From c619351e277d3c4755980df55fc2510ce197e43c Mon Sep 17 00:00:00 2001 From: ChenfeiP Date: Fri, 13 Dec 2024 13:59:03 -0600 Subject: [PATCH 1/3] Fixed bug #4857 --- .../S06.03.VolumesByShells.PTP08.pg | 69 +++++++------------ 1 file changed, 26 insertions(+), 43 deletions(-) diff --git a/OpenProblemLibrary/Rochester/setIntegrals20Volume/S06.03.VolumesByShells.PTP08.pg b/OpenProblemLibrary/Rochester/setIntegrals20Volume/S06.03.VolumesByShells.PTP08.pg index 35b8e6f21b..d9357c700e 100644 --- a/OpenProblemLibrary/Rochester/setIntegrals20Volume/S06.03.VolumesByShells.PTP08.pg +++ b/OpenProblemLibrary/Rochester/setIntegrals20Volume/S06.03.VolumesByShells.PTP08.pg @@ -18,62 +18,45 @@ DOCUMENT(); loadMacros( "PGstandard.pl", - "weightedGrader.pl", - "PGcourse.pl" + "PGML.pl", # Ensure PGML is loaded + "PGcourse.pl", + 'parserPopUp.pl' ); +$showPartialCorrectAnswers = 1; -install_weighted_grader(); +$popup1 = PopUp([ '?', 'dx', 'dy' ], 'dy'); +$popup2 = PopUp([ '?', 'dx', 'dy' ], 'dx'); +## $dropdown1 = DropDown([ 'Red', 'Blue', 'Green' ], 'Green'); +# $dropdown2 = DropDown([ 'Red', 'Blue', 'Green' ], 'Red', placeholder => 'Select One'); -$showPartialCorrectAnswers = 1; +Context("Numeric"); +Context()->variables->add(y => "Real"); -TEXT(beginproblem()); +$a = random(2,9,1); -$a=random(2,9,1); +$integrandWashers = Formula("pi*($a * y**2)**2"); +$integrandShells = Formula("2*pi*x*(1 - sqrt(x / $a))"); +$volume = Formula("pi*$a**2/5"); -BEGIN_TEXT +BEGIN_PGML +The volume of the solid obtained by rotating the region bounded by +[``` x = [$a] y^2, y = 1, x = 0, ```] +about the [`y`]-axis can be computed using the method of disks via an integral: -The volume of the solid obtained by rotating the region bounded by -\[ x=$a y^2, \ y = 1, \ x = 0, \] -about the \(y\)-axis can be computed using the method of disks via an integral -$BR -$BCENTER -\( \displaystyle V = \int_a^b \) \{ans_rule(40) \} \{pop_up_list(['?','dx','dy']) \} -$ECENTER -$BR -with limits of integration \( a = \) \{ans_rule( 5) \} and \( b = \) \{ans_rule( 5) \}. -$BR -$BR -The volume of this solid can also be computed using cylindrical shells via an integral -$BR -$BCENTER -\( \displaystyle -V = \int_\alpha^\beta \) \{ans_rule(40) \} \{pop_up_list(['?','dx','dy']) \} -$ECENTER -$BR -with limits of integration \( \alpha = \) \{ans_rule( 5) \} and \( \beta = \) \{ans_rule( 5) \}. -$BR -$BR -In either case, the volume is \( V = \) \{ans_rule( 25) \} cubic units. +>>[``V = \int_a^b``] [________]{$integrandWashers} [________]{$popup1}<< -END_TEXT +with limits of integration [` a = `] [_____]{0} and [` b = `] [_____]{1}. +The volume of this solid can also be computed using cylindrical shells via an integral: -$integrandWashers = "pi*($a * y**2)**2"; -$integrandShells = "2*pi*x*(1 - sqrt(x / $a))"; -$volume = "pi*$a**2/5"; -WEIGHTED_ANS(fun_cmp($integrandWashers, vars=>['x','y'],limits=>[[0,0],[1,10]]), 30); -WEIGHTED_ANS(str_cmp("dy"), 