Natural Growth Patterns

Calculus 2 · Axiom Academy

One rule behind a population, a half-life, and a bank balance Colonies of bacteria, decaying isotopes, and continuously-compounded money look like wildly different things — yet they share a single mathematical heartbeat. In every case, how fast the quantity changes is set by how much of it there is right now . Big amount, fast change; small amount, slow change. Watch the rule build the curve. At each step the little arrow shows the instantaneous rate — and notice it grows taller as the curve climbs . Stacking those proportional increments traces out the smooth solution . The arrow is the curve's own slope — and it's always k times the current height. That self-feeding loop is exponential growth. One dial, every behavior: turn k The shape is always the same exponential — only the constant k changes. Drag the dial: push k positive and the same curve becomes runaway growth ; pull it negative and it becomes decay that fades toward zero but never quite reaches it; sit at k=0 and nothing changes at all. Same equation, same curve family — the sign and size of k alone decide growth, decay, or standing still. Read the rule straight off the curve Here's the proof you can touch. Drag the point along a growth curve; at every spot the dashed tangent shows the slope , and the bar shows the height y . Their ratio never moves — it stays pinned at k . That's exactly what says.

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