Problem

(i) $f(x) = x – 4 + \cos 5x \quad x > 0$

(a) Show that the equation $f(x) = 0$ has a root $\alpha$ in the interval $[2.5, 3.5]$

(b) Use linear interpolation once on the interval $[2.5, 3.5]$ to find an approximation to $\alpha$, giving your answer to $2$ decimal places.

(ii) $g(x) = \dfrac{1}{10}x^2 - 2 + \dfrac{1}{2x} \quad x > 0$

(a) Determine $g'(x)$.

The equation $g(x) = 0$ has a root $\beta$ in the interval $[6, 7]$

(b) Using $x_0 = 6$ as a first approximation to $\beta$, apply the Newton–Raphson procedure once to $g(x)$ to find a second approximation to $\beta$, giving your answer to $3$ decimal places.