When the Rate Depends on the Amount
Last timeDoing It With Arithmetic, and the Error You Buy
A quantity whose rate is proportional to how much of it there is multiplies by a fixed factor in every fixed stretch of time, and the rule describing it is its own rate.
Most rates in the sciences are not given as functions of the clock. They are given in
terms of the quantity itself. Interest depends on the balance, decays per second depend
on the number of atoms left, heat lost depends on how much hotter the object still is
than the room. In every case the thing that is changing is also what sets the speed of
the change.
That self-reference sounds like it should make the problem harder. It makes it easier,
because it can be solved completely, and the answer is the most frequently met rule in
applied mathematics.
- the proportionality constant: the fractional change per unit time. Positive gives growth, negative gives decay, and its units are one over time
- the amount at the start. The rate equation cannot determine it, which is the arbitrary constant from the lesson on the two questions, here carrying real information
- the limit of compounding a proportional increase over ever shorter intervals, about two point seven one eight. It is not chosen, it is what that limit equals
- the factor by which the starting amount has been multiplied by time t. It is a multiplier rather than an addition, which is the whole character of this situation
Where the number comes from
Take a quantity growing at one hundred per cent per unit time and apply the growth in
instalments. One instalment at the end gives a doubling. Four instalments of a quarter
each, with each instalment computed on the amount then present, give more, because the
later instalments are computed on an already grown amount.
| step | instalments in the period | fraction added each time | factor after the whole period | how far short of the limit | what happened |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 2 | 0.7183 | All at the end: the amount doubles and nothing compounds. |
| 2 | 4 | 0.25 | 2.4414 | 0.2769 | Quarterly. Each quarter earns on what the earlier quarters added. |
| 3 | 12 | 0.0833 | 2.613 | 0.1052 | Monthly, and most of the gain over yearly has already been had. |
| 4 | 52 | 0.0192 | 2.6926 | 0.0257 | Weekly. The remaining gain is under one per cent of the factor. |
| 5 | 365 | 0.0027 | 2.7146 | 0.0037 | Daily, and the last column shows how little is left to gain. |
The rule built from that number has the property that its rate is itself. Everything
else follows: multiply the input by a constant and the chain rule from the earlier
lesson says the rate picks up that constant, which gives exactly the proportionality
wanted, and multiplying by a starting amount scales both sides equally.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents