Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts a section of the curve $y = \cos(\sqrt{x})$ for $x \geq 0$, with $x$ measured in radians. The shaded area bounded by the curve, the coordinate axes and the line $x = p^2$, where $p > 0$, is called $R$. The area of $R$ is $1$.
(i)[6]
Apply the substitution $x = u^2$ to evaluate $\int_0^{p^2} \cos(\sqrt{x}) \, dx$. Hence show that $\sin p = \frac{3 - 2 \cos p}{2p}$.
(ii)[3]
Use the iterative formula $p_{n+1} = \sin^{-1}\!\left(\frac{3 - 2 \cos p_n}{2p_n}\right)$, starting from $p_1 = 1$, to determine $p$ correct to $2$ decimal places. Record the outcome of each iteration to $4$ decimal places.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Replace $x$ and $dx$ everywhere in the integral.” …