| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with complex derivative required |
| Difficulty | Standard +0.3 This is a straightforward Newton-Raphson question requiring differentiation of a function with fractional powers (standard FP1 technique), verification of a sign change, and one iteration of the formula. While the derivative requires care with the negative fractional power, this is routine for Further Maths students and involves no novel problem-solving—just methodical application of learned procedures. |
| Spec | 1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(1.1) = -1.6359604\), \(f(1.5) = 2.0141723\) | M1 | Attempts to evaluate both \(f(1.1)\) and \(f(1.5)\) and evaluates at least one correctly to awrt (or trunc.) 2 s.f. |
| Sign change (and \(f(x)\) is continuous) therefore a root/\(\alpha\) is between \(x=1.1\) and \(x=1.5\) | A1 | Both values correct to awrt (or trunc.) 2 s.f., sign change and conclusion. e.g. \(-1.63.. < 0 < 2.014..\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = x^3 - \frac{5}{2}x^{-\frac{3}{2}} + 2x - 3\) | M1 | \(x^n \rightarrow x^{n-1}\) for at least one term |
| \(\Rightarrow f'(x) = 3x^2 + \frac{15}{4}x^{-\frac{5}{2}} + 2\) | A1 | Correct derivative oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(1.1) = 3(1.1)^2 + \frac{15}{4}(1.1)^{-\frac{5}{2}} + 2\ (= 8.585)\) | M1 | Attempt to find \(f'(1.1)\). Accept \(f'(1.1)\) seen and their value |
| \(\alpha_2 = 1.1 - \left(\frac{"-1.6359604"}{" 8.585"}\right)\) | M1 | Correct application of Newton-Raphson |
| \(\alpha_2 = 1.291\) | A1 | cao |
# Question 2:
$$f(x) = x^3 - \frac{5}{2x^{\frac{3}{2}}} + 2x - 3$$
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(1.1) = -1.6359604$, $f(1.5) = 2.0141723$ | M1 | Attempts to evaluate both $f(1.1)$ and $f(1.5)$ and evaluates at least one correctly to awrt (or trunc.) 2 s.f. |
| Sign change (and $f(x)$ is continuous) therefore a root/$\alpha$ is between $x=1.1$ and $x=1.5$ | A1 | Both values correct to awrt (or trunc.) 2 s.f., sign change and conclusion. e.g. $-1.63.. < 0 < 2.014..$ |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = x^3 - \frac{5}{2}x^{-\frac{3}{2}} + 2x - 3$ | M1 | $x^n \rightarrow x^{n-1}$ for at least one term |
| $\Rightarrow f'(x) = 3x^2 + \frac{15}{4}x^{-\frac{5}{2}} + 2$ | A1 | Correct derivative oe |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(1.1) = 3(1.1)^2 + \frac{15}{4}(1.1)^{-\frac{5}{2}} + 2\ (= 8.585)$ | M1 | Attempt to find $f'(1.1)$. Accept $f'(1.1)$ seen and their value |
| $\alpha_2 = 1.1 - \left(\frac{"-1.6359604"}{" 8.585"}\right)$ | M1 | Correct application of Newton-Raphson |
| $\alpha_2 = 1.291$ | A1 | cao |
---2.
$$\mathrm { f } ( x ) = x ^ { 3 } - \frac { 5 } { 2 x ^ { \frac { 3 } { 2 } } } + 2 x - 3 , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [1.1,1.5].
\item Find f'(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 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2014 Q2 [7]}}