| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Limit of sum as integral |
| Difficulty | Moderate -0.5 This question tests understanding of spreadsheet formulas, sigma notation, and numerical error concepts, but requires only straightforward interpretation and basic calculations. Part (a) is routine notation translation, (b) requires recognizing numerical precision limits, (c) is a standard relative error calculation, and (d) asks for conceptual explanation of chopping error—all accessible with no novel problem-solving required. |
| Spec | 1.09f Trapezium rule: numerical integration |
| A | B | C | ||
| 1 | \(x\) | \(\mathrm { f } ( x )\) | ||
| 2 | 1 | 0.367879441 | 0.367879441 | |
| 3 | 2 | 0.018315639 | 0.38619508 | |
| 4 | 3 | 0.00012341 | 0.38631849 | |
| 5 | 4 | \(1.12535 \mathrm { E } - 07\) | 0.386318602 | |
| 6 | 5 | \(1.38879 \mathrm { E } - 11\) | 0.386318602 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | e i2 seen oe |
| Answer | Marks |
|---|---|
| i1 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1 | |
| 1.1 | must see i = oe | |
| 1 | (b) | e‒5² is (very) small |
| Answer | Marks |
|---|---|
| sum | B1 |
| Answer | Marks |
|---|---|
| [2] | 2.2a |
| 2.4 | eg [the numbers are different oe] but |
| Answer | Marks |
|---|---|
| and the spreadsheet displays to 9 dp | condone eg B6 is (very) |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (c) | ‒ 0.00239 to ‒ 0.0024 isw |
| [1] | 1.1 | NB ‒ 0.002390568491 |
| 1 | (d) | [the approximations are all zero], so the relative |
| Answer | Marks | Guidance |
|---|---|---|
| case oe | B1 | |
| [1] | 2.4 | allow the relative errors are all ±1 |
Question 1:
1 | (a) | e i2 seen oe
4
e i2 oe
i1 | M1
A1
[2] | 1.1
1.1 | must see i = oe
1 | (b) | e‒5² is (very) small
and doesn’t affect the 9th decimal place in the
sum | B1
B1
[2] | 2.2a
2.4 | eg [the numbers are different oe] but
both round to the same value at 9dp
and the spreadsheet displays to 9 dp | condone eg B6 is (very)
small
1 | (c) | ‒ 0.00239 to ‒ 0.0024 isw | B1
[1] | 1.1 | NB ‒ 0.002390568491
1 | (d) | [the approximations are all zero], so the relative
error is the number divided by itself in each
case oe | B1
[1] | 2.4 | allow the relative errors are all ±1
1
1.5
1.3143166
1.2295168
1.2115408
1.210783
1.2107817
1.21078171 Fig. 1 shows some spreadsheet output concerning the values of a function, $\mathrm { f } ( x )$.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | l | l | l }
\hline
& A & \multicolumn{1}{|c|}{B} & \multicolumn{1}{|c|}{C} & \\
\hline
1 & $x$ & \multicolumn{1}{|c|}{$\mathrm { f } ( x )$} & & \\
\hline
2 & 1 & 0.367879441 & 0.367879441 & \\
\hline
3 & 2 & 0.018315639 & 0.38619508 & \\
\hline
4 & 3 & 0.00012341 & 0.38631849 & \\
\hline
5 & 4 & $1.12535 \mathrm { E } - 07$ & 0.386318602 & \\
\hline
6 & 5 & $1.38879 \mathrm { E } - 11$ & 0.386318602 & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{table}
The formula in cell B2 is ==EXP(-(A2\^{}2)) and equivalent formulae are in cells B3 to B6.
The formula in cell C 2 is $= \mathrm { B } 2$.\\
The formula in cell C3 is $\quad = \mathrm { C } 2 + \mathrm { B } 3$.
Equivalent formulae are in cells C4 to C6.
\begin{enumerate}[label=(\alph*)]
\item Use sigma notation to express the formula in cell C5 in standard mathematical notation.
\item Explain why the same value is displayed in cells C 5 and C 6.
Now suppose that the value in cell C2 is chopped to 3 decimal places and used to approximate the value in cell C2.
\item Calculate the relative error when this approximation is used.
Suppose that the values in cells B4, B5 and B6 are chopped to 3 decimal places and used as approximations to the original values in cells B4, B5 and B6 respectively.
\item Explain why the relative errors in these approximations are all the same.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2019 Q1 [6]}}