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