| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with verification |
| Difficulty | Standard +0.3 This is a standard Further Pure 1 Newton-Raphson question with routine parts: showing uniqueness via derivative sign, locating root by substitution, applying the iterative formula twice, and verifying accuracy via sign change. All steps are algorithmic with no novel insight required, though it's slightly above average difficulty due to being Further Maths content and requiring careful execution across multiple parts. |
| Spec | 1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
$$f (x) = x^3 + 8x - 19.$$
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $f(x) = 0$ has only one real root. [3]
\item Show that the real root of $f(x) = 0$ lies between 1 and 2. [2]
\item Obtain an approximation to the real root of $f(x) = 0$ by performing two applications of the Newton-Raphson procedure to $f(x)$ , using $x = 2$ as the first approximation. Give your answer to 3 decimal places. [4]
\item By considering the change of sign of $f(x)$ over an appropriate interval, show that your answer to part (c) is accurate to 3 decimal places. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q37 [11]}}