| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Bounds using rectangles |
| Difficulty | Standard +0.8 This FP2 question on numerical integration requires understanding Riemann sums with a non-standard function (e^(-1/x)) and involves multiple conceptual steps: explaining the lower bound structure, constructing an upper bound, numerical evaluation, and finding N by analyzing the difference between bounds. The final part requires algebraic manipulation of exponential series and solving an inequality, which is more demanding than routine integration questions but remains within standard FP2 scope. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Given expression is sum of areas of rectangles of width \(\frac{1}{n}\), heights \(e^{-1/x}\) | B1 | For identifying rectangle widths and heights |
| Given integral is area under the curve which is clearly greater | B1 | For correct explanation of lower bound |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Upper bound \(= \frac{1}{n}\left(e^{-n} + e^{-\frac{n}{2}} + e^{-\frac{n}{3}} + \ldots + e^{-\frac{n}{n-1}} + e^{-1}\right)\) | M1 A1 | For using n upper rectangles soi by \(e^{-n}\) and \(e^{-1}\); For correct expression |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Lower bound = 0.104(31) | B1 | For correct value |
| Upper bound = 0.196(28) | B1 | For correct value – accept 0.197 |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{n}e^{-1} < 0.001\) | B1 | For a correct statement (includes <) |
| \(\Rightarrow n > \frac{1000}{e} = 367.879\) | M1 | For rearranging (ignore < > = and allow RHS = \(10^{\ell/m} e^{\pm 1}\)) |
| \(\Rightarrow\) least N = 368 | A1 | For correct value |
| [3] |
### (i)
Given expression is sum of areas of rectangles of width $\frac{1}{n}$, heights $e^{-1/x}$ | B1 | For identifying rectangle widths and heights
Given integral is area under the curve which is clearly greater | B1 | For correct explanation of lower bound
| [2] |
### (ii)
Upper bound $= \frac{1}{n}\left(e^{-n} + e^{-\frac{n}{2}} + e^{-\frac{n}{3}} + \ldots + e^{-\frac{n}{n-1}} + e^{-1}\right)$ | M1 A1 | For using n upper rectangles soi by $e^{-n}$ and $e^{-1}$; For correct expression
| [2] |
### (iii)
Lower bound = 0.104(31) | B1 | For correct value
Upper bound = 0.196(28) | B1 | For correct value – accept 0.197
| [2] |
### (iv)
$\frac{1}{n}e^{-1} < 0.001$ | B1 | For a correct statement (includes <)
$\Rightarrow n > \frac{1000}{e} = 367.879$ | M1 | For rearranging (ignore < > = and allow RHS = $10^{\ell/m} e^{\pm 1}$)
$\Rightarrow$ least N = 368 | A1 | For correct value
| [3] |
---\includegraphics{figure_4}
The diagram shows the curve $y = e^{-\frac{1}{x}}$ for $0 < x \leq 1$. A set of $(n - 1)$ rectangles is drawn under the curve as shown.
\begin{enumerate}[label=(\roman*)]
\item Explain why a lower bound for $\int_0^1 e^{-\frac{1}{x}} dx$ can be expressed as
$$\frac{1}{n}\left(e^{-n} + e^{-\frac{n}{2}} + e^{-\frac{n}{3}} + \ldots + e^{-\frac{n}{n-1}}\right).$$ [2]
\item Using a set of $n$ rectangles, write down a similar expression for an upper bound for $\int_0^1 e^{-\frac{1}{x}} dx$. [2]
\item Evaluate these bounds in the case $n = 4$, giving your answers correct to 3 significant figures. [2]
\item When $n > N$, the difference between the upper and lower bounds is less than 0.001. By expressing this difference in terms of $n$, find the least possible value of $N$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2012 Q4 [9]}}