1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams

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SPS SPS FM Pure 2025 September Q9
18 marks Standard +0.3
A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$
    1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
    2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]
  1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
    1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
    2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{\frac{x}{2}}$$ [2 marks]
    3. Use the recurrence relation $$x_{n+1} = e - e^{\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
    4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_9}
Pre-U Pre-U 9794/1 2010 June Q2
3 marks Standard +0.3
The equation \(x^3 - 5x + 3 = 0\) has a root between \(x = 0\) and \(x = 1\).
  1. The equation can be rearranged into the form \(x = g(x)\) where \(g(x) = px^3 + q\). State the values of \(p\) and \(q\). [1]
  2. By considering \(|g'(x)|\), show that the iterative form \(x_{n+1} = g(x_n)\) with a suitable starting value converges to the root between \(x = 0\) and \(x = 1\). [You are not required to find this root.] [2]
Pre-U Pre-U 9794/2 2016 June Q7
11 marks Moderate -0.3
  1. Use a change of sign to verify that the equation \(\cos x - x = 0\) has a root \(\alpha\) between \(x = 0.7\) and \(x = 0.8\). [2]
  2. Sketch, on a single diagram, the curve \(y = \cos x\) and the line \(y = x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), giving the coordinates of all points of intersection with the coordinate axes. [2]
An iteration of the form \(x_{n+1} = \cos(x_n)\) is to be used to find \(\alpha\).
  1. By considering the gradient of \(y = \cos x\), show that this iteration will converge. [3]
  2. On a copy of your sketch from part (ii), illustrate how this iteration converges to \(\alpha\). [2]
  3. Use a change of sign to verify that \(\alpha = 0.7391\) to 4 decimal places. [2]