7 Sarah uses the trapezium rule to find a sequence of approximations to \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } \mathrm { d } x\).
Her spreadsheet output is shown in Fig. 7.1.
\begin{table}[h]
| \(n\) | \(T _ { n }\) | difference | ratio |
| 1 | 0.43634681 | | |
| 2 | 0.5580694 | 0.121723 | |
| 4 | 0.60199843 | 0.043929 | 0.36089 |
| 8 | 0.61787073 | 0.015872 | 0.36132 |
| 16 | 0.62357601 | 0.005705 | 0.35945 |
| 32 | 0.62561716 | 0.002041 | 0.35777 |
\captionsetup{labelformat=empty}
\caption{Fig. 7.1}
\end{table}
- Write down the value of \(h\) used to find the approximation 0.62357601 .
- Without doing any further calculation, state the value of \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } \mathrm { d } x\) as accurately as you
can, justifying the precision quoted. - Explain what the values in the ratio column tell you about the order of convergence of this sequence of approximations.
Sarah carries out further work using the midpoint rule and Simpson’s rule. Her results are shown in Fig. 7.2.
\begin{table}[h]
| M | N | O | P | Q | R |
| 1 | \(n\) | \(T _ { n }\) | \(M _ { n }\) | \(S _ { 2 n }\) | difference | ratio |
| 2 | 1 | 0.43634681 | 0.679792 | 0.5986436 | | |
| 3 | 2 | 0.5580694 | 0.64592745 | 0.61664144 | 0.018 | |
| 4 | 4 | 0.60199843 | 0.63374304 | 0.6231615 | 0.00652 | 0.362269 |
| 5 | 8 | 0.61787073 | 0.62928129 | 0.62547777 | 0.00232 | 0.355253 |
| 6 | 16 | 0.62357601 | 0.62765831 | 0.62629755 | 0.00082 | 0.35392 |
| 7 | 32 | 0.62561716 | 0.62707259 | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 7.2}
- Write down an efficient spreadsheet formula for calculating \(S _ { 16 }\).
- Determine the missing values in row 7.
- Use extrapolation to determine the value of \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } d x\) as accurately as you can, justifying
the precision quoted.
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