Question 7 - A-Level Maths

OCR MEI Further Numerical Methods 2021 November — Question 7 15 marks

Exam Board	OCR MEI
Module	Further Numerical Methods (Further Numerical Methods)
Year	2021
Session	November
Marks	15
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Numerical integration
Type	Simpson's rule application
Difficulty	Standard +0.8 This Further Maths numerical methods question requires understanding trapezium rule mechanics (part a), error analysis and convergence theory (parts b-c), Simpson's rule formula derivation (part d), and Richardson extrapolation (part f). While individual parts are accessible, the combination of theoretical understanding of convergence orders, practical spreadsheet implementation, and multi-method comparison elevates this above standard A-level. The ratio analysis requiring knowledge that ~0.36 ≈ 1/4 indicates O(h²) convergence is non-trivial.
Spec	1.09f Trapezium rule: numerical integration

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]

\(n\)	\(T _ { n }\)	difference	ratio
1	0.43634681
2	0.5580694	0.121723
4	0.60199843	0.043929	0.36089
8	0.61787073	0.015872	0.36132
16	0.62357601	0.005705	0.35945
32	0.62561716	0.002041	0.35777

\captionsetup{labelformat=empty} \caption{Fig. 7.1}

\end{table}

Write down the value of \(h\) used to find the approximation 0.62357601 .
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.

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}

Write down an efficient spreadsheet formula for calculating \(S _ { 16 }\).
Determine the missing values in row 7.
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]

Show mark scheme Show mark scheme source

Question 7:

Answer	Marks	Guidance
7	(a)	isw or 0.0625 isw

1

Answer	Marks	Guidance
16	B1	2.2a

[1]

Answer	Marks	Guidance
7	(b)	by comparison of T and T

16 32

Answer	Marks	Guidance
0.6 is certain or 0.63 is probable	B1	2.2b

[1]

0.5 0.4764669

0.4528879

0.4293074

0.4057756

0.3823498

0.1116318

0.1111278

0.1110835

0.1110821

Answer	Marks	Guidance
7	(c)	r appears to be between 0.25 and 0.5

so order of convergence is between 1st and 2nd

Answer	Marks
order	B1
B1	2.2b

2.2b

Alternative

Answer	Marks
r > 0.25 so convergence slower than 2nd order	B1
r < 0.5 so convergence faster than 1st order	B1

[2]

Answer	Marks	Guidance
7	(d)	or soi

2𝑀𝑀𝑛𝑛+𝑇𝑇𝑛𝑛 4𝑇𝑇2𝑛𝑛−𝑇𝑇𝑛𝑛

3 3

Answer	Marks
= (2O5 + N5)/3 or =(4N6 ‒ N5)/3	M1
A1	1.1
1.1	must see =

[2]

Answer	Marks	Guidance
7	(e)	awrt 0.62658745

awrt 0.00029

Answer	Marks
awrt 0.354	B1

B1

Answer	Marks
B1	1.1

1.1 [3]

Answer	Marks	Guidance
7	(f)	S and difference from table used in

2n

extrapolation

awrt 0.62658745 and awrt 0.00029 used

𝑟𝑟

0 .62658745+ 0.00029 ×1−𝑟𝑟

awrt 0.62674355 to awrt 0.62675058

comparison with their S

64

Answer	Marks
0.6267 is secure	M1

A1

M1

Answer	Marks
A1	3.1a

1.1 3.2a

Answer	Marks
2.2b	eg their 0.62658745 and their 0.00029

may see more dp for difference

0.35 ≤ r ≤ 0.36

or 0.62675 is possible; allow 0.626746

the last two A marks are only available

if answers obtained from extrapolation

to infinity from S

Answer	Marks
64	If M0 allow SC2 for

awrt 0.626607 obtained

from

16×0.62658745−0.62629755

15 then SC1 for 0.627

obtained from comparison

with S

64 [6]

PMT

OCR (Oxford Cambridge and RSA Examinations)

