| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with verification |
| Difficulty | Standard +0.3 This is a standard Newton-Raphson question with routine components: showing uniqueness via derivative sign, applying one iteration of the formula (straightforward calculation), and verifying accuracy by showing the root lies in a small interval. All techniques are textbook exercises for FP1, requiring no novel insight, making it slightly easier than average A-level difficulty. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx1.09d Newton-Raphson method |
$$f(x) = x^3 + x - 3.$$
The equation $f(x) = 0$ has a root, $\alpha$, between 1 and 2.
\begin{enumerate}[label=(\alph*)]
\item By considering $f'(x)$, show that $\alpha$ is the only real root of the equation $f(x) = 0$. [3]
\item Taking 1.2 as your first approximation to $\alpha$, apply the Newton-Raphson procedure once to $f(x)$ to obtain a second approximation to $\alpha$. Give your answer to 3 significant figures. [2]
\item Prove that your answer to part (b) gives the value of $\alpha$ correct to 3 significant figures. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q3 [7]}}