| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.5 This question tests standard numerical differentiation formulas (forward and central difference) and error estimation using linear approximation. All parts involve direct application of memorized formulas with straightforward arithmetic. The conceptual explanation in part (c) is standard textbook knowledge about central differences having better accuracy. While it's a Further Maths topic, the execution requires no problem-solving or novel insight—just formula recall and calculation. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 2.75 | 3 | 3.25 |
| \(\mathrm { f } ( x )\) | 0.920799 | 1 | 1.072858 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | 1.072858−1 |
| Answer | Marks |
|---|---|
| 0.29143 cao | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1a |
| 1.1 | 0.072858 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (b) | 1.072858−0.920799 |
| Answer | Marks |
|---|---|
| 0.30412 cao | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1a |
| 1.1 | 0.152059 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (c) | central difference method is a 2nd order |
| Answer | Marks | Guidance |
|---|---|---|
| direction. | E1 | |
| [1] | 2.4 | must refer to both methods |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (d) | 0.024 × 0.182 soi |
| ±0.004368 or ±0.00437 or ± 0.0044 | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1a |
| 1.1 | may be embedded |
Question 2:
2 | (a) | 1.072858−1
oe
3.25−3
0.29143 cao | M1
A1
[2] | 1.1a
1.1 | 0.072858
NB implies M1
0.25
2 | (b) | 1.072858−0.920799
oe
3.25−2.75
0.30412 cao | M1
A1
[2] | 1.1a
1.1 | 0.152059
NB implies M1
0.5
2 | (c) | central difference method is a 2nd order
method whereas forward difference method
is 1st order oe
or central difference method uses values
either side of 3 oe whereas forward
difference method steps in the positive x-
direction. | E1
[1] | 2.4 | must refer to both methods
must refer to both methods
2 | (d) | 0.024 × 0.182 soi
±0.004368 or ±0.00437 or ± 0.0044 | M1
A1
[2] | 3.1a
1.1 | may be embedded
mark the final answer2 The table shows some values of $x$ and the associated values of $y = f ( x )$.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$x$ & 2.75 & 3 & 3.25 \\
\hline
$\mathrm { f } ( x )$ & 0.920799 & 1 & 1.072858 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate an estimate of $\frac { \mathrm { dy } } { \mathrm { dx } }$ at $x = 3$ using the forward difference method, giving your answer correct to $\mathbf { 5 }$ decimal places.
\item Calculate an estimate of $\frac { \mathrm { dy } } { \mathrm { dx } }$ at $x = 3$ using the central difference method, giving your answer correct to $\mathbf { 5 }$ decimal places.
\item Explain why your answer to part (b) is likely to be closer than your answer to part (a) to the true value of $\frac { \mathrm { dy } } { \mathrm { dx } }$ at $x = 3$.
When $x = 5$ it is given that $y = 1.4645$ and $\frac { \mathrm { dy } } { \mathrm { dx } } = 0.1820$, correct to 4 decimal places.
\item Determine an estimate of the error when $\mathrm { f } ( 5 )$ is used to estimate $\mathrm { f } ( 5.024 )$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2022 Q2 [7]}}