# Created by Nikola Kuzmanovski

########################################################################

DOCUMENT();      

loadMacros(
   "PGstandard.pl",
   "MathObjects.pl",
   "PGinfo.pl",
);

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

##############################################################
#
#  Setup
#
#
Context("Numeric");
Context()->variables->add(u => "Real");
Context()->variables->add(X => "Real");

$f = Formula("e^x");
$F = Formula("e^x");
$g = Formula("sin(x)");
$gD = $g->D('x');

$integrand = $f->substitute(x => $g) * $gD;
$integral = $F->substitute(x => $g);
$fu = $f->substitute(x => "u");
$Fu = $F->substitute(x => "u");


$lowerLimit = random(10, 19, 1);
$upperLimit = random(20, 29, 1);
$area = $integral->eval(x => $upperLimit) - $integral->eval(x => $lowerLimit);
##############################################################
#
#  Text
#
#

Context()->texStrings;
BEGIN_TEXT
This problem deals with the following integral
\[ \int $integrand dx .\]

Your job is to evaluate it using substitution.
We are going to use the variable \(u\) as the substitution variable.
Give a choice for \(u\) below.

$BR
$BCENTER
\(\displaystyle  u = \) \{ans_rule\}.
$ECENTER

$BR
Then
$BCENTER
\(\displaystyle  du = \) \{ans_rule\} \(dx\).
$ECENTER

$BR
With this we can change the integral \(\displaystyle  \int $integrand dx \) to an integral of the form \(\displaystyle \int g(u) du\) for some function \(g(u)\).
Input this function in the line below $BBOLD (note that the \(+C\) is included at the end, so you don't need to input it). $EBOLD.

$BR
$BCENTER
\(\displaystyle  \int $integrand dx = \int \) \{ans_rule\} \(du\).
$ECENTER

$BR
Next, find the general antiderivative $BBOLD (note that the \(+C\) is included at the end, so you don't need to input it). $EBOLD
$BR
$BCENTER
\(\displaystyle  \int $integrand dx = \) \{ans_rule\} \(+C\).
$ECENTER

$BR
Finally, compute
$BR
$BCENTER
\(\displaystyle  \int_{$lowerLimit}^{$upperLimit} $integrand dx = \) \{ans_rule\}.
$ECENTER

END_TEXT
Context()->normalStrings;

##############################################################
#
#  Answers
#
#

sub checkMult{
	my ( $correct, $student ) = @_;
	
	if($student->isConstant){
		return 0
	}
    
	my $context = Context()->copy;
	return 0 if $student == 0;
    
	$context->flags->set(no_parameters=>0);
	$context->variables->add('C0'=>'Parameter');
    
	my $c0 = Formula($context,'C0');
	$student = Formula($context,$student);
	$correct = Formula($context,"$c0 * $correct");
    
	return $correct == $student;
}

sub checkAdd{
	my ( $correct, $student ) = @_;
	
	if($student->isConstant){
		return 0
	}
    
	my $context = Context()->copy;
	return 0 if $student == 0;
    
	$context->flags->set(no_parameters=>0);
	$context->variables->add('C0'=>'Parameter');
    
	my $c0 = Formula($context,'C0');
	$student = Formula($context,$student);
	$correct = Formula($context,"$c0 + $correct");
    
	return $correct == $student;
}

sub checkMultAdd{
	my ( $correct, $student ) = @_;
	
	if($student->isConstant){
		return 0
	}
    
	my $context = Context()->copy;
	return 0 if $student == 0;
    
	$context->flags->set(no_parameters=>0);
	$context->variables->add('C0'=>'Parameter');
	$context->variables->add('C1'=>'Parameter');
    
	my $c0 = Formula($context,'C0');
	my $c1 = Formula($context,'C1');
	$student = Formula($context,$student);
	$correct = Formula($context,"$c1 + $c0 * $correct");
    
	return $correct == $student;
}


$studentU = Formula("");
$scoreU = 0.0;
$checkU = sub{
	if($inputs_ref->{previewAnswers}){
		return 0.0;
	}
	
