## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('5')


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;

#my ($a, $aaa, $answer);
$a= random(1, 5, 1);

TEXT(EV2(<<EOT));
Use part I of the Fundamental Theorem of Calculus to
find the derivative of
\[ f(x) = \int_{-$a}^{x} \sqrt{t^3+\{$a**3\}}\, dt \]
$BR \( f'(x) \) = \{ans_rule( 20)\} $BR
[NOTE:  Enter a function as your answer.  Make sure that your
syntax is correct, i.e. remember to put all the necessary (, ), etc. ]
EOT

$aaa=$a**3;
$answer="sqrt(x^3+$aaa)" ;

ANS(fun_cmp($answer));

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

## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('7')


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(1,9,1);
$p = random(4,15,1);


TEXT(EV2(<<EOT));
Use part I of the Fundamental Theorem of Calculus to
find the derivative of
\[ g(y) = \int_{$a}^{y} {t^{$p} \sin t}\, dt \]
$BR \( g'(y) \) = \{ans_rule( 20)\} $BR
[NOTE:  Enter a function as your answer.  Make sure that your
syntax is correct, i.e. remember to put all the necessary (, ), etc. ]
EOT

@ans=(fun_cmp("y^($p)*sin(y)", vars=>"y"));

ANS(@ans);

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

## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('13')


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(1,9,1);
$b = random(1,9,1);
$p = random(4,15,1);


TEXT(EV2(<<EOT));
Use part I of the Fundamental Theorem of Calculus to
find the derivative of
\[ y = \int_{$a}^{\sqrt{x}} {\frac{\cos t}{t^{$p}}} dt \]
$BR \( \frac{dy}{dx} \) = \{ans_rule( 40)\} $BR
[NOTE:  Enter a function as your answer.  Make sure that your
syntax is correct, i.e. remember to put all the necessary (, ), etc. ]
EOT

@ans=(fun_cmp("cos(sqrt(x))/(2*x^(($p+1)/2))", vars=>"x"));

ANS(@ans);

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

## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('14')

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;

my ($a, $b, $answer);
$a= random(1, 5, 1);
$b= random(2, 5, 1);

TEXT(EV2(<<EOT));
Use part I of the Fundamental Theorem of Calculus to
find the derivative of
\[ h(x) = \int_{-$a}^{\sin(x)} (\cos(t^$b)+t) dt \]

$BR \(h'(x) = \)\{ANS_RULE(1, 35)\} $BR
$BR $BR

EOT

$answer="cos(x)*(cos((sin(x))^$b)+sin(x))" ;

ANS(fun_cmp($answer));

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

## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('17')

DOCUMENT();

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

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

$a = random(1,3,1);
$b = random(16,32,4);
$c = random(4,7,1);

TEXT(EV2(<<EOT));

Evaluate the definite integral:
$BR
$PAR
   \( \displaystyle{ \int_{0}^{$a} $b x^{$c} \,dx} = \)
   \{ans_rule(35) \} 
$BR
$BR

EOT

$ans = $b*($a**($c+1))/($c+1);
ANS(num_cmp($ans));

ENDDOCUMENT();
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('18')

DOCUMENT();

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

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

$a = random(3,6,1);
$b = random(2,6,1);
$c = random(2,6,1);

TEXT(EV2(<<EOT));

Evaluate the definite integral:
$BR
$PAR
   \(\displaystyle{ \int_{1}^{$a} \left($b x^{-2} - $c\right) \,dx} = \)
   \{ans_rule(35) \}
$BR
$BR

EOT

$ans = $b + $c - $a*$c - $b/$a;
ANS(num_cmp($ans));

ENDDOCUMENT();
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('21')

DOCUMENT();

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

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

$a = random(1,4,1);
$b = random(12,26,1);
$c = random(2,6,1);

TEXT(EV2(<<EOT));

Evaluate the definite integral:
$BR
$PAR
   \( \displaystyle{ \int_{$a}^{$b} $c \sqrt{x} \,dx} = \)
   \{ans_rule(35) \}
$BR
$BR

