Short on time? This page is the recap. For the graph, the step-size table and worked examples, use the full study guide. BC-only material. Prerequisites (tangent lines 4.6, differential equations 7.1–7.4, concavity in Unit 5) are on the Calculus BC roadmap.
Recap
- Euler’s method approximates a particular solution of dy/dx = f(x, y) from a known point.
- Each step is a tangent-line approximation: move h in x and h × slope in y.
- After each step, find the new slope at the new point. That is what makes it better than one long tangent line.
- Use it when you cannot find, or are not asked for, an exact solution.
Key relationships
| Idea | Rule | Note |
|---|---|---|
| Step rule | xₙ₊₁ = xₙ + h, yₙ₊₁ = yₙ + h · f(xₙ, yₙ) | Slope at the point you leave |
| Number of steps | (target x − start x) ÷ h | From 0 to 1 with h = 0.25: 4 steps |
| Step backwards | Use h < 0 | Same rule, negative h |
| Accuracy | Smaller h usually means smaller error | Never exactly zero error |
| Concave up (d²y/dx² > 0) | Euler underestimates | Tangent segments lie below the curve |
| Concave down (d²y/dx² < 0) | Euler overestimates | Tangent segments lie above the curve |
| Second derivative | Differentiate dy/dx implicitly | Substitute dy/dx before evaluating |
Table layout
| Step | (xₙ, yₙ) | Slope | h × slope | New point |
|---|
One row per step. Fill the new point, then copy it into the next row.
Mistakes to avoid
- Slope at the new point instead of the old one.
- Forgetting h: adding the slope itself.
- Giving the change, not the new y-value.
- Rounding every step. Round once, at the end.
- Judging over/under from “increasing”. Use the sign of d²y/dx².
- Wrong number of steps for the given h.
Quick self-check
- For dy/dx = 3 − y with y(0) = 0, use two steps of 0.5 to approximate y(1). (Step 1: slope 3, y = 1.5. Step 2: slope 1.5, y = 2.25.)
- For dy/dx = xy with y(1) = 2, use one step of 0.1 to approximate y(1.1). (2 + 0.1 × 2 = 2.2)
- In question 2, d²y/dx² = y + x · dy/dx = y + x²y, which is 4 at (1, 2). Over or under? (Positive, so concave up: 2.2 is an underestimate.)
- Why is a smaller step usually more accurate? (Each tangent segment is shorter, so it has less room to drift away from the curving solution.)
Next: practice questions.