Growth and Decay Patterns

Calculus 1 · Axiom Academy

One rule runs through populations, money, and radioactivity: when a quantity changes at a rate proportional to its own size, it must be exponential. When a thing's growth feeds on its own size A bank balance, a bacterial colony, a lump of uranium — they look unrelated, yet each obeys the same simple rule: the bigger it already is, the faster it changes. That single sentence forces the shape of the answer. Watch it happen first, then drive it yourself. The rate of change is proportional to the current amount. The constant k sets how fast, and its sign sets the direction. Watch a growing quantity sweep in from the left. At every instant a little arrow shows its rate of change — and notice the arrow tilts up harder exactly as the curve climbs higher. The slope isn't fixed: it tracks the height. That feedback is what makes the growth exponential. Steeper because taller, taller because steeper — the arrow and the height rise together. That mutual feedback is the whole signature of exponential growth. One dial flips growth into decay Everything above had k positive. But the law y = y_0 e^ kt doesn't care about the sign of k . Drag the dial below from positive to negative and watch the very same formula turn an explosion into a fade. Right at k = 0 the rate is zero and the quantity just sits still. k > 0 grows, k < 0 decays, k = 0 holds steady — and the time to double (or halve) is , the same for every starting amount. Check the rule at any point you like

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