Staircase/cobweb diagram

3 It is given that \(\mathrm { F } ( x ) = 2 + \ln x\). The iteration \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\) is to be used to find a root, \(\alpha\), of the equation \(x = 2 + \ln x\).

1. Taking \(x _ { 1 } = 3.1\), find \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers correct to 5 decimal places.
2. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). Given that \(\alpha = 3.14619\), correct to 5 decimal places, use the values of \(e _ { 2 }\) and \(e _ { 3 }\) to make an estimate of \(\mathrm { F } ^ { \prime } ( \alpha )\) correct to 3 decimal places. State the true value of \(\mathrm { F } ^ { \prime } ( \alpha )\) correct to 4 decimal places.
3. Illustrate the iteration by drawing a sketch of \(y = x\) and \(y = \mathrm { F } ( x )\), showing how the values of \(x _ { n }\) approach \(\alpha\). State whether the convergence is of the 'staircase' or 'cobweb' type.

4. The curve with equation \(y = 2 + \ln ( 4 - x )\) meets the line \(y = x\) at a single point, \(x = \beta\).
a. Show that \(2 < \beta < 3\) \begin{figure}[h] \end{figure} Figure 2 shows the graph of \(y = 2 + \ln ( 4 - x )\) and the graph of \(y = x\).
A student uses the iteration formula $$x _ { n + 1 } = 2 + \ln \left( 4 - x _ { n } \right) , \quad n \in N ,$$ in an attempt to find an approximation for \(\beta\).
Using the graph and starting with \(x _ { 1 } = 3\),
b. determine whether the or not this iteration formula can be used to find an approximation for \(\beta\), justifying your answer.

4 Answer the whole of this question on the insert provided. The sketch shows the curve with equation \(y = \mathrm { F } ( x )\) and the line \(y = x\). The equation \(x = \mathrm { F } ( x )\) has roots \(x = \alpha\) and \(x = \beta\) as shown.

1. Use the copy of the sketch on the insert to show how an iteration of the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), with starting value \(x _ { 1 }\) such that \(0 < x _ { 1 } < \alpha\) as shown, converges to the root \(x = \alpha\).
2. State what happens in the iteration in the following two cases.
   1. \(x _ { 1 }\) is chosen such that \(\alpha < x _ { 1 } < \beta\).
   2. \(x _ { 1 }\) is chosen such that \(x _ { 1 } > \beta\). \section*{Jan 2006} 4
   3. \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-03_873_1259_274_484}
   4. (a) \(\_\_\_\_\) (b) \(\_\_\_\_\) \section*{Jan 2006}