Unit 5: Calculus
Topic 5.18: Euler's method (2nd order) (HL only) Questions
Practice 12 exam-style questions for IB Math AI SL Topic 5.18. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.
11 mark
2026
The Euler update for the displacement in a 2nd-order system is:
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2026
To apply Euler's method to a 2nd-order DE d²x/dt² = g(t, x, dx/dt), you first:
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2026
Reaching t = 2 from t = 0.5 with step length h = 0.25 takes how many Euler steps?
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2026
A spring d²x/dt² = −x has x₀ = 2, v₀ = 0. After one Euler step with h = 0.5, the velocity v₁ is:
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2026
Two Euler estimates of x(1) for the same 2nd-order DE use h = 0.5 and h = 0.1. Which is more accurate, and why?
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2026
A damped oscillator obeys d²x/dt² = −2x − 0.5(dx/dt). At t = 0, x = 4 and dx/dt = 1. Using v = dx/dt and Euler with h = 0.2, find x and v after ONE step.
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2026
Find the approximations to and after two steps.
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2026
In an electrical circuit the charge q (coulombs) obeys d²q/dt² = −2q − 3(dq/dt). At t = 0, q = 0 and dq/dt = 5. Using i = dq/dt and Euler with h = 0.25, estimate q at t = 0.5 s.
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2026
A spring obeys d²x/dt² = −x with x(0) = 0 and dx/dt = 3 at t = 0. Using v = dx/dt and Euler with h = 0.25, estimate the displacement at t = 1 s. (Watch the step count.)
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2026
A spring obeys d²x/dt² = −9x (x in cm, t in s), released from rest at x(0) = 5. Use Euler's method with h = 0.1 to estimate the displacement at t = 0.2 s.
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2026
A falling parcel has speed v (m/s) and dropped distance h (m) modelled by dh/dt = v and dv/dt = 9.8 − 0.25v, with h(0) = 0 and v(0) = 0. Use Euler with step 0.5 s to estimate the speed and distance fallen at t = 1 s.
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2026
For d²x/dt² = −x with x(0) = 1 and dx/dt = 0, the exact value is x(1) = cos 1 ≈ 0.5403. Euler with h = 0.5 gives x(1) ≈ 0.75 and with h = 0.25 gives x(1) ≈ 0.6289. Find the percentage error in each estimate and comment.
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