	my ($correctAns, $studentAns, $ansHash) = @_;
	my $score = 0.0;
	$studentAns = Formula("$studentAns");
	$studentAns = $studentAns->substitute(X => "x");
	$studentU = $studentAns;
	
	my $studentSub = $f->substitute(x => $studentAns) * $gD;
	
	if($integrand == $studentSub){
		$ansHash->{ans_message} = "This substitution will work!";
		$score = 1.0;
	}elsif(checkMult($integrand, $studentSub)){
		$ansHash->{ans_message} = "This substitution will almost work. The final antiderivative will have an extra multiplicative constant";
		$score = 0.9;
	}elsif(checkMult($correctAns, $studentAns)){
		$ansHash->{ans_message} = "Your answer is close to the correct one. You have an extra multiplicative constant.";
		$score = 0.8;
	}elsif(checkAdd($correctAns, $studentAns)){
		$ansHash->{ans_message} = "Your answer is close to the correct one. You have added an extra constant.";
		$score = 0.8;
	}elsif(checkMultAdd($correctAns, $studentAns)){
		$ansHash->{ans_message} = "Your answer is close to the correct one. You have added an extra constant and you have an extra multiplicative constant.";
		$score = 0.7;
	}else{
		$ansHash->{ans_message} = "Your answer won't work. Try reviewing the section on integration by substitution.";
	}
	
	
	$scoreU = $score;
	return $score;
};
ANS($g->cmp(checker => $checkU));

$studentDiv = Formula("");
$scoreDiv = 0.0;
$checkDiv = sub{
	if($inputs_ref->{previewAnswers}){
		return 0.0;
	}
	
	my ($correctAns, $studentAns, $ansHash) = @_;
	my $score = 0.0;
	$studentAns = Formula("$studentAns");
	$studentAns = $studentAns->substitute(X => "x");
	$studentDiv = $studentAns;
	
	if($studentU == Formula("")){
		$ansHash->{ans_message} = "This answer depends on the previous answer. You don't get any points for it because you didn't enter anything for the previous answer.";
		return $score;
	}
	
	my $studentDerivative = $studentU->D('x');
	
	if($studentDerivative == $studentAns){
		$ansHash->{ans_message} = "You differentiated correctly!";
		$score = 1.0;
	}elsif(checkAdd($studentDerivative, $studentAns)){
		$ansHash->{ans_message} = "It seems like you made some mistake while differentiating. You have some extra additive constant.";
		$score = 0.9;
	}elsif(checkMult($studentDerivative, $studentAns)){
		$ansHash->{ans_message} = "It seems like you made some mistake while differentiating. You have some extra multiplicative constant.";
		$score = 0.9;
	}elsif(checkMultAdd($studentDerivative, $studentAns)){
		$ansHash->{ans_message} = "It seems like you made some mistake while differentiating. You have some extra additive constant and an extra multiplicative constant.";
		$score = 0.8;
	}else{
		$ansHash->{ans_message} = "It seems that you didn't differentiate correctly. Try reviewing differentiation from Calculus I.";
	}
	
	return $score;
};
ANS($gD->cmp(checker => $checkDiv));

$studentSub = Formula("");
$scoreSub = 0.0;
$checkSub = sub{
	if($inputs_ref->{previewAnswers}){
		return 0.0;
	}
	
	my ($correctAns, $studentAns, $ansHash) = @_;
	my $score = 0.0;
	$studentAns = Formula("$studentAns");
	$studentSub = $studentAns;
	
	if($studentU == Formula("") || $studentDiv == Formula("")){
		$ansHash->{ans_message} = "This answer depends on the previous two answers. You don't get any points for it because you didn't enter anything for the previous answers.";
		return $score;
	}
	
	if($correctAns == $studentAns){
		$ansHash->{ans_message} = "Good work substituting!";
		$score = 1.0;
	}elsif(checkAdd($correctAns, $studentAns)){
		$ansHash->{ans_message} = "It seems like you made some mistake while substituting. You have some extra additive constant.";
		$score = 0.9;
	}elsif(checkMult($correctAns, $studentAns)){
		$ansHash->{ans_message} = "It seems like you made some mistake while substituting. You have some extra multiplicative constant.";
		$score = 0.9;
	}elsif(checkMultAdd($correctAns, $studentAns)){
		$ansHash->{ans_message} = "It seems like you made some mistake while substituting. You have some extra additive constant and an extra multiplicative constant.";
		$score = 0.8;
	}else{
		$ansHash->{ans_message} = "It seems that your answer is not correct. Try reviewing integration by substitution.";
	}
	
	return $score;
};
ANS($fu->cmp(checker => $checkSub));

