Differentiate a Function

Difference quotients

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Complete Code

Download file: DifferentiateFunction.pg

PG problem file

Explanation

DOCUMENT();

loadMacros('PGstandard.pl', 'PGML.pl', 'PGcourse.pl');

Preamble

These standard macros need to be loaded. 
Context()->variables->add(k => 'Real');
Context()->flags->set(
    reduceConstants         => 0,
    reduceConstantFunctions => 0,
    formatStudentAnswer     => 'parsed',
);

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

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

$ans1 = $fx;
$ans2 = $fx->substitute(k => $k);
$ans3 = $fx->substitute(x => $a * pi, k => $k);

Setup

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

The main difference between the eval and substitute methods is

  • eval returns a Real (a number)
  • substitute returns a Formula

For a constant answer either the eval method can be used in which case the answer would be a Real, or the substitute method can be used in which case the answer would be a constant Formula.

For example, the eval method could be used as in $ans3 = $fx->eval(x => $a * pi, k => $k) to obtain a Real which will display as a single number in decimal format. Note that the eval method requires values for all variables in the formula. For example, calling $fx->eval(k => $k) gives errors, because a value is not provided for the variable x.

The substitute method can be used instead which gives more control over how the answer will be displayed. In particular, the context flag reduceConstants controls whether simple constant expressions like 2 * 4 or 5 * pi will be reduced to numbers in decimal format, the flag reduceConstantFunctions controls whether or not expressions such as sqrt(3) or sin(3) are evaluated to numbers or left as a function evaluated at a number, and setting the context flag formatStudentAnswer => 'parsed' will prevent the student’s “Entered” answer from being reduced to a single number in decimal format, and will display constants like pi instead of an approximation being displayed.

It is important to note that the above rules do not apply to the numbers passed to the substitute method. So $a * pi will be converted into decimal form before it is used by the substitute method.

Note that neither the eval method or substitute method can be used to perform function composition. Only numbers can be plugged in, not formulas.

For more details see Eval Versus Substitute and Constants in Problems.

BEGIN_PGML
Suppose [`f(x) = [$f]`] where [`k`] is a constant.

a. [`f'(x) =`] [_____]{$ans1}

b. If [`k = [$k]`] then [`f'(x) =`] [_____]{$ans2}

c. If [`k = [$k]`] then [`f'([$a]\pi) =`] [_____]{$ans3}
END_PGML

Statement

This is the problem statement in PGML. 
BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

A solution should be provided here.