Euler's Method: Step Size and Error

Euler's method builds an approximate solution to dy/dx = f(x,y) one small step at a time. Starting at a known point, you use the slope f(x,y) there to draw a short tangent line segment, step forward by h along x, and land on a new estimated point. Then you repeat: recompute the slope at the new point, step again. The whole path is a chain of straight-line pieces pretending to be the curve.

Each step is really just the tangent line approximation from Unit 4, applied over and over. A tangent line only matches the true curve exactly at the point of tangency; everywhere else it drifts off, and how fast it drifts depends on concavity. If the true solution curves upward (concave up), the tangent line sits below it, so Euler's estimate undershoots. If the solution curves downward (concave down), the tangent line sits above it, and Euler overshoots. That per-step drift is the local error, and it accumulates across steps.

Halving the step size cuts each local error roughly in half too, since you're approximating the same bending curve with shorter, straighter pieces, but you're also taking twice as many steps, so the total error shrinks only roughly by half overall, not to zero. This is why Euler's method is called a first-order method: error scales proportionally with step size h, not with h² or faster. That slow, proportional shrinkage is exactly what motivates looking for better approximation schemes later, and it's also why real computation has to balance step size against rounding error and computing cost rather than just shrinking h forever.