| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 6 |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with derivative given or simple |
| Difficulty | Moderate -0.3 This is a straightforward application of the Newton-Raphson method with a simple cubic function. Part (i) requires only substitution to locate the root between consecutive integers, and part (ii) is a standard iterative procedure with f(x) and f'(x) both easy to compute. The question is slightly easier than average because it provides clear guidance (starting value, stopping criterion) and involves no conceptual challenges beyond executing the algorithm correctly. |
| Spec | 1.09d Newton-Raphson method |
**(i)** Evaluate at least f(1) and f(2) to inspect for a change of sign M1
State $n = 1$ (accept $1 < \alpha < 2$) A1 **[2]**
**(ii)** $f'(x) = 3x^2 - 1$ B1
Obtain $x_{n+1} = x_n - \frac{x_n^3 - x_n - 1}{3x_n^2 - 1}$ or $\frac{2x_n^3 + 1}{3x_n^2 - 1}$ B1
Use attempt at Newton Raphson formula with $x_0 = 1$ M1
$x_1 = 1.5$ first two iterates A1
$x_2 = 1.347826$
$x_3 = 1.325200$
$x_4 = 1.324718$
$x_5 = 1.324718$ final two iterates supporting 1.32 A1
$x = 1.32$ stated B1 **[6]**6 The equation $x ^ { 3 } - x - 1 = 0$ has exactly one real root in the interval $0 \leq x \leq 3$.\\
(i) Denoting this root by $\alpha$, find the integer $n$ such that $n < \alpha < n + 1$.\\
(ii) Taking $n$ as a first approximation, use the Newton-Raphson method to find $\alpha$, correct to 2 decimal places. You must show the result of each iteration correct to an appropriate degree of accuracy.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q6 [6]}}