Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts part of the curve $y = 8x + \frac{1}{2}e^x$. The shaded region $R$ is enclosed by the curve and the lines $x = 0$, $y = 0$ and $x = a$, with $a$ positive. The area of $R$ is $\frac{1}{2}$.
(i)[5]
Determine an equation that $a$ satisfies, and show that it may be expressed as $a = \sqrt{\frac{2 - e^a}{8}}$.
(ii)[2]
Use calculation to confirm that the equation $a = \sqrt{\frac{2 - e^a}{8}}$ has a root in the interval $0.2$ to $0.3$.
(iii)[3]
Apply the iterative formula $a_{n+1} = \sqrt{\frac{2 - e^{a_n}}{8}}$ to find this root correct to 2 decimal places. Record each iterative value 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: “Integrate so that the resulting terms are $4x^2$ and $\tfrac12 e^x$” …