| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson convergence failure |
| Difficulty | Standard +0.8 This question requires understanding of Newton-Raphson failure modes (not just mechanical application), geometric interpretation of why convergence fails beyond a stationary point, and careful calculation. Part (ii) demands conceptual insight into the method's limitations, which is beyond standard textbook exercises. The multi-part structure and need to explain failure conditions elevates this above routine application. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt use of product rule. Clearly get \(x = 1\) | M1, A1 | Allow substitution of \(x=1\) |
| (ii) Explain use of tangent for next approx. Tangents at successive approx. give \(x>1\) | B1, B1 | Not use of G.C. to show divergence. Relate to crossing \(x\)-axis; allow diagram |
| (iii) Attempt correct use of N-R with their derivative. Get \(x_2 = -1\). Get \(-0.6839, -0.5775, (-0.5672\ldots)\). Continue until correct to 3 d.p. Get \(-0.567\) | M1, A1, A1, M1, A1 | To 3 d.p. minimum. May be implied. cao |
**(i)** Attempt use of product rule. Clearly get $x = 1$ | M1, A1 | Allow substitution of $x=1$
**(ii)** Explain use of tangent for next approx. Tangents at successive approx. give $x>1$ | B1, B1 | Not use of G.C. to show divergence. Relate to crossing $x$-axis; allow diagram
**(iii)** Attempt correct use of N-R with their derivative. Get $x_2 = -1$. Get $-0.6839, -0.5775, (-0.5672\ldots)$. Continue until correct to 3 d.p. Get $-0.567$ | M1, A1, A1, M1, A1 | To 3 d.p. minimum. May be implied. cao5\\
\includegraphics[max width=\textwidth, alt={}, center]{15dd10f9-73d4-4107-bb45-7866f5470572-3_606_890_815_630}
The diagram shows the curve with equation $y = x \mathrm { e } ^ { - x } + 1$. The curve crosses the $x$-axis at $x = \alpha$.\\
(i) Use differentiation to show that the $x$-coordinate of the stationary point is 1 .\\
$\alpha$ is to be found using the Newton-Raphson method, with $\mathrm { f } ( x ) = x \mathrm { e } ^ { - x } + 1$.\\
(ii) Explain why this method will not converge to $\alpha$ if an initial approximation $x _ { 1 }$ is chosen such that $x _ { 1 } > 1$.\\
(iii) Use this method, with a first approximation $x _ { 1 } = 0$, to find the next three approximations $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$. Find $\alpha$, correct to 3 decimal places.
\hfill \mbox{\textit{OCR FP2 2008 Q5 [9]}}