Problem

7. $f(x) = x^2 - x - 3$

(a) Show that the equation $f(x) = 0$ has a root, $\alpha$, in the interval $[1, 2]$

(b) Starting with the interval $[1, 2]$, use interval bisection twice to show that $\alpha$ lies in the interval $[1.25, 1.5]$

(c) (i) Determine $f'(x)$

(ii) Using $1.375$ as a first approximation for $\alpha$, apply the Newton-Raphson process once to $f(x)$ to determine a second approximation for $\alpha$, giving your answer to 3 decimal places.

(d) Use linear interpolation once on the interval $[1.25, 1.5]$ to obtain a different approximation for $\alpha$, giving your answer to 3 decimal places.