OCR MEI Further Numerical Methods (Further Numerical Methods) 2024 June

1 The table shows some values of \(x\), together with the associated values of a function, \(\mathrm { f } ( x )\).

1. Use the information in the table to calculate the most accurate estimate of \(f ^ { \prime } ( 2 )\) possible.
2. Calculate an estimate of the error when \(f ( 2 )\) is used as an estimate of \(f ( 2.05 )\).

2 You are given that \(a = \tanh ( 1 )\) and \(b = \tanh ( 2 )\).
\(A\) is the approximation to \(a\) formed by rounding \(\tanh ( 1 )\) to 1 decimal place.
\(B\) is the approximation to \(b\) formed by rounding \(\tanh ( 2 )\) to 1 decimal place.

1. Calculate the following.
   • The relative error \(\mathrm { R } _ { \mathrm { A } }\) when \(A\) is used to approximate \(a\).
   • The relative error \(\mathrm { R } _ { \mathrm { B } }\) when \(B\) is used to approximate \(b\).
   • Calculate the relative error \(\mathrm { R } _ { \mathrm { C } }\) when \(\mathrm { C } = \frac { \mathrm { A } } { \mathrm { B } }\) is used to approximate \(\mathrm { c } = \frac { \mathrm { a } } { \mathrm { b } }\).
   • Comment on the relationship between \(R _ { A } , R _ { B }\) and \(R _ { C }\).

3 The equation \(x ^ { 2 } - \cosh ( x - 2 ) = 0\) has two roots, \(\alpha\) and \(\beta\), such that \(\alpha < \beta\).

1. Use the iterative formula $$x _ { n + 1 } = g \left( x _ { n } \right) \text { where } g \left( x _ { n } \right) = \sqrt { \cosh \left( x _ { n } - 2 \right) } \text {, }$$ starting with \(x _ { 0 } = 1\), to find \(\alpha\) correct to \(\mathbf { 3 }\) decimal places. The diagram shows the part of the graphs of \(\mathrm { y } = \mathrm { x }\) and \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) for \(0 \leqslant x \leqslant 7\).
   \includegraphics[max width=\textwidth, alt={}, center]{83a06341-74e9-4f47-9104-e8e0259e7dfa-3_760_657_753_246}
2. Explain why the iterative formula used to find \(\alpha\) cannot successfully be used to find \(\beta\), even if \(x _ { 0 }\) is very close to \(\beta\).
3. Use the relaxed iteration $$\mathrm { x } _ { \mathrm { n } + 1 } = ( 1 - \lambda ) \mathrm { x } _ { \mathrm { n } } + \lambda \mathrm { g } \left( \mathrm { x } _ { \mathrm { n } } \right) ,$$ with \(\lambda = - 0.21\) and \(x _ { 0 } = 6.4\), to find \(\beta\) correct to \(\mathbf { 3 }\) decimal places. In part (c) the method of relaxation was used to convert a divergent sequence of approximations into a convergent sequence.
4. State one other application of the method of relaxation.

4 Between 1946 and 2012 the mean monthly maximum temperature of the water surface of a lake in northern England has been recorded by environmental scientists. Some of the data are shown in Table 4.1. \begin{table}[h] \end{table} Table 4.2 shows a difference table for the data. \begin{table}[h] \end{table}

1. Complete the copy of the difference table in the Printed Answer Booklet.
2. Explain why a quadratic model may be appropriate for these data.
3. Use Newton's forward difference interpolation formula to construct an interpolating polynomial of degree 2 for these data. This polynomial is used to model the relationship between \(T\) and \(t\). Between 1946 and 2012 the mean monthly maximum temperature of the water surface of the lake was recorded as \(8.9 ^ { \circ } \mathrm { C }\) for October and \(7.5 ^ { \circ } \mathrm { C }\) for November.
4. Determine whether the model is a good fit for the temperatures recorded in October and November. A scientist recorded the mean monthly maximum temperature of the water surface of the lake in 2022. Some of the data are shown in Table 4.3. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 4.3}

   Month	May	June	July	August	September
   \(t =\) Time in months	0	1	2	3	4
   \(T =\) Mean temperature in \({ } ^ { \circ } \mathrm { C }\)	10.3	14.7	16.9	16.9	14.8

   \end{table}
5. Adapt the polynomial found in part (c) so that it can be used to model the relationship between \(T\) and \(t\) for the data in Table 4.3.

5 The root of the equation \(\mathrm { f } ( x ) = 0\) is being found using the method of interval bisection. Some of the associated spreadsheet output is shown in the table below. The formula in cell B2 is \(\quad = \mathrm { EXP } ( \mathrm { A } 2 ) - \mathrm { A } 2 ^ { \wedge } 2 - \mathrm { A } 2 - 2\).

1. Write down the equation whose root is being found.
2. Write down a suitable formula for cell E2. The formula in cell A3 is $$= \mathrm { IF } ( \mathrm {~F} 2 < 0 , \mathrm { E } 2 , \mathrm {~A} 2 )$$ .
3. Write down a similar formula for cell C3.
4. Complete row 6 of the table on the copy in the Printed Answer Booklet.
5. Without doing any calculations, write down the value of the root correct to the number of decimal places which seems justified. You must explain the precision quoted.
6. Determine how many more applications of the bisection method are needed such that the interval which contains the root is less than 0.0005 .

6 Table 6.1 shows some values of \(x\) and the associated values of a function, \(y = f ( x )\). \begin{table}[h] \end{table}

1. Explain why it is not possible to use the central difference method to calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 1\).
2. Use the forward difference method to calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 1\). A student uses the forward difference method to calculate a series of approximations to \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 2\) with different values of the step length, \(h\). These approximations are shown in Table 6.2, together with some further analysis. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 6.2}

   \(h\)	0.8	0.4	0.2	0.1	0.05	0.025	0.0125	0.00625
   approximation	0.130452	0.138647	0.143381	0.145942	0.147277	0.147959	0.148304	0.148477
   difference		0.008195	0.004734	0.002561	0.001335	0.000682	0.000345	0.000173
   ratio			0.577633	0.541099	0.521186	0.510762	0.505424	0.502723

   \end{table}
3. 1. Explain what the ratios of differences tell you about the order of the method in this case.
   2. Comment on whether this is unusual.
4. Determine the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 2\) as accurately as possible. You must justify the precision quoted.

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.