| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Rectangle bounds for definite integral |
| Difficulty | Standard +0.3 This is a straightforward application of rectangular bounds for integration. Part (i) is routine calculus with logarithmic differentiation. Parts (ii)-(iv) involve standard lower/upper bound techniques using rectangles, requiring only careful arithmetic and understanding of the basic principle that rectangles below the curve give lower bounds. The function x^x adds mild interest but the actual work is mechanical calculation and diagram sketching—slightly easier than average A-level. |
| Spec | 1.06b Gradient of e^(kx): derivative and exponential model1.07g Differentiation from first principles: for small positive integer powers of x4.08g Derivatives: inverse trig and hyperbolic functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y=x^x \Rightarrow \ln y = x\ln x \Rightarrow \frac{1}{y}\frac{dy}{dx}=1+\ln x\) | M1 | For differentiating \(\ln y\) OR \(x\ln x\) w.r.t. \(x\) |
| \(\frac{dy}{dx}=x^x(1+\ln x)=0 \Rightarrow \ln x=-1 \Rightarrow x=e^{-1}\) | A1 | For \((1+\ln x)\) seen or implied |
| A1 3 | For correct \(x\)-value from fully correct working AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(A > 0.2\times0.5^{0.5}+0.2\times0.7^{0.7}+0.1\times0.9^{0.9}\) | M1 | For areas of 3 lower rectangles |
| \(\Rightarrow A > 0.3881(858) > 0.388\) | A1 2 | For lower bound rounding to AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(A < 0.2\times0.7^{0.7}+0.2\times0.9^{0.9}+0.1\times1^1\) | M1 | For areas of 3 upper rectangles |
| \(\Rightarrow A < 0.4377(177) < 0.438\) | A1 2 | For upper bound rounding to 0.438 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| [Sketch with lower rectangles] | M1 | Consider rectangle of height \(f(e^{-1})\) |
| A1 | Use at least 1 lower rectangle, height \(f(e^{-1})\) | |
| [Sketch with upper rectangles] | B1 3 | Use at least 1 upper rectangle, height \(f(0)\) |
| SR If more than one rectangle is used for either bound, they must be shown correctly |
## Question 6(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=x^x \Rightarrow \ln y = x\ln x \Rightarrow \frac{1}{y}\frac{dy}{dx}=1+\ln x$ | M1 | For differentiating $\ln y$ OR $x\ln x$ w.r.t. $x$ |
| $\frac{dy}{dx}=x^x(1+\ln x)=0 \Rightarrow \ln x=-1 \Rightarrow x=e^{-1}$ | A1 | For $(1+\ln x)$ seen or implied |
| | A1 **3** | For correct $x$-value from fully correct working **AG** |
## Question 6(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $A > 0.2\times0.5^{0.5}+0.2\times0.7^{0.7}+0.1\times0.9^{0.9}$ | M1 | For areas of 3 lower rectangles |
| $\Rightarrow A > 0.3881(858) > 0.388$ | A1 **2** | For lower bound rounding to **AG** |
## Question 6(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $A < 0.2\times0.7^{0.7}+0.2\times0.9^{0.9}+0.1\times1^1$ | M1 | For areas of 3 upper rectangles |
| $\Rightarrow A < 0.4377(177) < 0.438$ | A1 **2** | For upper bound rounding to 0.438 |
## Question 6(iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| [Sketch with lower rectangles] | M1 | Consider rectangle of height $f(e^{-1})$ |
| | A1 | Use at least 1 lower rectangle, height $f(e^{-1})$ |
| [Sketch with upper rectangles] | B1 **3** | Use at least 1 upper rectangle, height $f(0)$ |
| **SR** If more than one rectangle is used for either bound, they must be shown correctly | | |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{debf6581-25ff-4692-bdfb-154675a3cdb0-3_608_1134_258_504}
The diagram shows the curve $y = \mathrm { f } ( x )$, defined by
$$f ( x ) = \begin{cases} x ^ { x } & \text { for } 0 < x \leqslant 1 , \\ 1 & \text { for } x = 0 . \end{cases}$$
(i) By first taking logarithms, show that the curve has a stationary point at $x = \mathrm { e } ^ { - 1 }$.
The area under the curve from $x = 0.5$ to $x = 1$ is denoted by $A$.\\
(ii) By considering the set of three rectangles shown in the diagram, show that a lower bound for $A$ is 0.388 .\\
(iii) By considering another set of three rectangles, find an upper bound for $A$, giving 3 decimal places in your answer.
The area under the curve from $x = 0$ to $x = 0.5$ is denoted by $B$.\\
(iv) Draw a diagram to show rectangles which could be used to find lower and upper bounds for $B$, using not more than three rectangles for each bound. (You are not required to find the bounds.)
\hfill \mbox{\textit{OCR FP2 2011 Q6 [10]}}