Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The equation $x^3 - x^2 - 6 = 0$ has a single real root, written as $\alpha$.
(i)[2]
Find by calculation the two consecutive integers that bracket $\alpha$.
(ii)[2]
Show that, if a sequence generated by the iterative formula $x_{n+1} = \sqrt{\left(x_n + \frac{6}{x_n}\right)}$ converges, then its limit is $\alpha$.
(iii)[3]
Use this iterative formula to find $\alpha$ correct to $3$ decimal places. Record the result of 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: “Evaluate or inspect the sign of $x^3-x^2-6$ for two integer values of $x$” …