Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
Within $0 < x < \pi$, the equation $\cot x = 1 - x$ has a single root, which is written as $\alpha$.
(i)[2]
Show by calculation that $\alpha$ exceeds $2.5$.
(ii)[2]
Show that, if a sequence of values in the interval $0 < x < \pi$ defined by the iterative formula $$x_{n+1} = \pi + \tan^{-1}\left(\frac{1}{1 - x_n}\right)$$ converges, then its limit is $\alpha$.
(iii)[3]
Use this iterative formula to find $\alpha$ correct to $3$ decimal places. Record each iteration to $5$ decimal places.
Worked solution & mark scheme
This 7-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Calculate the value of the relevant expression or expressions for $x=2.5$ and for another suitable value, e.g. $x=3$” …