| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Moderate -0.8 This question requires understanding spreadsheet formulas to identify the function and limits, then applying basic midpoint rule theory about over/under-estimation using a provided graph. Part (a) is straightforward interpretation (f(x) = x^(1/x), limits 1 to 2, strip width 0.5), and part (b) requires recalling that midpoint rule over-estimates when the function is concave down. No complex calculations or novel insights neededβpurely routine application of standard numerical methods concepts. |
| Spec | 1.09f Trapezium rule: numerical integration |
| B | C | D | |
| 3 | \(x\) | \(\mathrm { f } ( x )\) | \(\mathrm { M } _ { 1 }\) |
| 4 | 1.5 | 1.3103707 | 0.65518535 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | seen |
| Answer | Marks |
|---|---|
| β«1.25 π₯π₯ dπ₯π₯ | B1 |
| Answer | Marks |
|---|---|
| B1 | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | not necessarily as an integral |
| Answer | Marks | Guidance |
|---|---|---|
| [3] | β« π₯π₯ dπ₯π₯ | |
| 4 | (b) | the curve is concave down |
| Answer | Marks | Guidance |
|---|---|---|
| over-estimate | B1 | 2.4 |
Question 4:
4 | (a) | seen
1
π₯π₯
π₯π₯
correct limits identified in integral
1
1.75
π₯π₯
β«1.25 π₯π₯ dπ₯π₯ | B1
B1
B1 | 1.1
1.1
1.1 | not necessarily as an integral
all correct
if B1B0B0 allow B1 SCB1 for , where a and b are
1
ππ
numerical values symmetrical abouπ₯π₯t 1.5
ππ
[3] | β« π₯π₯ dπ₯π₯
4 | (b) | the curve is concave down
so the midpoint rule will give an
over-estimate | B1 | 2.4 | or the curve is increasing and the gradient of the curve is
decreasing so the midpoint rule will give an over-estimate
[1]4 A spreadsheet is used to approximate $\int _ { a } ^ { b } f ( x ) d x$ using the midpoint rule with 1 strip. The output is shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
& B & C & D \\
\hline
3 & $x$ & $\mathrm { f } ( x )$ & $\mathrm { M } _ { 1 }$ \\
\hline
4 & 1.5 & 1.3103707 & 0.65518535 \\
\hline
\end{tabular}
\end{center}
The formula in cell C4 is $= \mathrm { B } 4 \wedge ( 1 / \mathrm { B } 4 )$.\\
The formula in cell D4 is $= 0.5 ^ { * } \mathrm { C } 4$.
\begin{enumerate}[label=(\alph*)]
\item Write the integral in standard mathematical notation.
A graph of $y = f ( x )$ is included in the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{4023e87c-34b1-4abd-9acc-ede5e4d68c7f-04_789_1004_1199_235}
\item Explain whether 0.65518535 is an over-estimate or an under-estimate of $\int _ { a } ^ { b } f ( x ) d x$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2023 Q4 [4]}}