| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with complex derivative required |
| Difficulty | Moderate -0.3 This is a straightforward Newton-Raphson application requiring differentiation of fractional powers (standard FP1 skill) and one iteration of the formula. The derivative is routine, and the calculation involves no conceptual challenges—just careful arithmetic with a calculator. Slightly easier than average due to being a single iteration with no complications. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f'(x) = x^{\frac{1}{2}} - \frac{1}{2}x^{-\frac{3}{2}}\) | M1 A1 | (2) |
| (b) \(f(5) = -0.0807\); \(f'(5) = 0.4025\) | B1; M1 | |
| \(x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 5 - \frac{-0.0807}{0.4025}\) | M1 | |
| \(= 5.2(0)\) | A1 | Accept 5.2 |
| (4) [6] |
(a) $f'(x) = x^{\frac{1}{2}} - \frac{1}{2}x^{-\frac{3}{2}}$ | M1 A1 | (2)
(b) $f(5) = -0.0807$; $f'(5) = 0.4025$ | B1; M1 |
$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 5 - \frac{-0.0807}{0.4025}$ | M1 |
$= 5.2(0)$ | A1 | Accept 5.2
| (4) [6] |
**Notes:**
The B and M marks are implied by a correct answer only with no working or by $\frac{1}{10}(10\sqrt{5} - 13)$
(a) M for at least one of $\pm ax^{\frac{1}{2}}$ or $\pm bx^{-\frac{3}{2}}$; A for correct (equivalent) answer only
(b) B for awrt -0.0807; first M for attempting their $f'(5)$; M for correct formula and attempt to substitute; A for awrt 5.20, but accept 5.2
---3.
$$\mathrm { f } ( x ) = 2 x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } - 5 , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { f } ^ { \prime } ( x )$.
The equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [4.5, 5.5].
\item Using $x _ { 0 } = 5$ as a first approximation to $\alpha$, apply the Newton-Raphson procedure once to $\mathrm { f } ( x )$ to find a second approximation to $\alpha$, giving your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2013 Q3 [6]}}