| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Compare Newton-Raphson with linear interpolation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring standard application of Newton-Raphson and linear interpolation. Part (a) is simple substitution, part (b) requires differentiating a fractional power and applying the NR formula once, and part (c) uses the basic linear interpolation formula. All techniques are routine with no problem-solving insight needed, though it's slightly above average difficulty due to being Further Maths content. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09d Newton-Raphson method |
4.
$$\mathrm { f } ( x ) = x ^ { \frac { 3 } { 2 } } - 3 x ^ { \frac { 1 } { 2 } } - 3 , \quad x > 0$$
Given that $\alpha$ is the only real root of the equation $\mathrm { f } ( x ) = 0$,
\begin{enumerate}[label=(\alph*)]
\item show that $4 < \alpha < 5$
\item Taking 4.5 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 decimal places.\\[0pt]
\item Use linear interpolation once on the interval [4,5] to find another approximation to $\alpha$, giving your answer to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2014 Q4 [10]}}