The Triangle Building

Shaftesbury Road

Cambridge

CB2 8EA

OCR Customer Contact Centre

Education and Learning

Telephone: 01223 553998

Facsimile: 01223 552627

Email: general.qualifications@ocr.org.uk

www.ocr.org.uk

For staff training purposes and as part of our quality assurance programme your call may be

recorded or monitored

Question 7:
7 | (a) | isw or 0.0625 isw
1
16 | B1 | 2.2a
[1]
7 | (b) | by comparison of T and T
16 32
0.6 is certain or 0.63 is probable | B1 | 2.2b
[1]
0.5
0.4764669
0.4528879
0.4293074
0.4057756
0.3823498
0.1116318
0.1111278
0.1110835
0.1110821
0.1110821
7 | (c) | r appears to be between 0.25 and 0.5
so order of convergence is between 1st and 2nd
order | B1
B1 | 2.2b
2.2b
Alternative
r > 0.25 so convergence slower than 2nd order | B1
r < 0.5 so convergence faster than 1st order | B1
[2]
7 | (d) | or soi
2𝑀𝑀𝑛𝑛+𝑇𝑇𝑛𝑛 4𝑇𝑇2𝑛𝑛−𝑇𝑇𝑛𝑛
3 3
= (2*O5 + N5)/3 or =(4*N6 ‒ N5)/3 | M1
A1 | 1.1
1.1 | must see =
[2]
7 | (e) | awrt 0.62658745
awrt 0.00029
awrt 0.354 | B1
B1
B1 | 1.1
1.1
1.1
[3]
7 | (f) | S and difference from table used in
2n
extrapolation
awrt 0.62658745 and awrt 0.00029 used
𝑟𝑟
0 .62658745+ 0.00029 ×1−𝑟𝑟
awrt 0.62674355 to awrt 0.62675058
comparison with their S
64
0.6267 is secure | M1
A1
A1
A1
M1
A1 | 3.1a
1.1
1.1
1.1
3.2a
2.2b | eg their 0.62658745 and their 0.00029
may see more dp for difference
0.35 ≤ r ≤ 0.36
or 0.62675 is possible; allow 0.626746
the last two A marks are only available
if answers obtained from extrapolation
to infinity from S
64 | If M0 allow SC2 for
awrt 0.626607 obtained
from
16×0.62658745−0.62629755
15
then SC1 for 0.627
obtained from comparison
with S
64
[6]
PMT
OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
Cambridge
CB2 8EA
OCR Customer Contact Centre
Education and Learning
Telephone: 01223 553998
Facsimile: 01223 552627
Email: general.qualifications@ocr.org.uk
www.ocr.org.uk
For staff training purposes and as part of our quality assurance programme your call may be
recorded or monitored

Show LaTeX source

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]
\begin{center}
\begin{tabular}{ | l | l | l | c | }
\hline
$n$ & \multicolumn{1}{|c|}{$T _ { n }$} & difference & ratio \\
\hline
1 & 0.43634681 &  &  \\
\hline
2 & 0.5580694 & 0.121723 &  \\
\hline
4 & 0.60199843 & 0.043929 & 0.36089 \\
\hline
8 & 0.61787073 & 0.015872 & 0.36132 \\
\hline
16 & 0.62357601 & 0.005705 & 0.35945 \\
\hline
32 & 0.62561716 & 0.002041 & 0.35777 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 7.1}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $h$ used to find the approximation 0.62357601 .
\item 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.
\item 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]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
 & M & N & O & P & Q & R \\
\hline
1 & $n$ & $T _ { n }$ & $M _ { n }$ & $S _ { 2 n }$ & difference & ratio \\
\hline
2 & 1 & 0.43634681 & 0.679792 & 0.5986436 &  &  \\
\hline
3 & 2 & 0.5580694 & 0.64592745 & 0.61664144 & 0.018 &  \\
\hline
4 & 4 & 0.60199843 & 0.63374304 & 0.6231615 & 0.00652 & 0.362269 \\
\hline
5 & 8 & 0.61787073 & 0.62928129 & 0.62547777 & 0.00232 & 0.355253 \\
\hline
6 & 16 & 0.62357601 & 0.62765831 & 0.62629755 & 0.00082 & 0.35392 \\
\hline
7 & 32 & 0.62561716 & 0.62707259 &  &  &  \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 7.2}
\end{center}
\end{table}
\item Write down an efficient spreadsheet formula for calculating $S _ { 16 }$.
\item Determine the missing values in row 7.
\item 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]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2021 Q7 [15]}}

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