7 Fig. 7 shows the graph of $y = \mathrm { f } ( x )$ for values of $x$ from 0 to 1 .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{662c2d48-228a-4b94-a4b4-cdd31634ef21-6_693_673_390_696}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}

The following spreadsheet printout shows estimates of $\int _ { 0 } ^ { 1 } f ( x ) d x$ found using the midpoint and trapezium rules for different values of $h$, the strip width.

\begin{center}
\begin{tabular}{ | l | r | r | l | }
\hline
A & \multicolumn{1}{|c|}{A} & \multicolumn{1}{c|}{B} & \multicolumn{1}{c|}{C} \\
\hline
1 & $h$ & \multicolumn{1}{c|}{Midpoint} & Trapezium \\
\hline
2 & 1 & 1.99851742 & 1.751283839 \\
\hline
3 & 0.5 & 1.9638591 & 1.874900631 \\
\hline
4 & 0.25 & 1.95135259 & 1.919379864 \\
\hline
5 & 0.125 & 1.94682102 & 1.935366229 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Without doing any further calculation, write down the smallest possible interval which contains the value of the integral. Justify your answer.
\item (A) - Calculate the ratio of differences, $r$, for the sequence of estimates calculated using the trapezium rule.

\begin{itemize}
  \item Hence suggest a value for $r$ correct to 2 significant figures.
  \item Comment on your suggested value for $r$.\\
(B) - Use extrapolation to find an improved approximation to the value of the integral.
  \item State the value of the integral to two decimal places.
  \item Explain why this precision is secure.
\end{itemize}

Using a similar approach with the sequence of estimates calculated using the midpoint rule, the approximation to the integral from extrapolation was found to be 1.94427 correct to 5 decimal places.
\item Andrea uses the extrapolated midpoint rule value and the value found in part (ii) ( $B$ ) to write down an interval which contains the value of the integral. Comment on the validity of Andrea's method.
\item Use the values from the spreadsheet output to calculate an approximation to the integral using Simpson's rule with $h = 0.125$. Give your answer to 5 decimal places.

Approximations to the integral using Simpson's rule are given in the spreadsheet output below. The number of applications of Simpson's rule is given in column N.

\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
N & O & P & Q \\
\hline
$n$ & Simpson & differences & ratio \\
\hline
1 & 1.91610623 & 0.01810005 & 0.3584931 \\
\hline
2 & 1.93420628 & 0.00648874 & 0.3556525 \\
\hline
4 & 1.94069502 & 0.00230774 & 0.3544828 \\
\hline
8 & 1.94300275 & 0.00081805 & 0.3539885 \\
\hline
16 & 1.94382081 & 0.00028958 & 0.3537638 \\
\hline
32 & 1.94411039 & 0.00010244 & 0.3536568 \\
\hline
64 & 1.94421283 & $3.623 \mathrm { E } - 05$ &  \\
\hline
128 & 1.94424906 &  &  \\
\hline
 &  &  &  \\
\hline
\end{tabular}
\end{center}
\item Use the spreadsheet output to find the value of the integral as accurately as possible. Justify the precision quoted.

\section*{END OF QUESTION PAPER}

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\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Numerical Methods  Q7 [19]}}