0); -WEIGHTED_ANS(num_cmp(0,tol=>0,tolType=>"absolute"), 2); -WEIGHTED_ANS(num_cmp(1,tol=>0,tolType=>"absolute"), 3); +>>[``V = \int_\alpha^\beta``] [________]{$integrandShells} [________]{$popup2}<< -WEIGHTED_ANS(fun_cmp($integrandShells, vars=>['x','y'],limits=>[[0,10],[0,0]]), 30); -WEIGHTED_ANS(str_cmp("dx"), 0); -WEIGHTED_ANS(num_cmp(0,tol=>0,tolType=>"absolute"), 2); -WEIGHTED_ANS(num_cmp($a,tol=>0,tolType=>"absolute"), 3); -WEIGHTED_ANS(num_cmp($volume), 30); +with limits of integration [` \alpha = `] [_____]{0} and [` \beta = `] [_____]{$a}. +In either case, the volume is [` V = `] [________]{$volume} cubic units. +END_PGML -ENDDOCUMENT(); +ENDDOCUMENT(); From d00047d6f614ca1221109038a919a4c3f64931fa Mon Sep 17 00:00:00 2001 From: ChenfeiP Date: Fri, 13 Dec 2024 13:59:03 -0600 Subject: [PATCH 2/3] Fixed bug #4856 --- .../S06.03.VolumesByShells.PTP08.pg | 69 +++++++------------ 1 file changed, 26 insertions(+), 43 deletions(-) diff --git a/OpenProblemLibrary/Rochester/setIntegrals20Volume/S06.03.VolumesByShells.PTP08.pg b/OpenProblemLibrary/Rochester/setIntegrals20Volume/S06.03.VolumesByShells.PTP08.pg index 35b8e6f21b..d9357c700e 100644 --- a/OpenProblemLibrary/Rochester/setIntegrals20Volume/S06.03.VolumesByShells.PTP08.pg +++ b/OpenProblemLibrary/Rochester/setIntegrals20Volume/S06.03.VolumesByShells.PTP08.pg @@ -18,62 +18,45 @@ DOCUMENT(); loadMacros( "PGstandard.pl", - "weightedGrader.pl", - "PGcourse.pl" + "PGML.pl", # Ensure PGML is loaded + "PGcourse.pl", + 'parserPopUp.pl' ); +$showPartialCorrectAnswers = 1; -install_weighted_grader(); +$popup1 = PopUp([ '?', 'dx', 'dy' ], 'dy'); +$popup2 = PopUp([ '?', 'dx', 'dy' ], 'dx'); +## $dropdown1 = DropDown([ 'Red', 'Blue', 'Green' ], 'Green'); +# $dropdown2 = DropDown([ 'Red', 'Blue', 'Green' ], 'Red', placeholder => 'Select One'); -$showPartialCorrectAnswers = 1; +Context("Numeric"); +Context()->variables->add(y => "Real"); -TEXT(beginproblem()); +$a = random(2,9,1); -$a=random(2,9,1); +$integrandWashers = Formula("pi*($a * y**2)**2"); +$integrandShells = Formula("2*pi*x*(1 - sqrt(x / $a))"); +$volume = Formula("pi*$a**2/5"); -BEGIN_TEXT +BEGIN_PGML +The volume of the solid obtained by rotating the region bounded by +[``` x = [$a] y^2, y = 1, x = 0, ```] +about the [`y`]-axis can be computed using the method of disks via an integral: -The volume of the solid obtained by rotating the region bounded by -\[ x=$a y^2, \ y = 1, \ x = 0, \] -about the \(y\)-axis can be computed using the method of disks via an integral -$BR -$BCENTER -\( \displaystyle V = \int_a^b \) \{ans_rule(40) \} \{pop_up_list(['?','dx','dy']) \} -$ECENTER -$BR -with limits of integration \( a = \) \{ans_rule( 5) \} and \( b = \) \{ans_rule( 5) \}. -$BR -$BR -The volume of this solid can also be computed using cylindrical shells via an integral -$BR -$BCENTER -\( \displaystyle -V = \int_\alpha^\beta \) \{ans_rule(40) \} \{pop_up_list(['?','dx','dy']) \} -$ECENTER -$BR -with limits of integration \( \alpha = \) \{ans_rule( 5) \} and \( \beta = \) \{ans_rule( 5) \}. -$BR -$BR -In either case, the volume is \( V = \) \{ans_rule( 25) \} cubic units. +>>[``V = \int_a^b``] [________]{$integrandWashers} [________]{$popup1}<< -END_TEXT +with limits of integration [` a = `] [_____]{0} and [` b = `] [_____]{1}. +The volume of this solid can also be computed using cylindrical shells via an integral: -$integrandWashers = "pi*($a * y**2)**2"; -$integrandShells = "2*pi*x*(1 - sqrt(x / $a))"; -$volume = "pi*$a**2/5"; -WEIGHTED_ANS(fun_cmp($integrandWashers, vars=>['x','y'],limits=>[[0,0],[1,10]]), 30); -WEIGHTED_ANS(str_cmp("dy"), 0); -WEIGHTED_ANS(num_cmp(0,tol=>0,tolType=>"absolute"), 2); -WEIGHTED_ANS(num_cmp(1,tol=>0,tolType=>"absolute"), 3); +>>[``V = \int_\alpha^\beta``] [________]{$integrandShells} [________]{$popup2}<< -WEIGHTED_ANS(fun_cmp($integrandShells, vars=>['x','y'],limits=>[[0,10],[0,0]]), 30); -WEIGHTED_ANS(str_cmp("dx"), 0); -WEIGHTED_ANS(num_cmp(0,tol=>0,tolType=>"absolute"), 2); -WEIGHTED_ANS(num_cmp($a,tol=>0,tolType=>"absolute"), 3); -WEIGHTED_ANS(num_cmp($volume), 30); +with limits of integration [` \alpha = `] [_____]{0} and [` \beta = `] [_____]{$a}. +In either case, the volume is [` V = `] [________]{$volume} cubic units. +END_PGML -ENDDOCUMENT(); +ENDDOCUMENT(); From cab46677db809e7223c51a56e38a75b8e1d6843a Mon Sep 17 00:00:00 2001 From: ChenfeiP Date: Fri, 13 Dec 2024 14:32:14 -0600 Subject: [PATCH 3/3] Please consider those problems for addition to the OPL --- Contrib/UMN-Morris/MSAF/derivative_rules.pg | 140 +++++++++++++++ Contrib/UMN-Morris/MSAF/limitlaws.pg | 121 +++++++++++++ Contrib/UMN-Morris/MSAF/sin_cos_transition.pg | 166 ++++++++++++++++++ 3 files changed, 427 insertions(+) create mode 100644 Contrib/UMN-Morris/MSAF/derivative_rules.pg create mode 100644 Contrib/UMN-Morris/MSAF/limitlaws.pg create mode 100644 Contrib/UMN-Morris/MSAF/sin_cos_transition.pg diff --git a/Contrib/UMN-Morris/MSAF/derivative_rules.pg b/Contrib/UMN-Morris/MSAF/derivative_rules.pg new file mode 100644 index 0000000000..9e54c17054 --- /dev/null +++ b/Contrib/UMN-Morris/MSAF/derivative_rules.pg @@ -0,0 +1,140 @@ +# DESCRIPTION +# A scaffolded problem where students apply derivative rules one step at a time. +# ENDDESCRIPTION + +## KEYWORDS('derivative, differentiation, derivative rules') +## TitleText1('Derivative Rules Problem') +## DBsubject(Calculus - single variable) +## DBchapter(Differentiation) +## DBsection(Derivatives of polynomials and power functions) +## MO(1) +## Level(2) +## Static(1) +## Language(en) +## Author('Chenfei Peng, Mercredi Chasman') +## Institution('UMinnMorris, 12/13/2024') +## TitleText1(Calculus) +## EditionText1(6e) +## AuthorText1(Stewart) +## Section1(3.1) + +DOCUMENT(); +loadMacros( + "PGstandard.pl", + "MathObjects.pl", + "PGML.pl", + "PGcourse.pl", + 'scaffold.pl', + 