3 The method of False Position is used to find a sequence of approximations to the root of an equation. The spreadsheet output showing these approximations, together with some further analysis, is shown below.
| C | D | E | F | G | H | I | J |
| 4 | a | f(a) | b | \(\mathrm { f } ( b )\) | \(x _ { \text {new } }\) | \(\mathrm { f } \left( x _ { \text {new } } \right)\) | difference | ratio |
| 5 | 1 | -1.8248 | 2 | 17.2899 | 1.09547 | -1.80507 | | |
| 6 | 1.09547 | -1.80507 | 2 | 17.2899 | 1.18097 | -1.75418 | 0.08551 | |
| 7 | 1.18097 | -1.75418 | 2 | 17.2899 | 1.25641 | -1.66246 | 0.07544 | 0.88229 |
| 8 | 1.25641 | -1.66246 | 2 | 17.2899 | 1.32164 | -1.52781 | 0.06523 | 0.86458 |
| 9 | 1.32164 | -1.52781 | 2 | 17.2899 | 1.37672 | -1.35706 | 0.05508 | 0.84439 |
| 10 | 1.37672 | -1.35706 | 2 | 17.2899 | 1.42208 | -1.1642 | 0.04536 | 0.8236 |
| 11 | 1.42208 | -1.1642 | 2 | 17.2899 | 1.45853 | -0.96616 | 0.03646 | 0.80376 |
| 12 | 1.45853 | -0.96616 | 2 | 17.2899 | 1.48719 | -0.77825 | 0.02866 | 0.78598 |
| 13 | 1.48719 | -0.77825 | 2 | 17.2899 | | | | |
| 14 | | | | | | | | |
The formula in cell D5 is \(\quad = \mathrm { SINH } \left( \mathrm { C5 } ^ { \wedge } 2 \right) - \mathrm { C5 } ^ { \wedge } 3 - 2\).
- Write down the equation which is being solved.
The formula in cell C 6 is \(\quad = \mathrm { IF } ( \mathrm { H } 5 < 0 , \mathrm { G } 5 , \mathrm { C } 5 )\).
- Write down a similar formula for cell E6.
- Calculate the values which would be displayed in cells G13 and G14 to find further approximations to the root.
- Explain what the values in column J tell you about
- the order of convergence of this sequence of estimates,
- the speed of convergence of this sequence of estimates.