OCR MEI Further Numerical Methods (Further Numerical Methods) 2019 June

1 Fig. 1 shows some spreadsheet output concerning the values of a function, \(\mathrm { f } ( x )\). \begin{table}[h] \end{table} The formula in cell B2 is ==EXP(-(A2\^{}2)) and equivalent formulae are in cells B3 to B6. The formula in cell C 2 is \(= \mathrm { B } 2\).
The formula in cell C3 is \(\quad = \mathrm { C } 2 + \mathrm { B } 3\). Equivalent formulae are in cells C4 to C6.

1. Use sigma notation to express the formula in cell C5 in standard mathematical notation.
2. Explain why the same value is displayed in cells C 5 and C 6. Now suppose that the value in cell C2 is chopped to 3 decimal places and used to approximate the value in cell C2.
3. Calculate the relative error when this approximation is used. Suppose that the values in cells B4, B5 and B6 are chopped to 3 decimal places and used as approximations to the original values in cells B4, B5 and B6 respectively.
4. Explain why the relative errors in these approximations are all the same.

2 Fig. 2.1 shows the graph of \(y = x ^ { 2 } \mathrm { e } ^ { 2 x } - 5 x ^ { 2 } + 0.5\). \begin{figure}[h] \end{figure} There are three roots of the equation \(x ^ { 2 } \mathrm { e } ^ { 2 x } - 5 x ^ { 2 } + 0.5 = 0\). The roots are \(\alpha , \beta\) and \(\gamma\), where \(\alpha < \beta < \gamma\).

1. Explain why it is not possible to use the method of false position with \(x _ { 0 } = 0\) and \(x _ { 1 } = 1\) to find \(\beta\) and \(\gamma\). The graph of the function indicates that the root \(\gamma\) lies in the interval [0.6, 0.8]. Fig. 2.2 shows some spreadsheet output using the method of false position using these values as starting points. \begin{table}[h]

   	A	B	C	D	E	F
   1	a	f(a)	b	f(b)	approx
   2	0.6	-0.10476	0.8	0.469941	0.636457	-0.07876
   3	0.636457	-0.07876	0.8	0.469941	0.659931	-0.04748
   4	0.659931	-0.04748	0.8	0.469941	0.672783	-0.0249
   5	0.672783	-0.0249	0.8	0.469941	0.679184	-0.01211
   6	0.679184	-0.01211	0.8	0.469941	0.682218	-0.00567
   7	0.682218	-0.00567	0.8	0.469941	0.683623	-0.00261
   8	0.683623	-0.00261	0.8	0.469941	0.684266	-0.00119
   9	0.684266	-0.00119	0.8	0.469941	0.684559	-0.00054
   10	0.684559	-0.00054	0.8	0.469941	0.684692	-0.00025
   11	0.684692	-0.00025	0.8	0.469941	0.684753	-0.00011
   12	0.684753	-0.00011	0.8	0.469941	0.68478	\(- 5.1 \mathrm { E } - 05\)

   \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{table}
2. Without doing any further calculation, write down the smallest possible interval which is certain to contain \(\gamma\).
3. State what is being calculated in column F. The formula in cell A3 is \(\quad = \operatorname { IF } ( \mathrm { F } 2 < 0 , \mathrm { E } 2 , \mathrm {~A} 2 )\).
4. Explain the purpose of this formula in the application of the method of false position. The method of false position uses the same formula for obtaining new approximations as the secant method.
5. Explain how the method of false position differs from the secant method.
6. Give one advantage and one disadvantage of using the method of false position instead of the secant method.

3 In the first week of an outbreak of influenza, 9 patients were diagnosed with the virus at a medical practice in Pencaster. Records were kept of \(y\), the total number of patients diagnosed with influenza in week \(n\). The data are shown in Fig. 3. \begin{table}[h] \end{table}

1. Complete the difference table in the Printed Answer Booklet.
2. Explain why a cubic model is appropriate for the data.
3. Use Newton's method to find the interpolating polynomial of degree 3 for these data. In both week 6 and week 7 there were 145 patients in total diagnosed with influenza at the medical practice.
4. Determine whether the model is a good fit for these data.
5. Determine the maximum number of weeks for which the model could possibly be valid.

4 Fig. 4 shows the graph of \(y = x ^ { 5 } - 6 \sqrt { x } + 4\). \begin{figure}[h] \end{figure} There are two roots of the equation \(x ^ { 5 } - 6 \sqrt { x } + 4 = 0\). The roots are \(\alpha\) and \(\beta\), such that \(\alpha < \beta\).

1. Show that \(0 < \alpha < 1\) and \(1 < \beta < 2\).
2. Obtain the Newton-Raphson iterative formula $$x _ { n + 1 } = x _ { n } - \frac { x _ { n } ^ { \frac { 11 } { 2 } } - 6 x _ { n } + 4 \sqrt { x _ { n } } } { 5 x _ { n } ^ { \frac { 9 } { 2 } } - 3 }$$
3. Use the iterative formula found in part (b) with a starting value of \(x _ { 0 } = 1\) to obtain \(\beta\) correct to 6 decimal places.
4. Use the iterative formula found in part (b) with a starting value of \(x _ { 0 } = 0\) to find \(x _ { 1 }\).
5. Give a geometrical explanation of why the Newton-Raphson iteration fails to find \(\alpha\) in part (d).
6. Obtain the iterative formula $$x _ { n + 1 } = \left( \frac { x _ { n } ^ { 5 } + 4 } { 6 } \right) ^ { 2 }$$
7. Use the iterative formula found in part (f) with a starting value of \(x _ { 0 } = 0\) to obtain \(\alpha\) correct to 6 decimal places.

5 Fig. 5 shows spreadsheet output concerning the estimation of the derivative of a function \(\mathrm { f } ( x )\) at \(x = 2\) using the forward difference method. \begin{table}[h] \end{table}

1. Write down suitable cell formulae for
   • cell C3,
   • cell D4.
   • Explain what the entries in cells D4 to D8 tell you about the order of the convergence of the forward difference method.
   • Write the entry in cell A10 in standard mathematical notation.
   • Explain what the values displayed in cells D10 to D17 suggest about the values in cells B10 to B16.
   • Write down the value of the derivative of \(\mathrm { f } ( x )\) at \(x = 2\) to an accuracy that seems justified, explaining your answer.
   The formula in cell B2 is \(\quad = ( \mathrm { LN } ( \mathrm { SQRT } ( \mathrm { SINH } ( ( 2 + \mathrm { A } 2 ) \wedge 3 ) ) ) - \mathrm { LN } ( \mathrm { SQRT } ( \mathrm { SINH } ( 2 \wedge 3 ) ) ) ) / \mathrm { A } 2\) and equivalent formulae are entered in cells B3 to B17.
2. Write \(\mathrm { f } ( x )\) in standard mathematical notation. The value displayed in cell B17 is zero, even though the calculation results in a non-zero answer.
3. Explain how this has arisen.

6 The spreadsheet output in Fig. 6 shows approximations to \(\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x\) found using the midpoint rule, denoted by \(M _ { n }\), and the trapezium rule, denoted by \(T _ { n }\). \begin{table}[h] \end{table}

1. Write down an efficient spreadsheet formula for cell C3.
2. By first completing the table in the Printed Answer Booklet using the Simpson's rule, calculate the most accurate estimate of \(\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x\) that you can, justifying the precision quoted. \section*{END OF QUESTION PAPER}