1 A curve passes through the point \(( 0,1 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + x ^ { 2 } }$$ Starting at the point \(( 0,1 )\), use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 0.4\). Give your answer to five decimal places.

## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| First increment is $0.2$, so $y \approx 1.2$ | B1B1 | PI; variations possible here |
| Second increment is $0.2\sqrt{1+0.2^2}$ | M1 | |
| $\approx 0.203961$, so $y \approx 1.40396$ | A2,1F | A1 if accuracy lost; ft num error |
| **Total** | **5** | |

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1 A curve passes through the point $( 0,1 )$ and satisfies the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + x ^ { 2 } }$$

Starting at the point $( 0,1 )$, use a step-by-step method with a step length of 0.2 to estimate the value of $y$ at $x = 0.4$. Give your answer to five decimal places.

\hfill \mbox{\textit{AQA FP1 2009 Q1 [5]}}