Moderate -0.8 This is a straightforward numerical methods question requiring repeated application of Euler's method with a given step size. The calculation is routine and mechanical (2 steps only), requiring no conceptual insight beyond knowing the Euler formula y_{n+1} = y_n + h·f(x_n). The arithmetic is simple with the given function, making this easier than average A-level questions.
A curve passes through the point \((9, 6)\) and satisfies the differential equation
$$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2 + \sqrt{x}}$$
Use a step-by-step method with a step length of \(0.25\) to estimate the value of \(y\) at \(x = 9.5\). Give your answer to four decimal places.
[5 marks]
A curve passes through the point $(9, 6)$ and satisfies the differential equation
$$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2 + \sqrt{x}}$$
Use a step-by-step method with a step length of $0.25$ to estimate the value of $y$ at $x = 9.5$. Give your answer to four decimal places.
[5 marks]
\hfill \mbox{\textit{AQA FP1 2014 Q1 [5]}}