Exponential Growth & Decay

Calculus 1 · Axiom Academy

LESSON Exponential Growth & Decay When a quantity's rate of change is proportional to how much is present, one differential equation — and one curve — governs it all. 1. The Law: Rate Proportional to Amount The whole subject starts from one sentence: the faster something changes, the more of it there already is. In symbols, the rate of change is a constant multiple k of the current amount y . the governing differential equation Watch a quantity climb the curve y = e^ kt below. At every instant the steepness of the tangent equals — so where the curve is twice as tall, it is rising twice as fast. The slope and the height grow together; that is what looks like. 2. The Solution, and the Sign of k Which function has a derivative proportional to itself? Only the exponential. Solving — separate variables and integrate — gives the general solution, where C=y(0) is the starting amount. C = y_0 is the initial amount at t=0 Everything now rides on the sign of k . Below, two quantities leave the same initial value C : one with , one with . Same equation, opposite destinies. . The amount climbs ever faster — populations, compound interest, viral spread. . The amount fades toward zero but never reaches it — radioactivity, drug clearance, cooling. Check it satisfies the equation Differentiate y=Ce^ kt to get . The derivative really is k times the function — so Ce^ kt is exactly the solution the law demands. 3. Half-Life and Doubling Time

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