Intermediate Value Theorem

Calculus 1 · Axiom Academy

The Intermediate Value Theorem A temperature sensor reads 60°F at midnight and 80°F at noon — so it must have read every temperature in between. That guarantee is a theorem. It's 60°F at midnight and 80°F at noon, and the sensor never skips — so somewhere in those 12 hours it had to read exactly 70°F . Three moves turn that hunch into a guarantee. Watch the temperature sweep the day Drag through the 24 hours. The reading slides smoothly — from 60°F up to 80°F by noon and back — without ever jumping. That unbroken curve is what we mean by continuous . Pick a temperature — the theorem finds the moment Choose any target between the start (60°F) and noon (80°F). The reading starts below it and ends above it, so the curve has to cross your line — and we read off the exact time it happened. Break continuity — and the guarantee breaks What is the theorem really leaning on? Open a sudden jump in the reading at 6 AM and watch a whole band of temperatures get skipped — values the sensor never shows. No jump, no skipping: that's why continuous is the one word that can't be dropped. f continuous on [a, b] — the reading has no breaks or jumps (our temperature curve), the hypothesis Beat 3 showed you can't drop. N between f(a) and f(b) — any target temperature between the 60°F start and the 80°F noon. then some c in (a, b) with f(c) = N — there is a moment when the sensor reads exactly that, even though the theorem never tells you when to compute it.

This is the written version of the interactive lesson above. See the full Calculus 1 course.