5 Fig. 5 shows spreadsheet output concerning the estimation of the derivative of a function \(\mathrm { f } ( x )\) at \(x = 2\) using the forward difference method.
\begin{table}[h]
| A | B | C | D |
| 1 | h | estimate | difference | ratio |
| 2 | 0.1 | 6.3050005 | | |
| 3 | 0.01 | 6.0300512 | -0.274949 | |
| 4 | 0.001 | 6.0030018 | -0.027049 | 0.098379 |
| 5 | 0.0001 | 6.0003014 | -0.0027 | 0.099835 |
| 6 | 0.00001 | 6.0000314 | -0.00027 | 0.099983 |
| 7 | 0.000001 | 6.0000044 | \(- 2.7 \mathrm { E } - 05\) | 0.099994 |
| 8 | 1E-07 | 6.0000016 | \(- 2.71 \mathrm { E } - 06\) | 0.100352 |
| 9 | 1E-08 | 6.0000013 | \(- 3.02 \mathrm { E } - 07\) | 0.111457 |
| 10 | 1E-09 | 6.0000018 | \(4.885 \mathrm { E } - 07\) | -1.61765 |
| 11 | 1E-10 | 6.0000049 | \(3.109 \mathrm { E } - 06\) | 6.363636 |
| 12 | 1E-11 | 6.0000005 | \(- 4.44 \mathrm { E } - 06\) | -1.42857 |
| 13 | 1E-12 | 6.0005334 | 0.0005329 | -120 |
| 14 | 1E-13 | 5.9952043 | -0.005329 | -10 |
| 15 | 1E-14 | 6.1284311 | 0.1332268 | -25 |
| 16 | 1E-15 | 5.3290705 | -0.799361 | -6 |
| 17 | 1E-16 | 0 | -5.329071 | 6.666667 |
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{table}
- Write down suitable cell formulae for
- cell C3,
- cell D4.
- Explain what the entries in cells D4 to D8 tell you about the order of the convergence of the forward difference method.
- Write the entry in cell A10 in standard mathematical notation.
- Explain what the values displayed in cells D10 to D17 suggest about the values in cells B10 to B16.
- Write down the value of the derivative of \(\mathrm { f } ( x )\) at \(x = 2\) to an accuracy that seems justified, explaining your answer.
The formula in cell B2 is \(\quad = ( \mathrm { LN } ( \mathrm { SQRT } ( \mathrm { SINH } ( ( 2 + \mathrm { A } 2 ) \wedge 3 ) ) ) - \mathrm { LN } ( \mathrm { SQRT } ( \mathrm { SINH } ( 2 \wedge 3 ) ) ) ) / \mathrm { A } 2\) and equivalent formulae are entered in cells B3 to B17. - Write \(\mathrm { f } ( x )\) in standard mathematical notation.
The value displayed in cell B17 is zero, even though the calculation results in a non-zero answer.
- Explain how this has arisen.