public function testZeroError()
{
// f(x) = 13x² -92x + 96
$f = function ($x) {
return 13 * $x ** 2 - 92 * $x + 96;
};
/*
* h²
* Error term for the Midpoint Formula: - f⁽³⁾(ζ₁)
* 6
*
* where ζ₁ lies between x₀ - h and x₀ + h
*
* h²
* Error term for the Endpoint Formula: - f⁽³⁾(ζ₀)
* 3
*
* where ζ₀ lies between x₀ and x₀ + 2h
*/
// f'(x) = 26x - 92
// f''(x) = 26
// f⁽³⁾(x) = 0
// Thus, our error is zero in both formulas for our function $f
$f’ = function ($x) {
return 26 * $x - 92;
};
$n = 3;
$a = 0;
$b = 4;
// Check that the endpoint formula agrees with f'(x) at x = 0
$target = 0;
$expected = $f’($target);
$actual = ThreePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual);
// Check that the midpoint formula agrees with f'(x) at x = 2
$target = 2;
$expected = $f’($target);
$actual = ThreePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual);
// Check that the (backward) endpoint formula agrees with f'(x) at x = 4
$target = 4;
$expected = $f’($target);
$actual = ThreePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual);
}
