##DESCRIPTION
##KEYWORDS('derivatives', 'trigonometric functions', 'product rule')
##  differentiation of  x^n*sin(x)
##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;

$an = random(2,5,1);
$as = random(-1,1,2);
$a  = $an * $as;
$n  = random(2,8,1);

$funct1 = "$a*x^($n-1)*($n*sin(x)+ x*cos(x))";

TEXT(EV2(<<EOT));
Let \[ f(x) = $a x^ {$n} \sin(x) \]
$PAR
\( f'( x ) = \) \{ans_rule(40) \}
EOT

$ans = $funct1;
&ANS(function_cmp($ans));


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

##DESCRIPTION
##KEYWORDS('derivatives', 'logarithmic functions')
##  differentiation of log function a x^n*ln(x)
##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;

$an = random(2,5,1);
$as = random(-1,1,2);
$a  = $an * $as;
$n  = random(2,8,1);

$funct1 = "$a*x^($n-1)*($n*ln(x)+ 1)";

TEXT(EV2(<<EOT));
Let \[ f(x) = $a x^ {$n} \ln x \]
$PAR
\( f'( x ) = \) \{ans_rule(40) \}
EOT

$ans = $funct1;
&ANS(function_cmp($ans));


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

##DESCRIPTION
##KEYWORDS('derivatives', 'exponential functions', 'inverse trig functions')
##  differentiation of log function a e^x*arcsin(x)
##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;

$an = random(2,5,1);
$as = random(-1,1,2);
$a  = $an * $as;
$n  = random(2,8,1);

$funct1 = "$a  e^x *($n/sqrt(1-($n x)^2)+ arcsin($n x))";

BEGIN_TEXT
Let \[ f(x) = $a e^x \arcsin($n x) \]
$PAR
\( f'( x ) = \) \{ans_rule(40) \}
END_TEXT

$ans = $funct1;
ANS(fun_cmp($ans,domain=>[-.99/$n, .99/$n] ));


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

##DESCRIPTION
##KEYWORDS('derivatives', 'logarithmic functions')
##  differentiation of log function a sqrt(x)*ln(x)
##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;

$an = random(2,5,1);
$as = random(-1,1,2);
$a  = $an * $as;
$n  = random(2,8,1);

$funct1 = "$a*$n*x^($n-1)*(sec($a x^$n)^2)";

TEXT(EV2(<<EOT));
Let \[ f(x) =  \tan($a x^$n) \]
$PAR
\( f'( x ) = \) \{ans_rule(40) \}
EOT

$ans = $funct1;
&ANS(function_cmp($ans));


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

##DESCRIPTION
##KEYWORDS('derivatives', 'exponential functions', 'inverse trig functions', 'chain rule')
##  differentiation of arctan(e^nx)
##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;

$n  = random(2,8,1);

$funct1 = "(1/(1+e^(2*$n*x)))($n*e^($n x))";

TEXT(EV2(<<EOT));
Let \[ f(x) =  \arctan(e^{$n x}) \]
$PAR
\( f'( x ) = \) \{ans_rule(40) \}
EOT

$ans = $funct1;
&ANS(function_cmp($ans));


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

##DESCRIPTION
##KEYWORDS('derivatives', 'trigonometric functions', 'product rule')
##  differentiation of sqrt(x^2+a)*cos(x)
##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;

$an = random(2,5,1);
$as = random(-1,1,2);
$a  = $an * $as;
$n  = random(2,8,1);

$funct1 = "cos($n x)*(1/2)*(1/sqrt(x^2+$a))*(2 x)-$n sin($n x)sqrt(x^ 2+$a )";

TEXT(EV2(<<EOT));
Let \[ f(x) = \sqrt{x^ 2+$a} \cos($n x) \]
$PAR
\( f'( x ) = \) \{ans_rule(40) \}
EOT

$ans = $funct1;
&ANS(fun_cmp($ans,domain=>[sqrt($an),2*sqrt($an)]));


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

##DESCRIPTION
##KEYWORDS('derivatives', 'logarithmic functions','exponential functions','product rule')
##  differentiation of log function a exp(x^n)*ln(nx)
##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 = random(2,5,1);
$n  = random(2,8,1);

$funct1 = "e^{x^$a} *(1/x)+ln($n x) e^{x^{$a}} *$a*x^{$a-1}";

TEXT(EV2(<<EOT));
Let \[ f(x) = e^{x^{$a}} \ln($n x) \]
$PAR
\( f'( x ) = \) \{ans_rule(40) \}
EOT

