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,           # no decimals
    reduceConstantFunctions => 1,           # simplify 4 + 5 * 2?
    formatStudentAnswer     => 'parsed',    # no decimals
);

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

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

$ans1 = $fx;

$ans2 = $fx->substitute(k => $k);    # formula
# $ans2 = $fx->eval(k => $k); # gives errors, must eval to real

$ans3 = $fx->substitute(x => $a * pi, k => $k);    # formula
# $ans3 = $fx->eval(x => $a * pi, k => $k); # real

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 $ans2 = $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, $ans3 = $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.

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.