openwebwork · weshuu · May 29, 2024
diff --git a/Contrib/Wisconsin/EauClaire/calculus/AntiderivativePowerRuleIntroduction.pg b/Contrib/Wisconsin/EauClaire/calculus/AntiderivativePowerRuleIntroduction.pg
@@ -0,0 +1,327 @@
+## DESCRIPTION
+##   ****
+## ENDDESCRIPTION
+
+## KEYWORDS('Power Rule')
+
+## DBsubject('Calculus')
+## DBchapter('Derivatives')
+## DBsection('Power Rule')
+## Date('07/20/2024')
+## Author('W. Shull')
+## Institution('UWEC')
+## TitleText1('')
+## EditionText1('')
+## AuthorText1('')
+## Section1('')
+## Problem1('')
+
+DOCUMENT();        # This should be the first executable line in the problem.
+
+loadMacros(
+  "PGstandard.pl",
+  "MathObjects.pl",        # the new parser
+  "contextFraction.pl",
+  "parserMultiAnswer.pl",
+  "answerHints.pl",
+  "PGML.pl",
+  "scaffold.pl",
+  "PGcourse.pl", # Customization file for the course
+  "customCSS.pl",
+  "parserPopUp.pl",   # <------ NEEDED FOR DROPDOWN MENUS
+  "niceTables.pl",
+  "parserFormulaUpToConstant.pl",
+  "PG.pl",
+  "PGbasicmacros.pl",
+  "PGchoicemacros.pl",
+  "PGanswermacros.pl",
+  "scaffold.pl",
+  "PGauxiliaryFunctions.pl",
+  "PGgraphmacros.pl",
+  "freemanMacros.pl",
+  "AnswerFormatHelp.pl",
+  "Parser.pl",
+  "niceTables.pl",
+  "PGstandard.pl", # Standard macros for PG language
+   "MathObjects.pl",
+   "PGML.pl",
+   #"source.pl",        # allows code to be displayed on certain sites.
+   #"scaffold.pl",
+   "unionTables.pl",
+   "parserPopUp.pl",#NEEDED FOR DROPDOWN MENUS
+   "PGtikz.pl",
+  "contextFraction.pl",
+  "parserMultiAnswer.pl",
+"answerHints.pl",
+  "PGcourse.pl",           # Customization file for the course
+   "customCSS.pl", # <---- MAKES BOXES PURPLE
+   #"PGcourse.pl",# Customization file for the course
+);
+
+TEXT(beginproblem());
+$showPartialCorrectAnswers = 1;
+
+#$same_derivative = PopUp(
+#  ["Choose one", "yes","no"],
+#  "yes");
+
+Context("Numeric");
+Context()->noreduce('(-x)-y');
+Context()->noreduce('(-x)+y');
+Context()->variables->add(u=>"Real", h=>"Real", n=>"Real", k=>"Real", f=>"Real", g=>"Real");
+
+$a1=random(2,4);
+$a2=non_zero_random(-4,4);
+$a3=non_zero_random(-4,4);
+$a4=non_zero_random(-4,4);
+$a5=random(2,6);
+$a6=non_zero_random(-10,10);
+$a7=non_zero_random(-10,10);
+while($a7==$a6){$a7=non_zero_random(-10,10);}
+
+$f1=Formula("$a1*x^4+$a2*x^2")->reduce;
+$f1d=Formula("4*$a1*x^3+2*$a2*x")->reduce;
+$f2=Formula("$a3*x^5+$a4*x")->reduce;
+$f2d=Formula("5*$a3*x^4+$a4")->reduce;
+$f3=FormulaUpToConstant("$a5*x^3")->reduce;
+$f3d=Formula("3*$a5*x^2")->reduce;
+$f4=Formula("$a5*x^3+$a6")->reduce;
+$f5=Formula("$a5*x^3+$a7")->reduce;
+
+$power3=FormulaUpToConstant("(x^4)/4");
+#$y1=non_zero_random(-10,10);
+
+#$g1d->{limits}=[1,16]; #Which x values are tested? We only want x values that are guaranteed to be in the domain.