'parserPopUp.pl', +); + +$showPartialCorrectAnswers = 1; + +# Define popup options for each step of the derivative process +$popup1 = PopUp([ '?', 'FOIL', 'Sum/difference Rule', 'Power Rule', 'Constant Rule', 'Constant Multiple Rule', +'Simplify', 'Distribute/expand', 'Product Rule', 'Factor'], 'Distribute/expand'); +$popup2 = PopUp([ '?', 'FOIL', 'Sum/difference Rule', 'Power Rule', 'Constant Rule', 'Constant Multiple Rule', +'Simplify', 'Distribute/expand', 'Product Rule', 'Factor'], 'Sum/difference Rule'); +$popup3 = PopUp([ '?', 'FOIL', 'Sum/difference Rule', 'Power Rule', 'Constant Rule', 'Constant Multiple Rule', +'Simplify', 'Distribute/expand', 'Product Rule', 'Factor'], 'Constant Multiple Rule'); +$popup4 = PopUp([ '?', 'FOIL', 'Sum/difference Rule', 'Power Rule', 'Constant Rule', 'Constant Multiple Rule', +'Simplify', 'Distribute/expand', 'Product Rule', 'Factor'], 'Power Rule'); +$popup5 = PopUp([ '?', 'FOIL', 'Sum/difference Rule', 'Power Rule', 'Constant Rule', 'Constant Multiple Rule', +'Simplify', 'Distribute/expand', 'Product Rule', 'Factor'], 'Simplify'); + +Context("Numeric"); +$f=Formula("35x^4 - 44x^3 + 24x^2"); +BEGIN_PGML +Evaluate the following derivative without using the Product Rule, Quotient Rule, or Chain Rule. Use only one derivative rule at a time and order terms in descending expression order. Answer blanks should be filled with mathematical expressions, numbers, or mathematical symbols [`+,-,*,/,^`] as appropriate. +END_PGML + +Scaffold::Begin( + can_open => "when_previous_correct", + is_open => "correct_or_first_incorrect" +); + +Section::Begin('Step 1'); + +BEGIN_PGML + +[`` +\frac{d}{dx} \Big[ x^3(7x^2 - 11x + 8) \Big] = \frac{d}{dx} \Big[ ``] [__]{7x^5} - [__]{11x^4} + [__]{8x^3} [`` \Big] +``] + +*Which rule applies here?* +[________]{$popup1} +END_PGML + +Section::End(); + +Section::Begin('Step 2'); + +BEGIN_PGML + +[`` += \frac{d}{dx} \Big[ ``] [__]{7x^5} [`` \Big] - \frac{d}{dx} \Big[ ``] [__]{11x^4} [`` \Big] + \frac{d}{dx} \Big[ ``] [__]{8x^3} [`` \Big] +``] + +*Which rule applies here?* +[________]{$popup2} +END_PGML + +Section::End(); + +Section::Begin('Step 3'); + +BEGIN_PGML + +[`` += 7\frac{d}{dx} \Big[ ``] [__]{Formula("x^5")} [`` \Big] - 11\frac{d}{dx} \Big[ ``] [__]{Formula("x^4")} [`` \Big] + 8\frac{d}{dx} \Big[ ``] [__]{Formula("x^3")} [`` \Big] +``] + +*Which rule applies here?* +[________]{$popup3} +END_PGML + +Section::End(); + +Section::Begin('Step 4'); + +BEGIN_PGML + +[`` += 7 \Big( ``] [__]{5x^4} [`` \Big) - 11 \Big( ``] [__]{4x^3} [`` \Big) + 8 \Big( ``] [__]{3x^2} [`` \Big) +``] + +*Which rule applies here?