| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Standard +0.8 This Further Maths numerical methods question requires applying central difference formulas with multiple step sizes, then making a judgment about accuracy by comparing estimates—going beyond routine calculation to require analytical thinking about numerical error and convergence, though the technique itself is standard once learned. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.09g Numerical methods in context |
| \(x\) | 0.2 | 0.3 | 0.35 | 0.4 | 0.45 | 0.5 | 0.6 |
| \(\mathrm { f } ( x )\) | 0.789922 | 0.754628 | 0.749199 | 0.749997 | 0.756257 | 0.767523 | 0.804299 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (i) | Evidence of correct use of formula |
| Answer | Marks |
|---|---|
| dx 0.03594.. 0.06447.. 0.07058 | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | M |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (ii) | E |
| Answer | Marks |
|---|---|
| S | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 2.4 | E.g. The difference between |
| Answer | Marks | Guidance |
|---|---|---|
| h | 0.2 | 0.1 |
| Answer | Marks | Guidance |
|---|---|---|
| dx | 0.03594.. | 0.06447.. |
Question 4:
4 | (i) | Evidence of correct use of formula
h 0.2 0.1 0.05
dy
dx 0.03594.. 0.06447.. 0.07058 | M1
CA1
A1
A1
[4] | 1.1
I
1.1
1.1
1.1 | M
Answers to 3 or more significant
figures
4 | (ii) | E
0.07 is reasonable
P
Accept any correct answer based on differences
S | B1
E1
[2] | 1.1
2.4 | E.g. The difference between
successive approximations has
reduced by roughly one fifth, so
further reduction in h is unlikely
to generate an answer larger than
0.075
h | 0.2 | 0.1 | 0.05
dy
dx | 0.03594.. | 0.06447.. | 0.070584 The table below gives values of a function $y = \mathrm { f } ( x )$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0.2 & 0.3 & 0.35 & 0.4 & 0.45 & 0.5 & 0.6 \\
\hline
$\mathrm { f } ( x )$ & 0.789922 & 0.754628 & 0.749199 & 0.749997 & 0.756257 & 0.767523 & 0.804299 \\
\hline
\end{tabular}
\end{center}
(i) Calculate three estimates of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at $x = 0.4$ using the central difference method.\\
(ii) State the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at $x = 0.4$ to an appropriate degree of accuracy. Justify your answer.
\hfill \mbox{\textit{OCR MEI Further Numerical Methods Q4 [6]}}