Problem

2. $f(x) = 7x - \dfrac{1}{2}x - \dfrac{5}{3x} \quad x > 0$

(a) Show that the equation $f(x) = 0$ has a root, $\alpha$, in the interval $[2.8, 2.9]$

(b) (i) Find $f'(x)$.

(ii) Hence, using $x_0 = 2.8$ as a first approximation to $\alpha$, apply the Newton-Raphson procedure once to $f(x)$ to calculate a second approximation to $\alpha$, giving your answer to 3 decimal places.

(c) Use linear interpolation once on the interval $[2.8, 2.9]$ to find another approximation to $\alpha$. Give your answer to 3 decimal places.