| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson error analysis |
| Difficulty | Standard +0.8 This is a multi-part FP2 Newton-Raphson question requiring derivation of the iterative formula (routine), sketching with geometric insight about divergence conditions, numerical computation, and verification of a quadratic convergence relationship. The error analysis in part (iv) requires understanding of convergence theory beyond standard application, making it moderately challenging for Further Maths but still within expected FP2 scope. |
| Spec | 1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| \(x_{n+1} = x_n - \frac{x_n^3 - k}{3x_n^2}\) | M1 | For correct \(\frac{f(x)}{f'(x)}\) seen (x or \(x_n\)) |
| \(\Rightarrow x_{n+1} = \frac{2x_n^3+k}{3x_n^2}\) | A1 | For simplification to AG (\(x_n\) and \(x_{n+1}\) required) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | For correct curve with \(\alpha\) (\(\sqrt[3]{k}\)) and \(-k\) marked | |
| M1 | For a suitable tangent shown | |
| A1 | with \(x_1\) and \(x_2\) marked such that \( | \alpha - x_2 |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha = \sqrt[3]{100}\) | B1 | For correct \(\alpha\) (Condone \(x = ...\)) |
| \(x_2 = 4.66667\) | B1 | For correct \(x_2\) (to at least 5dp) |
| \(x_3 = 4.64172\) | B1 | For correct \(x_3\) (to at least 5dp) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | For calculating \(e_1, e_2, e_3\) from \(\alpha\) or something better than \(x_3\); All correct 5 dp | |
| \(e_1 = -0.35841, \quad e_2 = -0.02508, \quad e_3 = -0.00013\) | A1 | |
| \(\frac{e_2^3}{e_1^2} = -0.00012\) | A1 | For obtaining \(-0.00012\); SC2 for consistently without –ve signs |
| [3] |
### (i)
$x_{n+1} = x_n - \frac{x_n^3 - k}{3x_n^2}$ | M1 | For correct $\frac{f(x)}{f'(x)}$ seen (x or $x_n$)
$\Rightarrow x_{n+1} = \frac{2x_n^3+k}{3x_n^2}$ | A1 | For simplification to AG ($x_n$ and $x_{n+1}$ required)
| [2] |
### (ii)
| B1 | For correct curve with $\alpha$ ($\sqrt[3]{k}$) and $-k$ marked
| M1 | For a suitable tangent shown
| A1 | with $x_1$ and $x_2$ marked such that $|\alpha - x_2| > |\alpha - x_1|$
| [3] |
Curve looks like cubic with one pt of inflection (g not nec. 0) at y axis
### (iii)
$\alpha = \sqrt[3]{100}$ | B1 | For correct $\alpha$ (Condone $x = ...$)
$x_2 = 4.66667$ | B1 | For correct $x_2$ (to at least 5dp)
$x_3 = 4.64172$ | B1 | For correct $x_3$ (to at least 5dp)
| [3] |
### (iv)
| M1 | For calculating $e_1, e_2, e_3$ from $\alpha$ or something better than $x_3$; All correct 5 dp
$e_1 = -0.35841, \quad e_2 = -0.02508, \quad e_3 = -0.00013$ | A1 |
$\frac{e_2^3}{e_1^2} = -0.00012$ | A1 | For obtaining $-0.00012$; SC2 for consistently without –ve signs
| [3] |
---It is given that $f(x) = x^3 - k$, where $k > 0$, and that $\alpha$ is the real root of the equation $f(x) = 0$. Successive approximations to $\alpha$, using the Newton-Raphson method, are denoted by $x_1, x_2, \ldots, x_n, \ldots$.
\begin{enumerate}[label=(\roman*)]
\item Show that $x_{n+1} = \frac{2x_n^3 + k}{3x_n^2}$. [2]
\item Sketch the graph of $y = f(x)$, giving the coordinates of the intercepts with the axes. Show on your sketch how it is possible for $|x_2 - x_1|$ to be greater than $|x_1|$. [3]
\end{enumerate}
It is now given that $k = 100$ and $x_1 = 5$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Write down the exact value of $\alpha$ and find $x_2$ and $x_3$ correct to 5 decimal places. [3]
\item The error $e_n$ is defined by $e_n = \alpha - x_n$. By finding $e_1$, $e_2$ and $e_3$, verify that $e_3 \approx \frac{e_2^2}{e_1}$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2012 Q5 [11]}}