Exponential growth and decay (§6.8)

Anything that follows an exponential law of growth or decay looks like

A = P e^kt,

where k is the relative growth rate, a term that can be explained using calculus. However, you can replace e with any other valid base (2, 10, whatever), so long as you change k to match. A different choice of the base can make the correct value of k either more or less obvious.

For example, if a quantity doubles in size every H years, then its size after t years is

A = P 2^t/H,

where P is the original size. If instead the quantity goes to half its size every h years, then its size after t years is

A = P 2^−t/h.

(In both of these formulas, you can use different units of time than years, as long as you do so for both t and H or h.) In these formulas, H is called the doubling time, and h is called the halflife.

If an object is placed in an environment at constant temperature, it will cool down or heat up to reach the environment's temperature. This temperature will neither grow nor decay exponentially; but according to Isaac Newton's law of cooling and heating, the difference in temperature will undergo exponential decay:

A − T = (P − T)e^kt,

where T is the temperature of the environment. You may prefer to solve for A:

A = T + (P − T)e^kt.

The book's version of this formula uses different variable names, but it is essentially the same.

Exponential decay is one thing, but exponential growth forever is unrealistic. In the model of logistic growth, there is a carrying capacity beyond which a population cannot grow. In this case, there is still an exponential growth, but it is the ratio of the population to the remaining capacity that grows exponentially:

A/(C − A) = P/(C − P) ⋅ e^kt,

where C is the carrying capacity. This looks rather different if we solve for A:

A = CPe^kt ÷ (C − P + Pe^kt).

The book has a slightly different formula, using not only different variable names but also constants with a different meaning, and dividing both sides of the fraction by e^kt (which makes it easier to solve for t).

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