| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Standard +0.3 This is a straightforward application of standard numerical differentiation formulas (forward and central difference) with given tabulated values. Parts (a) and (b) require simple substitution into formulas, part (c) is a geometric interpretation, and part (d) tests basic understanding that central difference is generally more accurate. This is routine bookwork for Further Maths numerical methods with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 1.5 | 2 | 2.5 |
| \(y\) | 1.682137 | 2.094395 | 2.318559 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (a) | oe soi |
| Answer | Marks |
|---|---|
| 2.5−2 | M1 |
| A1 | 1.1 |
| 1.1 | allow B2 for correct answer unsupported |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (b) | oe soi |
| Answer | Marks |
|---|---|
| 2.5−1.5 | M1 |
| A1 | 1.1 |
| 1.1 | allow B2 for correct answer unsupported |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (c) | chord joining point on curve where x = 1.5 to |
| Answer | Marks | Guidance |
|---|---|---|
| approximation to the gradient of the tangent] | B1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (d) | central difference method is (usually) a |
| Answer | Marks | Guidance |
|---|---|---|
| x-direction only oe | B1 | 2.2b |
Question 3:
3 | (a) | oe soi
2.318559−2.094395
0.448328 isw or 0.44833 isw
2.5−2 | M1
A1 | 1.1
1.1 | allow B2 for correct answer unsupported
[2]
3 | (b) | oe soi
2.318559−1.682137
0.636422 isw or 0.63642 isw
2.5−1.5 | M1
A1 | 1.1
1.1 | allow B2 for correct answer unsupported
[2]
3 | (c) | chord joining point on curve where x = 1.5 to
point on curve where x = 2.5
and tangent to the curve at
x = 2 drawn;
[and gradient of chord identified as an
approximation to the gradient of the tangent] | B1 | 1.1
[1]
3 | (d) | central difference method is (usually) a
second order method whereas forward
difference is (usually) a first order method,
so answer to part (b) probably more accurate
or central difference method uses x-values
on both sides of 2 oe whereas forward
difference method uses a step in the positive
x-direction only oe | B1 | 2.2b | must mention both methods
must mention both methods
[1]3 The diagram shows the graph of $y = f ( x )$ for values of $x$ from 1 to 3.5.\\
\includegraphics[max width=\textwidth, alt={}, center]{4023e87c-34b1-4abd-9acc-ede5e4d68c7f-03_945_1248_312_244}
The table shows some values of $x$ and the associated values of $y$.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$x$ & 1.5 & 2 & 2.5 \\
\hline
$y$ & 1.682137 & 2.094395 & 2.318559 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use the forward difference method to calculate an approximation to $\frac { \mathrm { dy } } { \mathrm { dx } }$ at $x = 2$.
\item Use the central difference method to calculate an approximation to $\frac { \mathrm { dy } } { \mathrm { dx } }$ at $x = 2$.
\item On the copy of the diagram in the Printed Answer Booklet, show how the central difference method gives the approximation to $\frac { \mathrm { dy } } { \mathrm { dx } }$ at $x = 2$ which was found in part (b).
\item Explain whether your answer to part (a) or your answer to part (b) is likely to give a better approximation to $\frac { \mathrm { dy } } { \mathrm { dx } }$ at $x = 2$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2023 Q3 [6]}}