Growth Rate Explorer

Differential Equations · Axiom Academy

Two quantities start at the exact same value. One grows by the same fixed AMOUNT each step; the other grows by the same fixed PERCENTAGE. Watch how differently that one choice plays out. Same start, wildly different endings Two populations both start at 100 . One grows linearly — it adds the same fixed amount every step. The other grows exponentially — it multiplies by the same fixed factor every step. Same starting point, same number of steps. Watch what that one difference in HOW they grow does to where they end up. Press play. The linear bar climbs by a steady +20 every step. The exponential bar climbs by every step — slower at first, then explosive. By step 15, the linear bar reaches 400 — the exponential bar reaches over 800 . Multiplying compounds; adding doesn't. The growth constant decides when exponential takes over Here the linear line is fixed at y = 100 + 30t . Drag k in the exponential curve y = 100e^ kt and watch the crossover point — where the exponential curve overtakes the line — slide left and right. A smaller k pushes the crossover far to the right; a larger k drags it back toward the start. The exponential ALWAYS wins eventually — k only decides how long linear gets to lead. The rate of change is the real difference Forget the curves for a second — look at the SLOPE, the rate of change , at a single moment t . Drag t and compare: the linear rate never changes. The exponential rate keeps climbing, because it's tied to the current value.

This is the written version of the interactive lesson above. See the full Differential Equations course.