+
+#SRAND($psvn); #FIX RANDOM VARIABLES ACROSS MULTIPLE PROBLEMS
+
+###################################
+# Main text
+###########################################
+#  The scaffold
+Scaffold::Begin(
+    can_open => "when_previous_correct",
+    is_open => "first_incorrect"
+);
+###########################################
+Section::Begin("Derivatives");
+
+Context()->texStrings;
+BEGIN_PGML
+
+We have lots of practice taking derivatives. Given [`f(x)=x^3`], for example, we know that [`f'(x)=3x^2`].
+
+But if we were given [`f'(x)=3x^2`] and not told [`f(x)`], could we have figured it out?
+
+Let's take a few more derivatives for practice:
+
+[`\frac d{dx}\left([$f1]\right)=`] [_]{$f1d}
+
+[`\frac d{dx}\left([$f2]\right)=`] [_]{$f2d}
+
+[`\frac d{dx}\left([$f4]\right)=`] [_]{$f3d}
+
+[`\frac d{dx}\left([$f5]\right)=`] [_]{$f3d}
+
+END_PGML
+Section::End();
+
+Section::Begin("Antiderivatives");
+
+BEGIN_PGML
+
+Notice that the last two functions we looked at had the same derivative! So if we are told that [`f'(x)=[$f3d]`], there is more than one possibility for [`f(x)`].
+
+In fact there are infinitely many possibilities! Any function of the form [`f(x)=[$f3]`], where [`C`] is any real number, has this derivative (since the derivative of [`C`] is 0). So we can say [`[$f4]`] is _an_ antiderivative of [`[$f3d]`], but not _the_ antiderivative.
+
+The full family of antiderivatives, [`[$f3]`], is often called the *general antiderivative*. You can write it by taking any antiderivative and simply adding [`+C`] to the end.
+
+Find the general antiderivative of [`[$f1d]`]: [_]{FormulaUpToConstant("$f1")->reduce}
+
+Find the general antiderivative of [`[$f2d]`]: [_]{FormulaUpToConstant("$f2")->reduce}
+
+END_PGML
+Section::End();
+
+Section::Begin("Back to \(x^3\) and \(3x^2\)");
+
+BEGIN_PGML
+
+Let's revisit [`f'(x)=3x^2`] and [`f(x)=x^3`]. In other words
+
+[``\frac d{dx}\big(x^3\big)=3x^2``]
+
+Let's divide both sides by 3:
+
+[``\frac13\cdot\frac d{dx}\Big(``][_]{Formula("x^3")}[``\Big)=``] [_]{Formula("x^2")}
+
+END_PGML
+Section::End();
+
+Section::Begin("Slight rearrangement");
+
+BEGIN_PGML
+
+Now let's rewrite the left side slightly:
+
+[``\frac13\cdot\frac d{dx}\big(x^3\big)=x^2``]
+
+[``\frac d{dx}\Big(``][_]{Formula("(x^3)/3")}[``\Big)=x^2``]
+
+END_PGML
+Section::End();
+
+Section::Begin("Another exponent");
+
+BEGIN_PGML
+
+Let's do something similar with some other exponents. First, a derivative:
+
+[``\frac d{dx}\big(x^4\big)=``] [_]{Formula("4*(x^3)")}
+
+END_PGML
+Section::End();
+
+Section::Begin("Another antiderivative");
+
+BEGIN_PGML
+
+[_]{1/4}[``\cdot\frac d{dx}\big(x^4\big)=x^3``]
+
+[``\frac d{dx}\bigg(``][_]{Formula("(x^4)/4")}[``\bigg)=x^3``]
+
+END_PGML
+Section::End();
+
+Section::Begin("Continuing the pattern");
+
+BEGIN_PGML
+
+Do the same for the next few exponents:
+
+[``\frac d{dx}\bigg(``][_]{Formula("(x^5)/5")}[``\bigg)=x^4``]
+
+[``\frac d{dx}\bigg(``][_]{Formula("(x^6)/6")}[``\bigg)=x^5``]
+
+[``\frac d{dx}\bigg(``][_]{Formula("(x^7)/7")}[``\bigg)=x^6``]
+
+[``\frac d{dx}\bigg(``][_]{Formula("(x^8)/8")}[``\bigg)=x^7``]
+
+END_PGML
+Section::End();
+
+Section::Begin("General antiderivatives");
+
+BEGIN_PGML
+
+Don't forget: Each function has an infinite family of antiderivatives, not just one. Enter the general antiderivatives of each function in the table below.