* +[________]{$popup4} +END_PGML + +Section::End(); + +Section::Begin('Step 5'); + +BEGIN_PGML +Finally, simplify: + +[`` += ``] [__]{$f} + + +END_PGML + +Section::End(); + +Scaffold::End(); + +BEGIN_PGML_SOLUTION +In this solution, we apply the following derivative rules step by step to compute the derivative . + +- **Step 1**: We **Distribute/Expand** the product to put it in a form we can differentiate. +- **Step 2**: We use the **Sum/Difference Rule** to break the derivative into individual terms. +- **Step 3**: We apply the **Constant Multiple Rule**, which allows us to pull out constants from the derivatives. +- **Step 4**: We then apply the **Power Rule** to differentiate each power x^n. +- **Step 5**: Finally, we **Simplify** the expression to get the final result. + +Final result: 35x^4 - 44x^3 + 24x^2 +END_PGML_SOLUTION + + +ENDDOCUMENT(); diff --git a/Contrib/UMN-Morris/MSAF/limitlaws.pg b/Contrib/UMN-Morris/MSAF/limitlaws.pg new file mode 100644 index 0000000000..176f39cacd --- /dev/null +++ b/Contrib/UMN-Morris/MSAF/limitlaws.pg @@ -0,0 +1,121 @@ +# DESCRIPTION +# A scaffolded problem where students use one limit law or operation at a time to compute a limit. +# ENDDESCRIPTION + +## DBsubject(Calculus - single variable) +## DBchapter(Limits and continuity) +## DBsection(Rules of limits - basic) +## KEYWORDS('calculus','limits','Limit') +## Level(2) +## MO(1) +## Static(1) +## Language(en) +## TitleText1('Calculus') +## EditionText1('6') +## AuthorText1('Stewart') +## Section1('2.3') +## Problem1('') +## Author('Chenfei Peng, Mercredi Chasman') +## Institution('UMinnMorris, 12/13/2024') + + +DOCUMENT(); +loadMacros( + "PGstandard.pl", + "MathObjects.pl", + "PGML.pl", + "PGcourse.pl", + 'scaffold.pl', + 'parserPopUp.pl', +); + +$showPartialCorrectAnswers = 1; +$popup1 = PopUp([ '?', 'Sum/difference law', 'Constant Multiple Law', 'Constant Law', 'Power Law', +'Quotient Law', 'Product Law', 'Linear Law'], 'Sum/difference law'); +$popup2 = PopUp([ '?', 'Sum/difference law', 'Constant Multiple Law', 'Constant Law', 'Power Law', +'Quotient Law', 'Product Law', 'Linear Law'], 'Constant Multiple Law'); +$popup3 = PopUp([ '?', 'Sum/difference law', 'Constant Multiple Law', 'Constant Law', 'Power Law', +'Quotient Law', 'Product Law', 'Linear Law'], 'Power Law'); +$popup4 = PopUp([ '?', 'Sum/difference law', 'Constant Multiple Law', 'Constant Law', 'Power Law', +'Quotient Law', 'Product Law', 'Linear Law'], 'Linear Law'); + + +Context("Numeric"); + +BEGIN_PGML +Use limit laws to compute the following limit. Use only one rule at a time. Answer blanks should be filled with mathematical expressions, numbers, or mathematical symbols [`+,-,*,/,^`] as appropriate. +END_PGML + +Scaffold::Begin( + can_open => "when_previous_correct", + is_open => "correct_or_first_incorrect" +); + +Section::Begin('Step 1'); + +BEGIN_PGML +[`` +\lim_{x \to 3} \Big[7x^2 + 11x \Big] = \lim_{x \to 3}\Big[ ``] [___]{7x^2} [``\Big]``] [___]{str_cmp("+")} [``\lim_{x \to 3}\Big[ ``] [_]{11x} [``\Big]``] + +*Which limit law applies here?* +[________]{$popup1} +END_PGML + +Section::End(); + +Section::Begin('Step 2'); +BEGIN_PGML + +[`` += ``] [___]{7} [`` \lim_{x \to 3}\Big[ ``] [___]{Formula("x^2")} [``\Big]``] + [_]{11} [``\lim_{x \to 3}\Big[ ``] [_]{x} [``\Big]``] + +*Which limit law applies here?