One Constant, and Everything It Decides

You'll learnto read the growth constant's sign and size correctly, to get a doubling time or a half-life out of it with a logarithm instead of a division, and to keep the constant solution that separating by dividing quietly throws away.

Every quantity whose rate of change is proportional to itself obeys one equation and has one solution, so the whole topic reduces to reading a single constant correctly. Its sign says grow or decay. Its size sets the doubling time and the half-life, through a logarithm and never through a division. And its units are one over time, which settles the two readings that cost marks every year: a reciprocal time cannot be a count of anything, and it cannot be a duration either. One more habit travels with it — separating the equation means dividing by the quantity, and the case that division discards is a solution too.

8 STEPS · 6 QUICK CHECKS · THE CONSTANT IS A RATE, NOT AN AMOUNT AND NOT A DURATION · v1

dy/dt = ky ⇒ y = y₀ e^(kt) ⇒ the sign is the direction, the size is ln 2 divided by it

Before you start

What you're looking at

Quantity against time. The horizontal axis is always time and the vertical axis is always the thing being modelled — a population, a mass, a temperature — so the two axes carry different units and the panels are not drawn at equal scale.

The question

Given one constant, what does the model do? Its sign, its size and its units each answer a different exam question, and none of the three answers is a number of years.

Watch for

The teal line in step 6 and the teal line in step 7. Both are constant solutions that the separation removed by dividing, and both come back at one particular value of the constant.

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