Stage 36 · Limits, Derivatives & Their Applications
36.2.3 Letting the Gap Shrink to Zero
36.2 How Fast Things Change: From Average to Instant
Point 3 of 5
36.2.3 Letting the Gap Shrink to Zero
Core ideaAs the two points slide together, the secant tips over into a tangent, and the average rate squeezes into the instant rate as Δx→0.
Module goal. Use a limit to squeeze "the average pace over a stretch" down into "the pace at this very instant," leading naturally to the heart of the derivative.