7 A student is using a spreadsheet to find approximations to \(\int _ { 0 } ^ { 1 } f ( x ) d x\) using the midpoint rule, the trapezium rule and Simpson's rule. Some of the associated spreadsheet output with \(n = 1\) and \(n = 2\), is shown in Table 7.1. \begin{table}[h] \end{table}

1. Complete the copy of Table 7.1 in the Printed Answer Booklet. Give your answers correct to 5 decimal places.
2. State the value of \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\) as accurately as possible. You must justify the precision quoted. The student calculates some more approximations using Simpson's rule. These approximations are shown in the associated spreadsheet output, together with some further analysis, in Table 7.2. The values of \(S _ { 2 }\) and \(S _ { 4 }\) have been blacked out, together with the associated difference and ratio. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 7.2}

   n	\(\mathrm { S } _ { 2 n }\)	difference	ratio
   1
   2
   4	0.674353	-0.0209
   8	0.665199	-0.00915	0.438059
   16	0.661297	-0.0039	0.426286
   32	0.659675	-0.00162	0.415762
   64	0.659015	-0.00066	0.406785

   \end{table}
3. The student checks some of her values with a calculator. She does not obtain 0.406785 when she calculates \(- 0.00066 \div ( - 0.00162 )\). Explain whether the value in the spreadsheet, or her value, is a more precise approximation to the ratio of differences in this case.
4. 1. State the order of convergence of the values in the ratio column. You must justify your answer.
   2. Explain what the values in the ratio column tell you about the order of the method in this case.
   3. Comment on whether this is unusual.
5. Determine the value of \(\int _ { 0 } ^ { 1 } f ( x ) d x\) as accurately as you can. You must justify the precision quoted.