How Wrong Is One Step?
Euler's step is a guess, so it carries some error. Because the population equation has a known exact solution, we can measure that error exactly and watch how it grows.
The error of a single step is just how far the estimate landed from the truth:
e₁ = p(t₁) − p₁ (exact − estimate)
The true value minus the Euler estimate. Usually we don't know the exact answer — here we luckily do.
Drag Δt and watch the gap. The red line is one Euler step; the green dot on the curve is the truth; the orange gap between them is the error:
The red line is one Euler step: start at p₀ and follow the slope for Δt. It lands at p₁, but the true answer is the green dot on the curve. The orange gap is the error e₁ = exact − estimate — and it grows as Δt grows (local error).
The error of a single step is called the local error, and for Euler it's linear: halve Δt, and one step's error roughly halves too. Bigger step → bigger miss. The obvious fix is coming next: take more, smaller steps.