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]
\includegraphics[alt={},max width=\textwidth]{87bb8eb7-b725-48b0-b32b-0bfce624cd91-08_576_881_315_333}
\captionsetup{labelformat=empty}
\caption{Fig. 6.1}
\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 \(b\).
- 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}
- 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) - Write a suitable formula for cell C3.
- 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}
- State the method being used.
- 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.
- Determine whether Liren is correct.