OCR MEI Further Numerical Methods (Further Numerical Methods) Specimen

1

1. Solve the following simultaneous equations. $$\begin{aligned} & x + \quad y = 1
   & x + 0.99 y = 2 \end{aligned}$$
2. The coefficient 0.99 is correct to two decimal places. All other coefficients in the equations are exact. With the aid of suitable calculations, explain why your answer to part (i) is unreliable.

2 The following spreadsheet printout shows the bisection method being applied to the equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { x } - x ^ { 2 } - 2\). Some values of \(\mathrm { f } ( x )\) are shown in columns B and D.

1. The formula in cell A 3 is \(= \mathrm { IF } ( \mathrm { F } 2 > 0\), A2, E2). State the purpose of this formula.
2. The formula in cell C 3 is \(= \mathrm { IF } ( \mathrm { F } 2 > 0 , \ldots , \ldots )\). What are the missing cell references?
3. In which row is the magnitude of the maximum possible error (mpe) less than \(5 \times 10 ^ { - 7 }\) for the first time?

3 The equation \(\sinh x + x ^ { 2 } - 1 = 0\) has a root, \(\alpha\), such that \(0 < \alpha < 1\).

1. Verify that the iteration \(x _ { r + 1 } = \frac { 1 - \sinh x _ { r } } { x _ { r } }\) with \(x _ { 0 } = 1\) fails to converge to this root.
2. Use the relaxed iteration \(x _ { r + 1 } = ( 1 - \lambda ) x _ { r } + \lambda \left( \frac { 1 - \sinh x _ { r } } { x _ { r } } \right)\) with \(\lambda = \frac { 1 } { 4 }\) and \(x _ { 0 } = 1\) to find \(\alpha\) correct to 6 decimal places.

4 The table below gives values of a function \(y = \mathrm { f } ( x )\).

1. Calculate three estimates of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 0.4\) using the central difference method.
2. State the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 0.4\) to an appropriate degree of accuracy. Justify your answer.

5 A vehicle is moving in a straight line. Its velocity at different times is recorded and shown below. The velocities are recorded to 5 significant figures and the times may be assumed to be exact. It is suggested initially that a quadratic model may be appropriate for this situation.

1. Given that the vehicle is modelled as a particle with constant mass, what assumption about the net force acting on the vehicle leads to a quadratic model?
2. Find Newton's interpolating polynomial of degree 2 to model this situation. Write your answer in the form \(v = a t ^ { 2 } + b t + c\).
3. Comment on whether this model appears to be appropriate.
4. Use this model to find an approximation to the distance travelled over the interval \(5 \leq t \leq 15\). Further investigation suggests that a cubic model may be more appropriate.
5. What technique would you use to fit a cubic model to the data in the table?

6 The secant method is to be used to solve the equation \(x - \ln ( \cos x ) - 1 = 0\).

1. Show that starting with \(x _ { 0 } = - 1\) and \(x _ { 1 } = 0\) leads to the method failing to find the root between \(x = 0\) and \(x = 1\). The spreadsheet printout shows the application of the secant method starting with \(x _ { 0 } = 0\) and \(x _ { 1 } = 1\). Successive approximations to the root are in column E.

   	A	B	C	D	E
   1	\(x _ { n }\)	\(\mathrm { f } \left( x _ { n } \right)\)	\(x _ { n + 1 }\)	\(\mathrm { f } \left( x _ { n + 1 } \right)\)	\(x _ { n + 2 }\)
   2	0	-1	1	0.6156265	0.6189549
   3	1	0.6156265	0.6189549	-0.175846	0.7036139
   4	0.6189549	-0.1758461	0.7036139	-0.025245	0.7178053
   5	0.7036139	-0.0252451	0.7178053	0.0011619	0.7171808
   6	0.7178053	0.0011619	0.7171808	- 7.4 E -06	0.7171848
   7	0.7171808	-7.402E-06	0.7171848	-2.16E-09	0.7171848
   8	0.7171848	-2.16E-09	0.7171848	3.997 E -15	0.7171848

2. What feature of column B shows that this application of the secant method has been successful?
3. Write down a suitable spreadsheet formula to obtain the value in cell E2. Some analysis of convergence is carried out, and the following spreadsheet output is obtained.

   	A	B	C	D	E	F	G	H
   1	\(x _ { n }\)	\(\mathrm { f } \left( x _ { n } \right)\)	\(x _ { n + 1 }\)	\(\mathrm { f } \left( x _ { n + 1 } \right)\)	\(x _ { n + 2 }\)
   2	0	-1	1	0.6156265	0.6189549	0.084659	0.167629	1.980053
   3	1	0.6156265	0.6189549	-0.175846	0.7036139	0.0141913	-0.044	-3.10054
   4	0.6189549	-0.1758461	0.7036139	-0.025245	0.7178053	-0.0006244	-0.00633	10.13727
   5	0.7036139	-0.0252451	0.7178053	0.0011619	0.7171808	3.953 E -06	0.000292	73.83899
   6	0.7178053	0.0011619	0.7171808	- 7.4 E -06	0.7171848	1.154 E -09	-1.8E-06
   7	0.7171808	-7.402E-06	0.7171848	- 2.16 E -09	0.7171848	- 2.109 E -15
   8	0.7171848	- 2.16 E -09	0.7171848	3.997 E -15	0.7171848

   The formula in cell F2 is =E3-E2. The formula in cell G2 is =F3/F2. The formula in cell H2 is =F3/(F2\^{}2).
4. (A) Explain the purpose of each of these three formulae.
   (B) Explain the significance of the values in columns G and H in terms of the rate of convergence of the secant method.
5. Explain why the values in cells F6 and F7 are not 0 . [Question 7 is printed overleaf.]

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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