Problem

The equation of a curve is $y = f(x)$, where $f(x) = (2x - 1)\sqrt{3x - 2} - 2$. The following points lie on the curve. Non-exact values have been given correct to $5$ decimal places.

$A(2, 4),\ B(2.0001, k),\ C(2.001, 4.00625),\ D(2.01, 4.06261),\ E(2.1, 4.63566),\ F(3, 11.22876)$

(a) Find the value of $k$. Give your answer correct to $5$ decimal places.

The table shows the gradients of the chords $AB$, $AC$, $AD$ and $AF$.

Chord	$AB$	$AC$	$AD$	$AE$	$AF$
Gradient of chord	$6.2501$	$6.2511$	$6.2608$		$7.2288$

(b) Find the gradient of the chord $AE$. Give your answer correct to $4$ decimal places.

(c) Deduce the value of $f'(2)$ using the values in the table.