The Exponential Model

Calculus 2 · Axiom Academy

When the rate of change is proportional to the amount, one equation — — governs growth, decay, doubling time, and half-life. The model starts from one sentence: the rate of change of y equals a constant times y itself . How fast something changes depends on how much of it there is. k is the constant of proportionality (the relative growth rate) 2. Solving by Separation of Variables To solve we separate the variables : gather every y on one side, every t on the other, then integrate. The solution that emerges is the exponential law. One constant decides everything: its sign sets growth versus decay, and its size |k| sets the speed. Both curves below leave the same starting amount y_0 — only the sign of k differs. y increases exponentially. Population growth, compound interest, an unchecked bacterial culture. y decreases exponentially. Radioactive decay, a cooling object, a drug clearing the bloodstream. 4. Doubling Time and Half-Life Exponential processes have a signature rhythm: a fixed amount of time always multiplies the quantity by the same factor. For growth that factor is 2 (the doubling time ); for decay it is (the half-life ). Doubling time ( k>0 ): reach 2y_0 Half-life ( k<0 ): reach y_0/2 k = 0.1 per hour doubling time hours. From one proportionality statement you derived the entire exponential model — its solution, the role of k , and the timing laws it forces. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Calculus 2 course.