5 The root of the equation \(\mathrm { f } ( x ) = 0\) is being found using the method of interval bisection. Some of the associated spreadsheet output is shown in the table below.
| 1 | A | B | C | D | E | F |
| 1 | a | f(a) | \(b\) | f(b) | c | \(\mathrm { f } ( c )\) |
| 2 | 2 | -0.6109 | 3 | 6.08554 | 2.5 | 1.43249 |
| 3 | 2 | -0.6109 | 2.5 | 1.43249 | 2.25 | 0.17524 |
| 4 | 2 | -0.6109 | 2.25 | 0.17524 | 2.125 | -0.2677 |
| 5 | 2.125 | -0.2677 | 2.25 | 0.17524 | 2.1875 | -0.0598 |
| 6 | | | | | | |
The formula in cell B2 is \(\quad = \mathrm { EXP } ( \mathrm { A } 2 ) - \mathrm { A } 2 ^ { \wedge } 2 - \mathrm { A } 2 - 2\).
- Write down the equation whose root is being found.
- Write down a suitable formula for cell E2.
The formula in cell A3 is
$$= \mathrm { IF } ( \mathrm {~F} 2 < 0 , \mathrm { E } 2 , \mathrm {~A} 2 )$$
.
- Write down a similar formula for cell C3.
- Complete row 6 of the table on the copy in the Printed Answer Booklet.
- Without doing any calculations, write down the value of the root correct to the number of decimal places which seems justified. You must explain the precision quoted.
- Determine how many more applications of the bisection method are needed such that the interval which contains the root is less than 0.0005 .