#DESCRIPTION
#KEYWORDS('continuity', 'theory')
# properties of continuous and differentiable functions -- theory
#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");

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

# continuity (for real numbers)

@questions = ();
@answers = ();
qa(~~@questions, ~~@answers,
EV2( "If \( f(x) \) is a  continuous function and the sequence \(a_{1}, a_{2}, a_{3}, ...\) converges to
a finite limit, then the sequence \(f(a_{1}), f(a_{2}), f(a_{3}), ...\) also converges to a limit."),
"T",

EV2( "If \( f(x) \) is a  continuous function and the sequence \(f(a_{1}), f(a_{2}), f(a_{3}), ...\) converges to
a finite limit, then the sequence \(a_{1}, a_{2}, a_{3}, ...\) also converges to a limit."),
"F",

# continuous functions and max and min

EV2( "Every continuous function has a maximum value."),
"F",

EV2( "Every continuous function whose domain is a bounded, closed
interval has a maximum value."),
"T",

EV2( "If a continuous function has a maximum value then it also has a minimum value."),
"F",

EV2( "Every continuous function
whose domain is a bounded, closed interval and which has a
maximum value also has a minimum value."),
"T",

EV2( "If a continuous function \(f(x)\) has a maximum value on an interval then the
function \( -f(x) \) has a minimum on that same interval."),
"T",

EV2( "If a continuous function has a maximum value then its domain must be a
bounded, closed interval."),
"F",

# differentiable functions and max-min
EV2( "Every differentiable function is continuous."),
"T",

EV2( "Every continuous function is differentiable."),
"F",

EV2( "Every differentiable function has a maximum value."),
"F",

EV2( "Every differentiable function whose domain is a bounded, closed
interval has a maximum value."),
"T",

EV2( "If a differentiable function has a maximum value then it also has a minimum value."),
"F",

EV2( "Every differentiable function
whose domain is a bounded, closed interval and which has a
maximum value also has a minimum value."),
"T",

EV2( "If a differentiable function \(f(x)\) has a maximum value on an interval then the
function \( -f(x) \) has a minimum on that same interval."),
"T",

EV2( "If a differentiable function has a maximum value then its domain must be a
bounded, closed interval."),
"F",

EV2( "If the linear approximation of a differentiable function is increasing at a point \( a \)
then the function is also increasing near the point \( a \)."),
"T",

EV2( "If a function is increasing near a point \(a \) then its linear approximation
 at \( a \) cannot be decreasing."),
"T",

EV2( "If the linear approximation of a differentiable function is decreasing at a point \( a \)
then the function could be constant near the point \( a \)."),
"F",

EV2( "If the linear approximation of a differentiable function is constant at a point \( a \) then
the function could be increasing near the point \( a \)."),
"T",

EV2( "If the linear approximation of a differentiable function is constant at a point \( a \) then
the function could be decreasing near the point \( a \)."),
"T",
);
$thisCourse = $inputs_ref->{'course'};
TEXT(EV2(<<EOT));
Enter a T or an F in each answer space below
to indicate whether the corresponding statement  is true or
false. $PAR A good technique is to think of several examples, especially examples
which might show that the statement is false! $PAR
For reference you can find
some definitions \{ &htmlLink("${htmlURL}continuitydefinitions.html","here")\}.
$PAR
You must get all of the answers correct to receive credit.
EOT

@slice = NchooseK(scalar(@questions),2);

TEXT(
&match_questions_list(@questions[@slice])
);
ANS(str_cmp([ @answers[@slice] ] ));

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