7 Fig. 7 shows the graph of \(y = \mathrm { f } ( x )\) for values of \(x\) from 0 to 1 . \begin{figure}[h] \end{figure} The following spreadsheet printout shows estimates of \(\int _ { 0 } ^ { 1 } f ( x ) d x\) found using the midpoint and trapezium rules for different values of \(h\), the strip width.

1. Without doing any further calculation, write down the smallest possible interval which contains the value of the integral. Justify your answer.
2. (A) - Calculate the ratio of differences, \(r\), for the sequence of estimates calculated using the trapezium rule.
   • Hence suggest a value for \(r\) correct to 2 significant figures.
   • Comment on your suggested value for \(r\).
     (B) - Use extrapolation to find an improved approximation to the value of the integral.
   • State the value of the integral to two decimal places.
   • Explain why this precision is secure.
   Using a similar approach with the sequence of estimates calculated using the midpoint rule, the approximation to the integral from extrapolation was found to be 1.94427 correct to 5 decimal places.
3. Andrea uses the extrapolated midpoint rule value and the value found in part (ii) ( \(B\) ) to write down an interval which contains the value of the integral. Comment on the validity of Andrea's method.
4. Use the values from the spreadsheet output to calculate an approximation to the integral using Simpson's rule with \(h = 0.125\). Give your answer to 5 decimal places. Approximations to the integral using Simpson's rule are given in the spreadsheet output below. The number of applications of Simpson's rule is given in column N.

   N	O	P	Q
   \(n\)	Simpson	differences	ratio
   1	1.91610623	0.01810005	0.3584931
   2	1.93420628	0.00648874	0.3556525
   4	1.94069502	0.00230774	0.3544828
   8	1.94300275	0.00081805	0.3539885
   16	1.94382081	0.00028958	0.3537638
   32	1.94411039	0.00010244	0.3536568
   64	1.94421283	\(3.623 \mathrm { E } - 05\)
   128	1.94424906

5. Use the spreadsheet output to find the value of the integral as accurately as possible. Justify the precision quoted. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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