$studentIntegral = Formula("");
$scoreIntegral = 0.0;
$checkIntegral = sub{
	if($inputs_ref->{previewAnswers}){
		return 0.0;
	}
	
	my ($correctAns, $studentAns, $ansHash) = @_;
	my $score = 0.0;
	$studentAns = Formula("$studentAns");
	$studentAns = $studentAns->substitute(X => "x");
	$studentIntegral = $studentAns;

	
	if(checkAdd($correctAns, $studentAns)){
		$ansHash->{ans_message} = "Great Job!";
		$score = 1.0;
	}elsif(checkMultAdd($correctAns, $studentAns)){
		$ansHash->{ans_message} = "It seems like you made some mistake while integrating. You have some extra multiplicative constant.";
		$score = 0.9;
	}elsif(checkAdd($Fu, $studentAns)){
		$ansHash->{ans_message} = "You forgot to switch the u at the end.";
		$score = 0.8;
	}elsif(checkMultAdd($Fu, $studentAns)){
		$ansHash->{ans_message} = "You forgot to switch the u at the end and you have an extra multiplicative constant.";
		$score = 0.7;
	}else{
		$ansHash->{ans_message} = "It seems that your answer is not correct. Try reviewing integration by substitution.";
	}
	
	$scoreIntegral = $score;
	return $score;
};
ANS($integral->cmp(checker => $checkIntegral));

$studentFTC = Formula("");
$scoreFTC = 0.0;
$checkFTC = sub {
	if($inputs_ref->{previewAnswers}){
		return 0.0;
	}
	
	my ($correctAns, $studentAns, $ansHash) = @_;
	my $score = 0.0;
	$studentAns = Formula("$studentAns");
	$studentIntegral = $studentIntegral->substitute(u => "x");
	
	if($studentU == Formula("") || $studentDiv == Formula("") || $studentSub == Formula("")){
		$ansHash->{ans_message} = "This answer depends on the previous three answers. You don't get any points for it because you didn't enter anything for the previous answers.";
		$scoreFTC = 0.0;
		return $score;
	}
	
	if($scoreIntegral == 1.0 && $correctAns == $studentAns){
		$score = 1.0;
		$scoreFTC = $score;
		$ansHash->{ans_message} = "You computed the antiderivative correctly and used the Fundamental Theorem of Calculus correctly!";
		return $score;
	} 

	eval{
                my $lowerEval = $studentIntegral->eval(x => $lowerLimit);
				my $upperEval = $studentIntegral->eval(x => $upperLimit);
		
		if ($upperEval - $lowerEval == $studentAns){
			$ansHash->{ans_message} = "You didn't compute the antiderivative correctly. However, you used the Fundamental Theorem of Calculus correctly.";
			$score = 1.0;
		}elsif($lowerEval - $upperEval == $studentAns){
			$ansHash->{ans_message} = "You switched the bounds of integration.";
			$score = 0.9;
		}elsif($upperEval + $lowerEval == $studentAns){
			$ansHash->{ans_message} = "You need to subtract the lower bound, not add it.";
			$score = 0.9;
		}else{
			$ansHash->{ans_message} = "It seems that your answer is not correct. Try reviewing the Fundamental Theorem of Calculus.";
		}
		
		1;
	} or do{
		$ansHash->{ans_message} = "It seems that the answer you gave for the antiderivative doesn't evaluate at the bounds of the definite integral";
		$ansHash->{ans_message} = $ansHash->{ans_message} . " Try double checking your work for the antiderivative.";
		$ansHash->{ans_message} = $ansHash->{ans_message} . " Check for things like division by 0 or negatives under a square root.";
	};
	
	return $score;
};
ANS($area->cmp(checker => $checkFTC));


ENDDOCUMENT();