| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply basic Euler method for differential equations |
| Difficulty | Standard +0.3 This is a straightforward application of two standard numerical methods (Euler's method and a mid-point variant) with explicit formulas provided. Students simply substitute given values into the formulas with h=0.1, requiring only basic arithmetic and function evaluation. No conceptual insight or problem-solving is needed beyond following the given procedures, making it slightly easier than average. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(y(2,1) = y(2) + 0.1[2^2 - 1] = 1+0.1 \times 3 = 1.3\) | M1A1, A1 | Total: 3 |
| (b) \(y(2.2) = y(2) + 2(0.1)[f(2.1, y(2.1))]\) | M1 | |
| \(\ldots = 1+2(0.1)[2.1^2 - 1.3^2]\) | A1/ | |
| \(\ldots = 1+0.2 \times 2.72 = 1.544\) | A1 | Total: 3 |
**(a)** $y(2,1) = y(2) + 0.1[2^2 - 1] = 1+0.1 \times 3 = 1.3$ | M1A1, A1 | Total: 3 | Ft on cand's answer to (a)
**(b)** $y(2.2) = y(2) + 2(0.1)[f(2.1, y(2.1))]$ | M1 | |
$\ldots = 1+2(0.1)[2.1^2 - 1.3^2]$ | A1/ | | Ft on cand's answer to (a)
$\ldots = 1+0.2 \times 2.72 = 1.544$ | A1 | Total: 3 | CAO1 The function $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = x ^ { 2 } - y ^ { 2 }$$
and
$$y ( 2 ) = 1$$
\begin{enumerate}[label=(\alph*)]
\item Use the Euler formula
$$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with $h = 0.1$, to obtain an approximation to $y ( 2.1 )$.
\item Use the formula
$$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with your answer to part (a), to obtain an approximation to $y ( 2.2 )$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2008 Q1 [6]}}