OCR MEI Further Numerical Methods (Further Numerical Methods) 2021 November

1

1. 1. Determine the relative error when
      • 1.414214 is used to approximate \(\sqrt { 2 }\),
2. \(1.414214 ^ { 2 }\) is used to approximate 2.
   (ii) Write down the relationship between your answers to part (a)(i).
3. Fig. 1 shows some spreadsheet output.
\begin{table}[h]

	A	B	C
1	2	1.414214	2

\captionsetup{labelformat=empty} \caption{Fig. 1}
\end{table} The formula in cell B1 is = SQRT (A1)
and the formula in cell C 1 is \(\quad = \mathrm { B } 1 \wedge 2\).
Ben evaluates \(1.414214 ^ { 2 }\) on his calculator and obtains 2.000001238 . He states that this shows that the value displayed in cell C1 is wrong. Explain whether Ben is correct.

2 The table shows some values of \(x\) and the associated values of \(\mathrm { f } ( x )\).

1. Complete the difference table in the Printed Answer Booklet.
2. Explain why the data may be interpolated by a polynomial of degree 2.
3. Use Newton's forward difference interpolation formula to obtain a polynomial of degree 2 for the data.

3 The method of False Position is used to find a sequence of approximations to the root of an equation. The spreadsheet output showing these approximations, together with some further analysis, is shown below. The formula in cell D5 is \(\quad = \mathrm { SINH } \left( \mathrm { C5 } ^ { \wedge } 2 \right) - \mathrm { C5 } ^ { \wedge } 3 - 2\).

1. Write down the equation which is being solved. The formula in cell C 6 is \(\quad = \mathrm { IF } ( \mathrm { H } 5 < 0 , \mathrm { G } 5 , \mathrm { C } 5 )\).
2. Write down a similar formula for cell E6.
3. Calculate the values which would be displayed in cells G13 and G14 to find further approximations to the root.
4. Explain what the values in column J tell you about
   • the order of convergence of this sequence of estimates,
   • the speed of convergence of this sequence of estimates.

4 The table shows some values of \(x\) and the associated values of \(\mathrm { f } ( x )\).

1. Calculate four estimates of the derivative of \(\mathrm { f } ( x )\) at \(x = 4\).
2. Without doing any further calculation, state the value of \(f ^ { \prime } ( 4 )\) as accurately as you can, justifying the precision quoted.

5 When Nina does the weekly grocery shopping she models the total cost by adding up the cost of each item in her head as she goes along. To simplify matters she rounds the cost of each item to the nearest pound. One week Nina buys 48 items.

1. Calculate the maximum possible error in Nina's model in this case. Nina estimated the total cost of her shopping to be \(\pounds 92\). The actual cost is \(\pounds 90.23\).
2. Explain whether this is consistent with Nina’s model. The next week her husband, Kareem, does the weekly shopping. He models the total cost by chopping the cost of each item to the nearest pound as he goes along. On this occasion Kareem buys 52 items.
3. Calculate the expected error in Kareem's model in this case. Using his model Kareem estimates the total cost as \(\pounds 76\). The total cost of the shopping is \(\pounds 103.24\).
4. Explain how such a large error could arise. The next week Kareem buys \(n\) items.
5. Write down a formula for the maximum possible error when Kareem uses his model to estimate the total cost of his shopping.
6. Explain how Kareem's model could be adapted so that his formula gives the same expected error as Nina's model when they are both used to estimate the total cost of the shopping.

6 The equation \(0.5 \ln x - x ^ { 2 } + x + 1 = 0\) has two roots \(\alpha\) and \(\beta\), such that \(0 < \alpha < 1\) and \(1 < \beta < 2\).

1. Use the Newton-Raphson method with \(x _ { 0 } = 1\) to obtain \(\beta\) correct to \(\mathbf { 6 }\) decimal places. Fig. 6.1 shows part of the graph of \(y = 0.5 \ln x - x ^ { 2 } + x + 1\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{945883ad-c153-4c51-83d3-978e4c769ed5-06_1112_1156_529_354} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
   \end{figure}
2. On the copy of Fig. 6.1 in the Printed Answer Booklet, illustrate the Newton-Raphson method working to obtain \(x _ { 1 }\) from \(x _ { 0 } = 1\). Beth is trying to find \(\alpha\) correct to 6 decimal places.
3. Suggest a reason why she might choose the Newton-Raphson method instead of fixed point iteration. Beth tries to find \(\alpha\) using the Newton-Raphson method with a starting value of \(x _ { 0 } = 0.5\). Her spreadsheet output is shown in Fig. 6.2. \begin{table}[h]

   \(r\)	\(\mathrm { x } _ { \mathrm { r } }\)
   0	0.5
   1	- 0.40343
   2	\#NUM!

   \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{table}
4. Explain how the display \#NUM! has arisen in the cell for \(x _ { 2 }\). Beth decides to use the iterative formula $$x _ { n + 1 } = g \left( x _ { n } \right) = \sqrt { 0.5 \ln \left( x _ { n } \right) + x _ { n } + 1 }$$
5. Determine the outcome when Beth uses this formula with \(x _ { 0 } = 0.5\).
6. 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 \(\lambda = - 0.041\) and \(x _ { 0 } = 0.5\) to obtain \(\alpha\) correct to \(\mathbf { 6 }\) decimal places.

7 Sarah uses the trapezium rule to find a sequence of approximations to \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } \mathrm { d } x\).
Her spreadsheet output is shown in Fig. 7.1. \begin{table}[h] \end{table}

1. Write down the value of \(h\) used to find the approximation 0.62357601 .
2. Without doing any further calculation, state the value of \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } \mathrm { d } x\) as accurately as you
   can, justifying the precision quoted.
3. Explain what the values in the ratio column tell you about the order of convergence of this sequence of approximations. Sarah carries out further work using the midpoint rule and Simpson’s rule. Her results are shown in Fig. 7.2. \begin{table}[h]

   	M	N	O	P	Q	R
   1	\(n\)	\(T _ { n }\)	\(M _ { n }\)	\(S _ { 2 n }\)	difference	ratio
   2	1	0.43634681	0.679792	0.5986436
   3	2	0.5580694	0.64592745	0.61664144	0.018
   4	4	0.60199843	0.63374304	0.6231615	0.00652	0.362269
   5	8	0.61787073	0.62928129	0.62547777	0.00232	0.355253
   6	16	0.62357601	0.62765831	0.62629755	0.00082	0.35392
   7	32	0.62561716	0.62707259

   \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{table}
4. Write down an efficient spreadsheet formula for calculating \(S _ { 16 }\).
5. Determine the missing values in row 7.
6. Use extrapolation to determine the value of \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } d x\) as accurately as you can, justifying
   the precision quoted.
   [0pt] [6]