| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 9 |
| 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 application of Newton-Raphson with standard calculus. Part (a) is routine sign-checking, part (b) requires differentiating powers of x (rewriting √x as x^{1/2}), and part (c) is a single iteration of the Newton-Raphson formula. While it's Further Maths content, the derivative is standard and only one iteration is required, making this easier than an average A-level question overall. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Attempt evaluation of \(f(1.1)\) and \(f(1.2)\) | M1 | Looking for sign change |
| \(f(1.1) = 0.30875\), \(f(1.2) = -0.28199\); change of sign \(\Rightarrow\) root in interval | A1 | awrt \(0.3\) and \(-0.3\) and indication of sign change |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(x) = \frac{3}{2}x^{-\frac{1}{2}} - 9x^{-\frac{3}{2}}\) | M1 A1 A1 | Multiply by power and subtract 1 from power for evidence of differentiation for first M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(1.1) = 0.30875\), \(f'(1.1) = -6.37086...\) | B1 B1 | awrt \(0.309\) B1; awrt \(-6.37\) B1 if answer incorrect |
| \(x_1 = 1.1 - \frac{0.30875...}{-6.37086...}\) | M1 | Evidence of Newton-Raphson for M1 |
| \(= 1.15\) (to 3 sig.figs.) | A1 | Evidence of Newton-Raphson and awrt \(1.15\) award 4/4 |
## Question 5:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Attempt evaluation of $f(1.1)$ and $f(1.2)$ | M1 | Looking for sign change |
| $f(1.1) = 0.30875$, $f(1.2) = -0.28199$; change of sign $\Rightarrow$ root in interval | A1 | awrt $0.3$ and $-0.3$ and indication of sign change |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = \frac{3}{2}x^{-\frac{1}{2}} - 9x^{-\frac{3}{2}}$ | M1 A1 A1 | Multiply by power and subtract 1 from power for evidence of differentiation for first M1 |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(1.1) = 0.30875$, $f'(1.1) = -6.37086...$ | B1 B1 | awrt $0.309$ B1; awrt $-6.37$ B1 if answer incorrect |
| $x_1 = 1.1 - \frac{0.30875...}{-6.37086...}$ | M1 | Evidence of Newton-Raphson for M1 |
| $= 1.15$ (to 3 sig.figs.) | A1 | Evidence of Newton-Raphson and awrt $1.15$ award 4/4 |
---5.
$$f ( x ) = 3 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 20$$
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [1.1,1.2].
\item Find $\mathrm { f } ^ { \prime } ( x )$.
\item Using $x _ { 0 } = 1.1$ 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 2009 Q5 [9]}}