| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Derive Newton-Raphson formula |
| Difficulty | Moderate -0.3 This is a standard Newton-Raphson question testing routine application of the method. Part (i) is a straightforward derivation using coordinate geometry (equation of tangent line), part (ii) requires describing the iterative process with a sketch, and part (iii) involves one iteration with hyperbolic functions. While FP2 content, this is textbook material requiring no novel insight—slightly easier than average due to its procedural nature and low mark allocation. |
| Spec | 1.09d Newton-Raphson method4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials |
\includegraphics{figure_3}
A curve with no stationary points has equation $y = f(x)$. The equation $f(x) = 0$ has one real root $\alpha$, and the Newton-Raphson method is to be used to find $\alpha$. The tangent to the curve at the point $(x_1, f(x_1))$ meets the $x$-axis where $x = x_2$ (see diagram).
\begin{enumerate}[label=(\roman*)]
\item Show that $x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}$. [3]
\item Describe briefly, with the help of a sketch, how the Newton-Raphson method, using an initial approximation $x = x_1$, gives a sequence of approximations approaching $\alpha$. [2]
\item Use the Newton-Raphson method, with a first approximation of 1, to find a second approximation to the root of $x^2 - 2\sinh x + 2 = 0$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q3 [7]}}