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June 2016 p31 q6
1885
(i) By sketching a suitable pair of graphs, show that the equation \(5e^{-x} = \sqrt{x}\) has one root.
(ii) Show that, if a sequence of values given by the iterative formula \(x_{n+1} = \frac{1}{2} \ln\left(\frac{25}{x_n}\right)\) converges, then it converges to the root of the equation in part (i).
(iii) Use this iterative formula, with initial value \(x_1 = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Solution
(i) To show that the equation \(5e^{-x} = \sqrt{x}\) has one root, sketch the graphs of \(y = 5e^{-x}\) and \(y = \sqrt{x}\). The intersection point of these graphs represents the root. The exponential function \(5e^{-x}\) decreases rapidly, while \(\sqrt{x}\) increases slowly. They intersect at one point, indicating one root.
(ii) The iterative formula is \(x_{n+1} = \frac{1}{2} \ln\left(\frac{25}{x_n}\right)\). Rearrange this to show it is equivalent to \(5e^{-x} = \sqrt{x}\). If the sequence converges, it must satisfy the original equation, thus converging to the root.
(iii) Using the iterative formula with \(x_1 = 1\):
