Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram displays the curves $y = \cos x$ and $y = \dfrac{k}{1 + x}$, where $k$ is a constant, for $0 \leq x \leq \frac{1}{2}\pi$. The curves meet at the point for which $x = p$.
(a)[5]
Show that $p$ obeys the equation $\tan p = \dfrac{1}{1 + p}$.
(b)[3]
Apply the iterative formula $p_{n+1} = \tan^{-1}\!\left(\dfrac{1}{1 + p_n}\right)$ to work out the value of $p$ correct to $3$ decimal places. Show each iteration to $5$ decimal places.
(c)[2]
Hence determine the value of $k$ correct to $2$ decimal places.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State that $\cos p = \frac{k}{1+p}$” …