| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Topic | Fixed Point Iteration |
| Type | Compare iteration convergence |
| Difficulty | Challenging +1.2 This question tests standard fixed-point iteration theory with routine differentiation and convergence criteria (|F'(α)| < 1). While it requires multiple steps and understanding of convergence rates, the techniques are straightforward applications of A-level calculus and numerical methods without requiring novel insight or complex problem-solving. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09e Iterative method failure: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) \( | e_{r+1} | = 0.34^r |
| Answer | Marks | Guidance |
|---|---|---|
| \(10^{-10} | e_1 | > 0.34^r |
| M1 Attempt to solve \(10^{-10} | e_1 | = F'(1.9)^r |
(i) $F_1'(x) = x - \dfrac{3}{4}x^2$
B1
$F_2'(x) = \dfrac{1}{3}(2x^2 - 4x + 7)^{-\frac{2}{3}}(4x-4)$
M1 Attempt chain rule
A1 Must include any necessary brackets
$F_3'(x) = \dfrac{-4(x^2-2x)-(7-4x)(2x-2)}{(x^2-2x)^2}$
M1 Attempt quotient rule (allow sign muddles in numerator). Could also differentiate partial fractions
A1 [5] No need to simplify (isw if done incorrectly)
(ii) $F_1'(1.9) = -0.8075$
$F_2'(1.9) = 0.340...$
$F_3'(1.9) = 50.9...$
M1 Attempt $F'(1.9)$ for all three functions
$F_1$ and $F_2$ converge
B1* State correct condition for convergence – could be for acceptance or rejection, but must have modulus sign oe
A1d* [4] Identify $F_1$ and $F_2$. Need $F'(1.9)$ correct for all 3 functions
Faster convergence is $F_2$ because the magnitude of the gradient is smallest near the root. WWW
A1 Must have magnitude soi, not just 'gradient smaller'. Need $F'(1.9)$ correct for all 3 functions. M1B0A0A1 is possible
(iii) $|e_{r+1}| = 0.34^r|e_1|$
M1 Attempt to apply general statement to this question e.g. $e_2 = 0.34 \times e_1$
$10^{-10}|e_1| > 0.34^r|e_1|$
M1 Attempt to solve $10^{-10}|e_1| = F'(1.9)^r|e_1|$. Allow index of $r$ or $r-1$. Could use a more accurate value for $a$. Allow any numerical value for $e_1$, inc 1.9. Allow any $F'(1.9)$ as long as $|F'(1.9)| < 1$
so $r > \dfrac{-10}{\log 0.34} > 21.34$, so 22 iterations.
A1 [3] Obtain 21/22/23 depending on index and method used. If numerical $e_1$ used, it must have been correct. No credit for answer only11 The cubic equation $x ^ { 3 } - 2 x ^ { 2 } + 4 x - 7 = 0$ has a single root $\alpha$, close to 1.9 , which can be found using an iteration of the form $x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)$. Three possible functions that can be used for such an iteration are
$$\mathrm { F } _ { 1 } ( x ) = \frac { 7 } { 4 } + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 } , \quad \mathrm {~F} _ { 2 } ( x ) = \sqrt [ 3 ] { 2 x ^ { 2 } - 4 x + 7 } , \quad \mathrm {~F} _ { 3 } ( x ) = \frac { 7 - 4 x } { x ^ { 2 } - 2 x }$$
(i) Differentiate each of these functions with respect to $x$.\\
(ii) Without performing any iterations, and using $x = 1.9$, show that an iterative process based on only two of the given functions will converge.
Determine which one will do so more rapidly.
The sequence of errors, $e _ { n }$, is such that $e _ { n + 1 } \approx \mathrm {~F} ^ { \prime } ( \alpha ) e _ { n }$.\\
(iii) Using the iteration from part (ii) with the most rapid convergence, estimate the number of iterations required to reduce the magnitude of the error from $\left| e _ { 1 } \right|$ in the first term to less than $10 ^ { - 10 } \left| e _ { 1 } \right|$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2014 Q11 [12]}}