DifferentiateFunction1

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Differentiating and Evaluating a Function

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This PG code shows how to create a function using MathObjects, differentiate it, and evaluate it.


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PG problem file Explanation

Problem tagging data

Problem tagging:

DOCUMENT(); 

loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"AnswerFormatHelp.pl",
"unionLists.pl",
);

TEXT(beginproblem());

Initialization: We load unionLists.pl to create an enumerated list in the Main Text section.

Context("Numeric")->variables->add(k=>"Real");
Context()->flags->set(
  reduceConstants=>0, # no decimals
  reduceConstantFunctions=>1, # combine 4+5*2?
  formatStudentAnswer=>'parsed', # no decimals
);

$a = random(6,9,1);
$k = random(3,5,1);

$f = Formula("k x^2");
$fx = $f->D('x');

@answer = ();

$answer[0] = $fx;

$answer[1] = $fx->substitute(k=>$k);

$answer[2] = $fx->substitute(x=>$a*pi,k=>$k);

Setup: The partial differentiation operator is ->D('x').

The main difference between eval() and substitute() is

  • eval() returns a Real (a number)
  • substitute() returns a Formula
Since plugging a particular number $k into the Formula $f returns a Formula $k x, if we had used the eval method $answer[1] = $fx->eval(k=>$k); instead of the substitute method, we would get errors because $k x is a Formula, not a Real. Note: You cannot use eval or substitute to perform function composition, i.e., you can only plug in numbers, not formulas.

When the answer is a constant, we can use either the eval method, in which case the answer would be a Real, or the substitute method, in which case the answer would be a constant Formula. If you use the eval method, $answer[2] = $fx->eval(x=>$a*pi,k=>$k); the answer will be a Real and will display as a single number in decimal format. If you use the substitute method instead, you have more control over how the answer will be displayed. In particular, the context flag reduceConstants controls whether the answer will be reduced to a single number in decimal format, the flag reduceConstantFunctions controls whether or not expressions such as 4+5*2 are reduced to 14, and setting the context flag formatStudentAnswer=>'parsed' will prevent the student's answer from being reduced to a single number in decimal format and will also display pi instead of 3.14159...

For more details, see eval versus substitute, formatting correct answers, and constants in problems.

Context()->texStrings;
BEGIN_TEXT
Suppose \( f(x) = $f \) where \( k \) is a 
constant.  
\{ BeginList("OL",type=>"a") \}

$ITEM \( f'(x) = \) 
\{ ans_rule(20) \}
\{ AnswerFormatHelp("formulas") \}

$ITEMSEP
$ITEM If \( k = $k \) then \( f'(x) = \)
\{ ans_rule(20) \}
\{ AnswerFormatHelp("formulas") \}

$ITEMSEP
$ITEM If \( k = $k \) then \( f'($a\pi) = \)
\{ ans_rule(20) \}
\{ AnswerFormatHelp("formulas") \}

\{ EndList("OL") \}
END_TEXT
Context()->normalStrings;

Main Text: We use an ordered list to display the three parts to this question. The $ITEMSEP command puts extra separation between items. This list is provided by the macro unionLists.pl.

$showPartialCorrectAnswers = 1;

foreach my $i (0..2) {
  ANS( $answer[$i]->cmp() );
}

Answer Evaluation:

Context()->texStrings;
BEGIN_SOLUTION
Solution explanation goes here.
END_SOLUTION
Context()->normalStrings;

COMMENT("MathObject version.");

ENDDOCUMENT();

Solution:

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