6 Table 6.1 shows some values of \(x\) and the associated values of a function, \(y = f ( x )\).
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 6.1}
| \(x\) | 1.5 | 1 | 2 |
| \(\mathrm { f } ( x )\) | 0.84089 | 1 | 1.18921 |
\end{table}
- Explain why it is not possible to use the central difference method to calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 1\).
- Use the forward difference method to calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 1\).
A student uses the forward difference method to calculate a series of approximations to \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 2\) with different values of the step length, \(h\).
These approximations are shown in Table 6.2, together with some further analysis.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 6.2}
| \(h\) | 0.8 | 0.4 | 0.2 | 0.1 | 0.05 | 0.025 | 0.0125 | 0.00625 |
| approximation | 0.130452 | 0.138647 | 0.143381 | 0.145942 | 0.147277 | 0.147959 | 0.148304 | 0.148477 |
| difference | | 0.008195 | 0.004734 | 0.002561 | 0.001335 | 0.000682 | 0.000345 | 0.000173 |
| ratio | | | 0.577633 | 0.541099 | 0.521186 | 0.510762 | 0.505424 | 0.502723 |
- Explain what the ratios of differences tell you about the order of the method in this case.
- Comment on whether this is unusual.
- Determine the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 2\) as accurately as possible. You must justify the precision quoted.