4 Fig. 4.1 shows part of the graph of \(y = e ^ { x } - x ^ { 2 } - x - 1.1\). \begin{figure}[h] \end{figure} The equation \(\mathrm { e } ^ { x } - x ^ { 2 } - x - 1.1 = 0\) has a root \(\alpha\) such that \(1 < \alpha < 2\).
Ali is considering using the Newton-Raphson method to find \(\alpha\). Ali could use a starting value of \(x _ { 0 } = 1\) or a starting value of \(x _ { 0 } = 2\).

1. Without doing any calculations, explain whether Ali should use a starting value of \(x _ { 0 } = 1\) or a starting value of \(x _ { 0 } = 2\), or whether using either starting value would work equally well. Ali is also considering using the method of fixed point iteration to find \(\alpha\). Ali could use a starting value of \(x _ { 0 } = 1\) or a starting value of \(x _ { 0 } = 2\). Fig. 4.2 shows parts of the graphs of \(y = x\) and \(y = \ln \left( x ^ { 2 } + x + 1.1 \right)\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f8c78ab8-7daa-4d6f-9bb5-093a46c590b8-04_819_1011_1818_255} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
2. Without doing any calculations, explain whether Ali should use a starting value of \(x _ { 0 } = 1\) or a starting value of \(x _ { 0 } = 2\) or whether either starting value would work equally well. Ali used one of the above methods to find a sequence of approximations to \(\alpha\). These are shown, together with some further analysis in the associated spreadsheet output in Fig. 4.3. \begin{table}[h]

   		M	N	O
   	\(r\)	\(\mathrm { X } _ { \mathrm { r } }\)
   4	0	2
   5	1	1.879008	- 0.121
   6	2	1.858143	- 0.021	0.172
   7	3	1.857565	\(- 6 \mathrm { E } - 04\)	0.028
   8	4	1.857564	\(- 4 \mathrm { E } - 07\)	\(8 \mathrm { E } - 04\)
   9	5	1.857564	\(- 2 \mathrm { E } - 13\)	\(6 \mathrm { E } - 07\)

   \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{table} The formula in cell N5 is =M5-M4
   and the formula in cell O6 is =N6/N5
   equivalent formulae are in cells N6 to N9 and O7 to O9 respectively.
3. State what is being calculated in the following columns of the spreadsheet.
   1. Column N
   2. Column O
4. Explain whether the values in column O suggest that Ali used the Newton-Raphson method or the iterative formula \(\mathrm { x } _ { \mathrm { n } + 1 } = \ln \left( \mathrm { x } _ { \mathrm { n } } ^ { 2 } + \mathrm { x } _ { \mathrm { n } } + 1.1 \right)\) to find this sequence of approximations to \(\alpha\).