6 Fig. 6.1 shows the graph of \(y = \mathrm { e } ^ { 3 x } - 11 x - 0.5\) for \(- 0.5 \leqslant x \leqslant 1\). \begin{figure}[h] \end{figure} The equation \(\mathrm { e } ^ { 3 x } - 11 x - 0.5 = 0\) has two roots, \(\alpha\) and \(\beta\), such that \(\alpha < \beta\). Dennis is going to use the method of interval bisection with starting values denoted by \(a\) and \(b\).

1. Explain why the method of interval bisection starting with \(a = 0\) and \(b = 1\) may not be used to find either \(\alpha\) or \(\beta\). Dennis uses the method of interval bisection starting with \(a = 0\) and \(b = 0.5\) to find \(\alpha\). Some spreadsheet output is shown in Fig. 6.2. \begin{table}[h]

   	A	B	C	D	E	F
   1	a	f(a)	b	f(b)	\(x _ { \text {new } }\)	\(\mathrm { f } \left( x _ { \text {new } } \right)\)
   2	0	0.5	0.5	-1.51831	0.25	-1.133
   3	0	0.5	0.25	-1.133	0.125	-0.42
   4	0	0.5	0.125	-0.42001	0.0625	0.01873
   5	0.0625	0.01873	0.125	-0.42001	0.09375	-0.2065

   \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{table} Dennis states that the formula in cell B2 is $$= \operatorname { EXP } \left( 3 ^ { * } \mathrm {~A} 1 \right) - 11 \mathrm {~A} 2 - 0.5$$ Dennis has made two errors.
2. Write a correct version of Dennis's formula for cell B2. The formula in cell A3, which is correct, is
   = IF(F2 > 0, E2, A2)
3. Write a suitable formula for cell C3.
4. Use the information in Fig. 6.2 to
   • find the value of \(\alpha\) as accurately as possible,
   • state the maximum possible error in this estimate.
   Liren uses a different method to find a sequence of estimates of the value of \(\beta\) using a spreadsheet. The output, together with some further analysis, is shown in Fig. 6.3. \begin{table}[h]

   	A	B	C	D
   1	\(x\)	f(x)	difference	ratio
   2	0.4	-1.5799
   3	0.6	-1.0504
   4	0.99671	8.4245
   5	0.64398	-0.6809	-0.35273
   6	0.67036	-0.4026	0.026378	-0.0748
   7	0.70852	0.08386	0.038164	1.44682
   8	0.70194	-0.0075	-0.00658	-0.1724
   9	0.70248	-0.0001	0.00054	-0.082
   10	0.70249	\(1.8 \mathrm { E } - 07\)	\(8.88 \mathrm { E } - 06\)	0.01646

   \captionsetup{labelformat=empty} \caption{Fig. 6.3}
   \end{table} The formula in cell A4 is $$= ( \mathrm { A } 2 * \mathrm {~B} 3 - \mathrm { A } 3 * \mathrm {~B} 2 ) / ( \mathrm { B } 3 - \mathrm { B } 2 )$$
5. State the method being used.
6. Explain what the values in column D tell you about the order of convergence of this sequence of estimates. Liren states that \(\beta = 0.70249\) correct to 5 decimal places.
7. Determine whether Liren is correct.