Calculus! #42: Early Transcendentals 3.8

Exponential Growth and Decay

This section is actually a brief introduction to differential equations. A differential equation is pretty much what it sounds like – it’s an equation that involves a function and its derivative. One of the simplest versions of that would be y=y’.

Also simple is y = ky’, where k is some constant. The book phrases this dy/dt = ky, but it’s the same thing.

What does this mathematical phrase mean? I like to read it as “the value of the function at a given point is proportional to the value of the derivative at that point.”

Eventually we’ll be studying differential equations here, and we’ll get into how to explicitly solve here. HOWEVER, you may remember a certain type of functions which has a derivative proportionate to itself: exponential functions!

Let’s take the function f(t)=e^{kt}. (Note, I used t as the variable since the book did, but I could’ve also used x or p or n or whatever)

If we take the derivative, we get ke^{kt}. So, we see that the derivative of such a function is equal to k*(itself). Nice.

You may notice that there is an entire type of function that obeys that rule. It’s of the form f(t) = Ce^{kt}, where C is some constant and k is some constant. You may also notice that the equation gets easy to solve at f(0), because the k disappears, meaning the equation simply reduces to f(0) = C.

This is very important. Why? As it happens, exponential equations of this form are easy to solve at t=0. But, equally handy, real life systems are often easy to understand at t=0. That is, it’s often easy to know what’s going on right at the beginning. Let’s get into some examples.

Population Growth

dP/dt = kP, where P is population, t is time, and k is some constant.

How do you read that? There are many ways, but I like to say it as “The change in population is proportional to the current population. Makes sense. If you have two rabbits breeding, there’ll be fewer offspring than if you have 20 rabbits breeding. For a simple system, the more individuals you have, the faster the population grows.

When we solve like we did last time, we get this:

P(t) = Ce^{kt}

I read that as “population grows exponentially with time at a rate of k.” If we know the initial population, we can easily solve for C by setting t=0.

Pretty simple, I think. The nice thing about differential equations is that you experience them in real life. If you have two cats and they have kittens and their kittens have kittens, the growth rate definitely feels exponential and depends on the number of cats you start with (P at t=0), the rate at which the breed per time unit (k), and the amount of time that has passed.

Radioactive Decay

Similarly, you can have a situation where the population is decreasing. The book gives the example of a substance losing mass over time. This is basically the same as rabbit breeding, except the rate is negative.

So, we have an equation of the form:

m(t) = {m_0}*e^{kt}

In this case, we know k must be negative, since m(t) has to decrease over time. You’ll notice that for negative k, as t increases, the right side of the equation becomes a smaller and smaller fraction of its initial value, approaching closer and closer to 0. Once again, this makes a lot of sense.

In the case of matter decaying, it also makes sense that it takes longer and longer to lose the same amount of mass. Why? Radioactive decay involves crazy quantum shit. Suffice it to say that it’s a probabilistic process. So, like with lottery tickets where you’re more likely to win if you have more tickets, with matter you’re more likely to get radioactive decay if you have more atoms.

With that in mind, the book actualy shows you how to calculate what’s called a “half-life.” A half-life is simply how much time it takes for half the the matter to decay away. The cute thing is that you can actually solve for this.

Since you know the initial mass, you can easily calculate half the mass. Then the equation simply becomes:

m(t) = m(0)/2 = m(0)e^{kt}

Assuming you know the rate of decay, that leaves only one value to discover – t. Using natural logs, you can easily solve. Pretty neat, right?

I remember being in chemistry class and wondering what the hell ln(2) had to do with figuring out how long it took for half of a chemical to react. The above equation explains!


The book goes on to give several more example, but I think by now you’ve probably got the basic idea. The simple exponential equation is incredibly useful and applies in lots of places. Of course, you have to remember that it’s a simplification many times. For example, our bunny population cannot grow infinitely in the real world because there would have to be infinite food. So, in a more realistic model, you’d have to include some modification. Or, in the case of banking, you might have to add some sort of cap, since so-called “Methuselah Trusts” are illegal in some places.

However, generally speaking, the Ce^kt format appears over and over in many fields. It’s basic shit, and you should have it as part of your working knowledge. When you see Ce^kt, you shouldn’t think “okay, the C is some constant, and then e is the base and k…” You should see it and think “ah, something’s growing exponentially!”


Next stop: Related Rates



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