EOT

$ans = (2*$c/3)*($b**(3/2) - $a**(3/2));
ANS(num_cmp($ans));

ENDDOCUMENT();
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'definite integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('31')


DOCUMENT();

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

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

$a = random(1,5,2);
$d = random(1,2,1);
$b = $a+$d;


TEXT(EV2(<<EOT));

Use Part 2 of the Fundamental Theorem of Calculus to decide if the definite 
integral exists and either evaluate the integral or enter DNE if it does not exist.
$BR
$PAR
   \[ \int_{$a \pi /2}^{$b \pi} {\sec x \tan x}\; dx \]
$BR
Integral =    \{ans_rule(35) \}
$BR
EOT



@ans = (num_cmp("DNE",strings =>["DNE"]));
ANS(@ans);

ENDDOCUMENT();
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Fundamental Theorem of Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'definite integrals', 'fundamental theorem of calculus')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/2/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('35')

DOCUMENT();

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

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

$a = random(1,3,1);
$d = random(2,7,1);
$b = $a+$d;
$c = random(2,6,1);

$p = random(2,5,1);

TEXT(EV2(<<EOT));

Use Part 2 of the Fundamental Theorem of Calculus to decide if the definite 
integral exists and either evaluate the integral or enter DNE if it does not exist.
$BR
$PAR
   \[ \int_{$a}^{$b} {$c^t} dt \]

   \{ans_rule(35) \} 
$BR
EOT

$ans = "1/ln($c)*($c**$b-$c**$a)";
ANS(num_cmp($ans));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('integrals', 'fundamental theorem of calculus', 'absolute max')
##  Find the absolute maximum of the integral of the function
##ENDDESCRIPTION

## BenBush tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Fundamental Theorem of Calculus')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('5.3')
## Problem1('55,56,62,66')

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;

$p1  = random(-5,-3,1);
$r1 = $p1 - 1;
$r2 = $p1 - 4;
$p0 = random($r2,$r1,1);
$p2 = $p1 + 1;
$p3 = random(-1,1,1);
$p4 = $p3 + 1;
$p5 = random(3,5,1);
$p6 = $p5 + 1;

$a = random(2,5,1);
$b = random(-5,-2,1);
while ($a == -$b) {$b = random(-5,-2,1)}

TEXT(EV2(<<EOT));
NOTE:  It will be easier to see the function f(x) if you use the display mode "typeset".  Keep in mind, though, that loading the problem into your computer using this display mode will take longer.
$PAR
Let
\[ f(x)  = \begin{cases} 
              0 \hbox {\quad if \quad} x < $p1 \\
              $a \hbox {\quad if \quad} $p1 \leq x < $p3 \\
              $b \hbox {\quad if \quad} $p3 \leq x < $p5 \\
              0 \hbox {\quad if \quad} x \geq $p5 \end{cases} \]
and \[ g(x)  = \int_{$p1}^{x} f(t) dt \]
$BR
Determine the value of each of the following: $BR
(a) \( g($p0) = \) \{ans_rule(5)\} $BR
(b) \( g($p2) = \) \{ans_rule(5)\} $BR
(c) \( g($p4) = \) \{ans_rule(5)\} $BR
(d) \( g($p6) = \) \{ans_rule(5)\} $BR
(e) The absolute maximum of \( g(x) \) occurs when \( x = \) \{ans_rule(5)\} and is the value \{ans_rule(5)\} 
$PAR
It may be helpful to make a graph of f(x) when answering these questions.
EOT

$ans1 = 0;
$ans2 = $a;
$ans3 = $a * ($p3 - $p1) + $b;
$ans4 = $a * ($p3 - $p1) + $b * ($p5 - $p3);
$ans5 = $p3;
$ans6 = $a * ($p3 - $p1);

ANS(num_cmp($ans1));
ANS(num_cmp($ans2));
ANS(num_cmp($ans3));
ANS(num_cmp($ans4));
ANS(num_cmp($ans5));
ANS(num_cmp($ans6));

ENDDOCUMENT();