Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts the curve $y = e^{-\frac{1}{2}x^2}\sqrt{(1 + 2x^2)}$ for $x \ge 0$, together with its maximum point $M$.
(i)[4]
Find the exact $x$-coordinate of $M$.
(ii)[3]
The values produced by the iterative formula $x_{n+1} = \sqrt{\ln(4 + 8x_n^2)}$, with starting value $x_1 = 2$, converge to a certain value $\alpha$. State an equation satisfied by $\alpha$ and hence show that $\alpha$ is the $x$-coordinate of a point on the curve where $y = 0.5$.
(iii)[3]
Use the iterative formula to determine $\alpha$ correct to $2$ decimal places. Give each iterate to $4$ decimal places.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Use the correct product or quotient rule and the chain rule at least once” …