OCR MEI Further Numerical Methods 2020 November — Question 4 10 marks

Exam BoardOCR MEI
ModuleFurther Numerical Methods (Further Numerical Methods)
Year2020
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicNumerical integration
TypeSimpson's rule application
DifficultyChallenging +1.2 This is a structured numerical methods question requiring trapezium rule calculation, understanding of Simpson's rule relationships, and Richardson extrapolation. While it involves multiple parts and the final extrapolation technique, each step is methodical and follows standard Further Maths numerical methods procedures without requiring novel insight. The spreadsheet context and systematic progression make it more accessible than typical proof-based Further Maths questions.
Spec1.09f Trapezium rule: numerical integration
4
  1. Use the trapezium rule with 1 strip to calculate an estimate of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your
    answer correct to six decimal places.
    [0pt] [2]
    Fig. 4 shows some spreadsheet output containing further approximations to this integral using the trapezium rule, denoted by \(T _ { n }\), and Simpson's rule, denoted by \(S _ { 2 n }\). \begin{table}[h]
    ABC
    1\(n\)\(T _ { n }\)\(S _ { 2 n }\)
    212.130135
    322.149378
    442.1347512.129862
    582.131084
    \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{table}
  2. Write down an efficient formula for cell C 4.
  3. Find the value of \(S _ { 4 }\), giving your answer correct to 6 decimal places.
  4. Without doing any further calculation, state the value of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\) as accurately as
    possible, justifying the precision quoted.
    [0pt] [2]
  5. Use the fact that Simpson's rule is a fourth order method to obtain an improved approximation to the value of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), stating the value of this integral to a precision which seems justified.
Question 4:
AnswerMarks Guidance
4(a) oe soi
1
2.๏ฟฝ2โˆš0271+037๏ฟฝ cao
AnswerMarks
2M1
A11.1
1.1
[2]
AnswerMarks Guidance
4(b) = (4*B5 โ€’ B4)/3
A11.1a
1.1use of
4๐‘‡๐‘‡2๐‘›๐‘› โ€’ ๐‘‡๐‘‡๐‘›๐‘›
all correct
3
[2]
AnswerMarks Guidance
4(c) soi
4ร—2.134751 โ€’2.149378
2.129875 cao
AnswerMarks
3M1
A11.1
1.1or use of composite rule with h =
0.25
[2]
AnswerMarks Guidance
4(d) comparison of S and S
4 8
AnswerMarks
2.1299 is secure or 2.12986 is probableE1
B13.1a
2.2b
[2]
AnswerMarks Guidance
4(e) oe
16 ร—2.129862 โ€’๐‘ก๐‘กโ„Ž๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’ 2.129875
15
AnswerMarks
2.12986 is certain or 2.129861 is possibleM1
A13.1a
2.2ballow M1 for
2.129862 โ€’๐‘ก๐‘กโ„Ž๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’ 2.129875
e g 2.129862+ 16
[2]
AnswerMarks
41.49682562605 
Question 4:
4 | (a) | oe soi
1
2.๏ฟฝ2โˆš0271+037๏ฟฝ cao
2 | M1
A1 | 1.1
1.1
[2]
4 | (b) | = (4*B5 โ€’ B4)/3 | M1
A1 | 1.1a
1.1 | use of
4๐‘‡๐‘‡2๐‘›๐‘› โ€’ ๐‘‡๐‘‡๐‘›๐‘›
all correct
3
[2]
4 | (c) | soi
4ร—2.134751 โ€’2.149378
2.129875 cao
3 | M1
A1 | 1.1
1.1 | or use of composite rule with h =
0.25
[2]
4 | (d) | comparison of S and S
4 8
2.1299 is secure or 2.12986 is probable | E1
B1 | 3.1a
2.2b
[2]
4 | (e) | oe
16 ร—2.129862 โ€’๐‘ก๐‘กโ„Ž๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’ 2.129875
15
2.12986 is certain or 2.129861 is possible | M1
A1 | 3.1a
2.2b | allow M1 for
2.129862 โ€’๐‘ก๐‘กโ„Ž๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’ 2.129875
e g 2.129862+ 16
[2]
4 | 1.49682562605
4
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with 1 strip to calculate an estimate of $\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x$, giving your\\
answer correct to six decimal places.\\[0pt]
[2]\\
Fig. 4 shows some spreadsheet output containing further approximations to this integral using the trapezium rule, denoted by $T _ { n }$, and Simpson's rule, denoted by $S _ { 2 n }$.

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
 & A & B & C &  \\
\hline
1 & $n$ & $T _ { n }$ & $S _ { 2 n }$ &  \\
\hline
2 & 1 &  & 2.130135 &  \\
\hline
3 & 2 & 2.149378 &  &  \\
\hline
4 & 4 & 2.134751 & 2.129862 &  \\
\hline
5 & 8 & 2.131084 &  &  \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{table}
\item Write down an efficient formula for cell C 4.
\item Find the value of $S _ { 4 }$, giving your answer correct to 6 decimal places.
\item Without doing any further calculation, state the value of $\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x$ as accurately as\\
possible, justifying the precision quoted.\\[0pt]
[2]
\item Use the fact that Simpson's rule is a fourth order method to obtain an improved approximation to the value of $\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x$, stating the value of this integral to a precision which seems justified.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2020 Q4 [10]}}
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