Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram illustrates the curve $y = \sqrt{x}\cos x$, for $0 \le x \le \tfrac{3}{2}\pi$, together with its lowest point $M$, where $x = a$. The shaded area between the curve and the $x$-axis is labelled $R$.
(a)[3]
Show that $a$ satisfies the equation $\tan a = \dfrac{1}{2a}$.
(b)[3]
The values generated by the iterative relation $a_{n+1} = \pi + \tan^{-1}\!\left(\dfrac{1}{2a_n}\right)$, beginning with $x_1 = 3$, approach $a$. Use the relation to find $a$ correct to 2 decimal places. Show each iteration to 4 decimal places.
(c)[6]
Find the volume of the solid formed when the region $R$ is turned fully about the $x$-axis. Give your answer in terms of $\pi$.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Apply the product rule accurately” …