Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts the graph of $y = 3 - e^{-\frac{1}{2}x}$. On the same diagram, sketch $y = |5x - 4|$, and demonstrate that $3 - e^{-\frac{1}{2}x} = |5x - 4|$ has precisely two real roots. The two roots of $3 - e^{-\frac{1}{2}x} = |5x - 4|$ are labelled $\alpha$ and $\beta$, with $\alpha < \beta$.
(a)[2]
On the diagram, sketch the graph of $y = |5x - 4|$, and establish that the equation $3 - e^{-\frac{1}{2}x} = |5x - 4|$ has exactly two real roots.
(b)[2]
By calculation, show that $\alpha$ is between $0.36$ and $0.37$.
(c)[3]
Apply the iterative formula $x_{n+1} = \frac{1}{5}\left(7 - e^{-\frac{1}{2}x_n}\right)$ to determine $\beta$ correct to $4$ significant figures. Record each iterate to $6$ significant figures.
Worked solution & mark scheme
This 7-mark question has a full step-by-step worked solution and mark scheme. One marking point: “A reasonably accurate sketch with the vertex placed on the positive $x$-axis” …