| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2021 |
| Session | November |
| 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 numerical differentiation question requiring calculation of forward differences using given data points, followed by observing convergence. While it requires understanding of numerical methods and precision, the calculations are routine and the pattern recognition in part (b) is accessible to students who have practiced this topic. |
| Spec | 1.07a Derivative as gradient: of tangent to curve |
| \(x\) | 4 | 4.0001 | 4.001 | 4.01 | 4.1 |
| \(\mathrm { f } ( x )\) | 4 | 4.0002386 | 4.0023871 | 4.0239468 | 4.2472072 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | or or |
| Answer | Marks |
|---|---|
| 2.386 (with h = 0.0001) | M1 |
| Answer | Marks |
|---|---|
| A1 | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | use of forward difference method |
| Answer | Marks |
|---|---|
| all four correct | may be implied by one |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (b) | comparison of last two estimates |
| 2.39 is secure or 2.386 is possible | M1 | |
| A1 | 1.1 | |
| 2.2b | if M0 allow SC1 for 2.39 is secure or |
Question 4:
4 | (a) | or or
4.2472072‒4 4.0239468‒4
0.1 or 0.01
4.0023871‒4 4.0002386‒4
2.470.2000172 (0w.0i0t0h1 h = 0.1)
2.39468 (with h = 0.01)
2.3871 (with h = 0.001)
2.386 (with h = 0.0001) | M1
A1
A1
A1 | 3.1a
1.1
1.1
1.1 | use of forward difference method
any two correct
any three correct
all four correct | may be implied by one
correct answer
[4]
4 | (b) | comparison of last two estimates
2.39 is secure or 2.386 is possible | M1
A1 | 1.1
2.2b | if M0 allow SC1 for 2.39 is secure or
2.386 is possible regardless of
justification
[2]4 The table shows some values of $x$ and the associated values of $\mathrm { f } ( x )$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 4 & 4.0001 & 4.001 & 4.01 & 4.1 \\
\hline
$\mathrm { f } ( x )$ & 4 & 4.0002386 & 4.0023871 & 4.0239468 & 4.2472072 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate four estimates of the derivative of $\mathrm { f } ( x )$ at $x = 4$.
\item Without doing any further calculation, state the value of $f ^ { \prime } ( 4 )$ as accurately as you can, justifying the precision quoted.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2021 Q4 [6]}}