Euler's method (1st order)
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Flip to reveal answersWhat is the Euler recurrence for dy/dx = f(x, y)?
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All 8 Flashcards — Euler's method (1st order)
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Question
What is the Euler recurrence for dy/dx = f(x, y)?
Answer
y_(n+1) = y_n + h·f(x_n, y_n) and x_(n+1) = x_n + h, starting from (x₀, y₀).
Question
How many Euler steps reach a target x from x₀ with step length h?
Answer
number of steps = (target − x₀) ÷ h. E.g. x = 1 to x = 2 with h = 0.25 is 4 steps.
Question
What does each Euler step actually do geometrically?
Answer
It moves along a short STRAIGHT line at the gradient measured at the current point — so the path only approximates the true curve.
Question
Is Euler's method exact?
Answer
No — it is an approximation. A smaller step length h gives more steps and a more accurate estimate, but never the exact value.
Question
Does Euler over- or under-estimate?
Answer
It depends on the curve's concavity: it tends to UNDER-estimate for a concave-up curve and OVER-estimate for a concave-down curve.
Question
Which gradient does each Euler step use?
Answer
The gradient f(xₙ, yₙ) at the START of the step (the point you are currently at), not at the new point.
Question
How do you find the percentage error of an Euler estimate?
Answer
percentage error = |approx − exact| ÷ exact × 100%, using the exact solution given in the question.
Question
What is the GDC route for Euler's method in AI?
Answer
Store the recurrence as a recursive sequence (or fill a table of n, xₙ, yₙ, gradient); a calculator is allowed on every AI paper.
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Full study notes for Euler's method (1st order)
Topic 5.16 hub
Euler's method (1st order) (HL only)
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