2 The table shows some values of \(x\) and the associated values of \(y = f ( x )\).
| \(x\) | 2.75 | 3 | 3.25 |
| \(\mathrm { f } ( x )\) | 0.920799 | 1 | 1.072858 |
- Calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 3\) using the forward difference method, giving your answer correct to \(\mathbf { 5 }\) decimal places.
- Calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 3\) using the central difference method, giving your answer correct to \(\mathbf { 5 }\) decimal places.
- Explain why your answer to part (b) is likely to be closer than your answer to part (a) to the true value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 3\).
When \(x = 5\) it is given that \(y = 1.4645\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = 0.1820\), correct to 4 decimal places.
- Determine an estimate of the error when \(\mathrm { f } ( 5 )\) is used to estimate \(\mathrm { f } ( 5.024 )\).