* +[________]{$popup2} +END_PGML +Section::End(); + +Section::Begin('Step 3'); +BEGIN_PGML + +[`` += 7 ( \lim_{x \to 3} \Big[ ``] [___]{x} [``\Big] )^2 + 11 \lim_{x \to 3} \Big[ ``] [_]{x} [``\Big]``] + +*Which limit law applies here?* +[________]{$popup3} +END_PGML +Section::End(); + +Section::Begin('Step 4'); +BEGIN_PGML + +[`` = 7 ( ``] [___]{3} [`` )^2 + 11 ( ``] [___]{3} [`` ) ``] + +*Which limit law applies here?* +[________]{$popup4} +END_PGML +Section::End(); + +Section::Begin('Step 5'); +BEGIN_PGML + +[`` = ``] [___]{96} + +END_PGML +Section::End(); + +Scaffold::End(); + + +BEGIN_PGML_SOLUTION +In this solution, we apply the limit laws step by step to simplify the expression and find the value of the limit. + +- **Step 1**: Apply the **Sum/Difference Law** to split the limit. +- **Step 2**: Apply the **Constant Multiple Law** to factor out the constants. +- **Step 3**: Use the **Power Law** to deal with the powers of \( x \). +- **Step 4**: Evaluate the limits by substitution. +- **Step 5**: Combine the results to get the final answer. + +Final result: 96. +END_PGML_SOLUTION + +ENDDOCUMENT(); diff --git a/Contrib/UMN-Morris/MSAF/sin_cos_transition.pg b/Contrib/UMN-Morris/MSAF/sin_cos_transition.pg new file mode 100644 index 0000000000..901281ca9d --- /dev/null +++ b/Contrib/UMN-Morris/MSAF/sin_cos_transition.pg @@ -0,0 +1,166 @@ +# DESCRIPTION +# A scaffolded problem where students simplify a trigonometric expression one step at a time +# ENDDESCRIPTION + +## KEYWORDS('') +## DBsubject(Trigonometry) +## DBchapter(Analytic trigonometry) +## DBsection(Using and proving general identities) +## TitleText1('Trigonometric Simplification Problem') +## MO(1) +## Level(3) +## Static(1) +## Language(en) +## EditionText1('') +## AuthorText1('') +## Section1('') +## Problem1('') +## Date('12/08/2024') +## Author('Mercredi Chasman, Chenfei Peng') +## Institution('UMinnMorris') + +DOCUMENT(); +loadMacros( + "PGstandard.pl", + "MathObjects.pl", + "PGML.pl", + "PGcourse.pl", + 'scaffold.pl', + 'parserPopUp.pl', +); + +$showPartialCorrectAnswers = 1; + +# Define popup options for each step of the trigonometric simplification process +$popup1 = PopUp([ '?', 'FOIL', 'Law of Exponents', 'Factor', 'Add/combine terms', 'Pythagorean Identity', +'Pythagorean Theorem', 'Cancel', 'Rewrite in terms of sin/cos'], 'Rewrite in terms of sin/cos'); +$popup2 = PopUp([ '?', 'FOIL', 'Law of Exponents', 'Factor', 'Add/combine terms', 'Pythagorean Identity', +'Pythagorean Theorem', 'Cancel', 'Rewrite in terms of sin/cos'], 'Add/combine terms'); +$popup3 = PopUp([ '?', 'FOIL', 'Law of Exponents', 'Factor', 'Add/combine terms', 'Pythagorean Identity', +'Pythagorean Theorem', 'Cancel', 'Rewrite in terms of sin/cos'], 'Law of Exponents'); +$popup4 = PopUp([ '?', 'FOIL', 'Law of Exponents', 'Factor', 'Add/combine terms', 'Pythagorean