Quick questions on Slope fields and Euler's method: VCE Specialist Mathematics Unit 4

Because the method holds the gradient fixed across each step, it drifts away from a curving solution, undershooting or overshooting depending on the concavity. Halving the step size roughly halves the error, so a smaller $h$ gives a better approximation at the cost of more steps.

Gradient at $(1, 2)$ is $f(1, 2) = 1 + 2 = 3$. Then

Gradient at $(1.1, 2.3)$ is $f(1.1, 2.3) = 1.1 + 2.3 = 3.4$. Then

The increment is $h\,f(x_n, y_n)$. Forgetting the factor $h$ overshoots massively.

What does a horizontal segment in a slope field indicate? [1 mark]

Write the Euler update formula for $y_{n+1}$. [1 mark]

For $\frac{dy}{dx} = 2x$ with $y(0) = 1$ and $h = 0.5$, find $y$ at $x = 0.5$. [2 marks]