| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply improved Euler method for differential equations |
| Difficulty | Standard +0.3 This is a straightforward application of two standard numerical methods (Euler and improved Euler) with explicit formulas provided. Students only need to substitute given values into the formulas and perform arithmetic calculations—no derivation, problem-solving insight, or conceptual understanding beyond basic substitution is required. While it's an FP3 topic, the execution is purely mechanical. |
| Spec | 1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| \(y(3.1) = 2.6238\) | M1A1, A1 | Condone greater accuracy; 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 2.6494(2\ldots) = 2.6494\) to 4dp | m1, A1 | PI: ft from (a), 4dp or better; PI: ft on \(0.1 \times 3.1 \times \ln[6.2 + \text{answer(a)}]\); CAO Must be 2.6494; 5 marks total |
**(a)**
$y_1 = 2 + 0.1 \times [3\ln(2 \times 3 + 2)] = 2 + 0.3\ln8 = 2.6238$ (to 4dp)
$y(3.1) = 2.6238$ | M1A1, A1 | Condone greater accuracy; 3 marks total
**(b)**
$k_1 = 0.1 \times 3\ln 8 = 0.6238(32\ldots)$
$k_2 = 0.1 \times f(3.1, 2.6238(32\ldots))$
$\ldots = 0.1 \times 3.1 \times \ln[6.2 + \text{answer(a)}]$
$[= 0.6750(1\ldots)]$
$y(3.1) = 2 + \frac{1}{2}[0.6238(3\ldots) + 0.6750(1\ldots)]$
$= 2.6494(2\ldots) = 2.6494$ to 4dp | m1, A1 | PI: ft from (a), 4dp or better; PI: ft on $0.1 \times 3.1 \times \ln[6.2 + \text{answer(a)}]$; CAO Must be 2.6494; 5 marks total
**Total: 8 marks**
---1 The function $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = x \ln ( 2 x + y )$$
and
$$y ( 3 ) = 2$$
\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 $y ( 3.1 )$, giving your answer to four decimal places.
\item Use the improved Euler formula
$$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$
where $k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)$ and $k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)$ and $h = 0.1$, to obtain an approximation to $y ( 3.1 )$, giving your answer to four decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2010 Q1 [8]}}