Identity', +'Pythagorean Theorem', 'Cancel', 'Rewrite in terms of sin/cos'], 'Pythagorean Identity'); +$popup5 = PopUp([ '?', 'FOIL', 'Law of Exponents', 'Factor', 'Add/combine terms', 'Pythagorean Identity', +'Pythagorean Theorem', 'Cancel', 'Rewrite in terms of sin/cos'], 'Factor'); +$popup6 = PopUp([ '?', 'FOIL', 'Law of Exponents', 'Factor', 'Add/combine terms', 'Pythagorean Identity', +'Pythagorean Theorem', 'Cancel', 'Rewrite in terms of sin/cos'], 'Cancel'); + +Context("Numeric"); +Context()->functions->enable("sin", "cos", "tan"); +$a = Formula("sin(x) + 1"); +$b = Formula("(sin(x) + 1)^2"); +$c = Formula("cos(x)^2"); +$d = Formula("1 - sin(x)^2"); +$e = Formula("1 - sin(x)"); + +BEGIN_PGML +Prove the following trigonometric identity one step at a time: + +[```\Big( \tan(x) + \sec(x) \Big)^2=\frac{1+\sin(x)}{1-\sin(x)} ```] +END_PGML + +Scaffold::Begin( + can_open => "when_previous_correct", + is_open => "correct_or_first_incorrect" +); + + +Section::Begin('Step 1'); + +BEGIN_PGML + +[`` +\Big( \tan(x) + \sec(x) \Big)^2 = \Big( ``] [________]{Formula("sin(x)")} [``/``] [________]{Formula("cos(x)")} [``+``] +[________]{1} [``/``] [________]{Formula("cos(x)")} [`` \Big)^2``] + +*Which applies here?* +[________]{$popup1} +END_PGML + +Section::End(); + +Section::Begin('Step 2'); + +BEGIN_PGML + + +[`` += \Big( ( ``] [________]{$a} [`` ) / \cos(x) \Big)^2``] + +*Which applies here?* +[________]{$popup2} +END_PGML + +Section::End(); + +Section::Begin('Step 3'); + +BEGIN_PGML + + +[`` += ``] [________]{$b} [``/``] [________]{$c} + +*Which applies here?* +[________]{$popup3} +END_PGML + +Section::End(); + +Section::Begin('Step 4'); + +BEGIN_PGML + +[`` += (\sin(x) + 1)^2 / ( ``] [________]{$d} [`` ) ``] + +*Which applies here?* +[________]{$popup4} +END_PGML + +Section::End(); + +Section::Begin('Step 5'); + +BEGIN_PGML + +[`` += (\sin(x) + 1)^2 / \big[ (1 + \sin(x)) ( ``] [________]{$e} [`` ) \big] ``] + +*Which applies here?* +[________]{$popup5} + +END_PGML + +Section::End(); + +Section::Begin('Step 6'); + +BEGIN_PGML + +[`` += ( ``] [________]{$a} [`` ) / ( ``] [________]{$e} [`` ) ``] + +*Which applies here?* +[________]{$popup6} + +END_PGML + +Section::End(); + +Scaffold::End(); + +BEGIN_PGML_SOLUTION +In this solution, we apply the following trigonometric identities and algebraic rules step by step to simplify the expression (tan(x) + sec(x))^2. + +- **Step 1**: We start by expressing tan(x) and sec(x) in terms of sine and cosine. +- **Step 2**: We simplify the expression by combining terms. +- **Step 3**: Apply the law of exponents to simplify the fraction. +- **Step 4**: Use the identity 1 - (sin(x))^2 = (cos(x))^2 to rewrite the denominator. +- **Step 5**: Factor the denominator into a product of binomials. +- **Step 6**: Cancel common terms to achieve the final simplified expression. + +Final result: (1 + sin(x)) / (1 - sin(x)) +END_PGML_SOLUTION + +ENDDOCUMENT();