)*g_rule($xg+.00001) ), strings=>['DNE'] ),  # we are expecting integer answers
'\( f($xg)g($xg) \)',
num_cmp(f_rule($xg)*g_rule($xg), strings=>['DNE'] ),

'\( \displaystyle \lim_{x\to $xg^-} [f( g(x) ) ] \) ',
num_cmp(rnd( f_rule( g_rule($xg-.00001) )), strings=>['DNE'] ) ,  # we are expecting integer answers
'\( \displaystyle \lim_{x\to $xg^+} [f( g(x) ) ] \) ',
num_cmp(rnd( f_rule( g_rule( $xg + .00001 ) ) ), strings=>['DNE']  ),  # we are expecting integer answers
'\( f( g($xg) ) \)',
num_cmp(f_rule( g_rule($xg) ), strings=>['DNE'] ),

#check for zeros in answering the division limits
'\( \displaystyle \lim_{x\to $xg^-} [f(x)/g(x) ] \) ',
( not rnd(g_rule($xg-.00001) ) and  rnd(f_rule($xg-.00001) ) ) ? num_cmp('DNE', strings=>['DNE'])        # if g is zero, but f is not
           : num_cmp(rnd( f_rule($xg-.00001)/g_rule($xg-.00001) , -3 ), tol =>.002, strings=>['DNE'] ),  # the chances of g being identically zero are small since the derivatives at defined points are never zero.
# we are expecting integer answers
'\( \displaystyle \lim_{x\to $xg^+} [f(x)/g(x) ] \) ',
( not rnd(g_rule($xg+.00001) ) and  rnd(f_rule($xg+.00001) ) ) ? num_cmp('DNE', strings=>['DNE'])
           : num_cmp(rnd( f_rule($xg+.00001)/g_rule($xg+.00001) , -3 ), tol =>.002, strings=>['DNE'] ) ,  
# we are expecting integer answers
'\( f($xg)/g($xg) \)',
(rnd(g_rule($xg)) ) ? num_cmp( rnd(f_rule($xg)/g_rule($xg), -3), tol =>.002, strings=>['DNE'] ) : num_cmp('DNE', strings=>['DNE']),

);

@slice = NchooseK(scalar(@questions) ,4);

BEGIN_TEXT
$BR
The graphs of \( f(x) \) and \( g(x) \) are given above.  Use them to evaluate each quantity below.
Write $BITALIC DNE$EITALIC if the limit or value does not exist (or if it's infinity).
$BR
\{ match_questions_list(@questions[@slice]  ) \}
END_TEXT


ANS(@answers[@slice]     );
ENDDOCUMENT();        # This should be the last executable line in the problem.