+
+[@
+DataTable(
+[
+    ["function","general antiderivative"],
+    ['\(x^3\)',ans_rule(10)],
+    ['\(x^4\)',ans_rule(10)],
+    ['\(x^5\)',ans_rule(10)],
+    ['\(x^6\)',ans_rule(10)],
+    ['\(x^7\)',ans_rule(10)],
+],
+align => '|c | c|',
+midrules => 1,
+tablecss => " border-spacing:0px 0px; border-radius: 5px; border-collapse:separate;"
+);
+@]*
+
+END_PGML
+
+ANS(FormulaUpToConstant("(x^4)/4")->cmp);
+ANS(FormulaUpToConstant("(x^5)/5")->cmp);
+ANS(FormulaUpToConstant("(x^6)/6")->cmp);
+ANS(FormulaUpToConstant("(x^7)/7")->cmp);
+ANS(FormulaUpToConstant("(x^8)/8")->cmp);
+
+Section::End();
+
+Section::Begin("Pattern in terms of \(n\)");
+
+BEGIN_PGML
+
+Based on the pattern, the general antiderivative of [`x^n`] is [_]{FormulaUpToConstant("(x^(n+1))/(n+1)")}
+
+END_PGML
+
+Section::End();
+
+Section::Begin("Indefinite integral notation");
+
+BEGIN_PGML
+
+We have a notation for the general antiderivative of any function [`f(x)`]. It looks like this:
+
+[```\int f(x)dx```]
+
+The [``\int``] is called the *integral* symbol, and an expression like this overall is called an *indefinite integral*. The function [`f(x)`], whose general antiderivative we want, is called the *integrand*. It is always sandwiched between the integral symbol and [`dx`], where [`x`] is the input variable of the function.
+
+So if [`F(x)`] is one of the antiderivatives of [`f(x)`] (in other words, if [`F'(x)=f(x)`]), then
+
+[```\int f(x)dx=F(x)+C```]
+
+We now have everything we need for the...
+
+[@ openDiv({ "class" => "importantFormula" }) @]*
+Power Rule:
+
+For any constant exponent [`n`]:
+
+>>[``\int x^n dx=``] [_]{FormulaUpToConstant("x^(n+1)/(n+1)")}<<
+
+as long as [`n\ne`] [_]{-1}
+
+[@ closeDiv() @]*
+
+
+END_PGML
+
+Section::End();
+
+Section::Begin("Basic antiderivative rules");
+
+BEGIN_PGML
+
+Here are some more basic rules, besides the power rule:
+
+[@ openDiv({ "class" => "importantFormula" }) @]*
+Sum and Difference Rules:
+
+>>[``\int\big(f(x)+g(x)\big)dx=\int f(x)dx+\int g(x)dx``]<<
+
+>>[``\int\big(f(x)-g(x)\big)dx=\int f(x)dx-\int g(x)dx``]<<
+
+[@ closeDiv() @]*
+
+[@ openDiv({ "class" => "importantFormula" }) @]*
+
+Constant Multiple Rule:
+
+>>[``\int k\cdot f(x)dx=k\int f(x)dx``]<<
+
+[@ closeDiv() @]*
+
+[@ openDiv({ "class" => "importantFormula" }) @]*
+Exponential Rule:
+
+>>[``\int e^xdx=e^x+C``]<<
+
+[@ closeDiv() @]*
+
+END_PGML
+
+Section::End();
+
+  Scaffold::End();
+Context()->normalStrings;
+
+$showPartialCorrectAnswers = 1;
+
+
+ENDDOCUMENT();