Moderate -0.3 This is a straightforward application of Euler's method with a simple trigonometric function. The question requires only mechanical step-by-step calculation (two steps) with clear instructions and no conceptual challenges—slightly easier than average due to its routine nature, though the numerical work and radian mode requirement prevent it from being trivial.
2 A curve satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x$$
where the angle \(2 x\) is measured in radians.
Starting at the point \(( 0.5,1 )\) on the curve, use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 0.7\). Give your answer to three significant figures.
(6 marks)
\(x = 0.5 \Rightarrow y' = \sin 1\) First increment is \(0.1 \sin 1\); \(x = 0.6 \Rightarrow y \approx 1.084\) Second increment is \(0.1 \sin 1.2\); \(x = 0.7 \Rightarrow y \approx 1.177 \approx 1.18\)
M1, m1A1, A1, m1, A1F
6 marks
Total: 6 marks
| $x = 0.5 \Rightarrow y' = \sin 1$ First increment is $0.1 \sin 1$; $x = 0.6 \Rightarrow y \approx 1.084$ Second increment is $0.1 \sin 1.2$; $x = 0.7 \Rightarrow y \approx 1.177 \approx 1.18$ | M1, m1A1, A1, m1, A1F | 6 marks | Max 4/6 if degrees used; PI by correct answer at end; Ft error in $y(0.6)$ |
**Total: 6 marks**
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2 A curve satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x$$
where the angle $2 x$ is measured in radians.\\
Starting at the point $( 0.5,1 )$ on the curve, use a step-by-step method with a step length of 0.1 to estimate the value of $y$ at $x = 0.7$. Give your answer to three significant figures.\\
(6 marks)
\hfill \mbox{\textit{AQA FP1 2005 Q2 [6]}}