$ans = $funct1;
&ANS(function_cmp($ans));


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

##DESCRIPTION
##  limits
##ENDDESCRIPTION

##KEYWORDS ('limit', 'rational function')

DOCUMENT();

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

TEXT(&beginproblem);
$showPartialCorrectAnswers = 1;

$c1 = random(2, 12, 1);
$c2 = random(2, 12, 1);
$c3 = random(2, 12, 1);

$k1 = random(2, 12, 1);
$k2 = random(2, 12, 1);
$k3 = random(2, 12, 1);

$p1 = random(3, 6, 1);
$p2 = random(1, 2, 1);

$q1 = random(3, 6, 1);
$q2 = random(1, 2, 1);

TEXT(EV3(<<'EOT'));
Evaluate
\[ \lim_{x \rightarrow \infty} \frac{$c1 x^$p1+$c2 x^$p2 + $c3}{$k1 x^$q1+$k2 x^$q2 + $k3}.\]
$PAR
If the limit is \( \infty \), enter 'INF', and if the
limit is \( -\infty \), then enter '-INF'.
$PAR
Limit = \{ ans_rule(20) \}
$BR
EOT
if ( $p1 > $q1)
   {
    $ans = "INF";
    &ANS(num_cmp($ans,strings=>["INF","-INF"]));
   }
   elsif ( $p1 == $q1 )
   {
    $ans = $c1/$k1;
    &ANS(num_cmp($c1/$k1));
   }
   else
   {
    $ans = 0;
    &ANS(strict_num_cmp(0,0));
   }

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##  limits
##ENDDESCRIPTION

##KEYWORDS ('limit', 'la Hospital')

DOCUMENT();

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

TEXT(&beginproblem);
$showPartialCorrectAnswers = 1;

$problem = random(1, 2, 1);

$k1 = random(2, 12, 1);
$p1 = random(3, 6, 1);
$q1 = random(3, 6, 1);

if ($problem ==1)
 {
   TEXT(EV3(<<'EOT'));
   Evaluate
   \[ \lim_{x \rightarrow \infty} \frac{e^{$p1 x}}{$k1 x^$q1}.\]
   $PAR
   If the limit is \( \infty \), enter 'INF', and if the
   limit is \( -\infty \), then enter '-INF'.
   $PAR
   Limit = \{ ans_rule(20) \}
   $BR
EOT
   $ans = "INF";
   &ANS(num_cmp($ans,strings=>["INF","-INF"]));
 }
   else 
 {
   TEXT(EV3(<<'EOT'));
   Evaluate
   \[ \lim_{x \rightarrow \infty} \frac{$k1 x^$q1.}{e^{$p1 x}}\]
   $PAR
   If the limit is \( \infty \), enter 'INF', and if the
   limit is \( -\infty \), then enter '-INF'.
   $PAR
   Limit = \{ ans_rule(20) \}
   $BR
EOT
   $ans = 0;
   &ANS(strict_num_cmp(0,0));
 }

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##  limits
##ENDDESCRIPTION

##KEYWORDS ('limit', 'la Hospital')

DOCUMENT();

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

TEXT(&beginproblem);
$showPartialCorrectAnswers = 1;

$problem = random(1, 2, 1);

$k1 = random(2, 12, 1);
$p1 = random(3, 6, 1);
$q1 = random(3, 6, 1);

if ($problem ==1)
 {
   TEXT(EV3(<<'EOT'));
   Evaluate
   \[ \lim_{x \rightarrow \infty} \frac{$k1 sqrt(x)}{ln($p1 x)}.\]
   $PAR
   If the limit is \( \infty \), enter 'INF', and if the
   limit is \( -\infty \), then enter '-INF'.
   $PAR
   Limit = \{ ans_rule(20) \}
   $BR
EOT
   $ans = "INF";
   &ANS(num_cmp($ans,strings=>["INF","-INF"]));
 }
   else 
 {
   TEXT(EV3(<<'EOT'));
   Evaluate
   \[ \lim_{x \rightarrow \infty} \frac{ln($p1 x)}{$k1 sqrt(x)}.\]
   $PAR
   If the limit is \( \infty \), enter 'INF', and if the
   limit is \( -\infty \), then enter '-INF'.
   $PAR
   Limit = \{ ans_rule(20) \}
   $BR
EOT
   $ans = 0;
   &ANS(strict_num_cmp(0,0));
 }

ENDDOCUMENT();