Calculus I

The Function That Is Its Own Slope

Or: Why one strange number between 2 and 3 refuses to change

Companion notebook. Play with this chapter’s ideas — sliders, animations, and drills — in notebooks/ch5_exponentials.ipynb.

By the end of this chapter you will be able to do four things.

You will know what $e^x$ is, and — more importantly — why the number $e \approx 2.718$ is the natural base for an exponential, and not some friendlier-looking number like $2$ or $10$.

You will know that $e^x$ is its own derivative, and you will know this is not a magic trick or a coincidence to memorize. It is the reason we picked $e$ in the first place. We are going to go looking for a function that is its own slope, and $e$ is what we find.

You will know how logarithms undo exponentials, and you will be able to prove — not assert, prove — that the slope of $\ln x$ is $1/x$.

And you will be able to point all of this at the real world: money that grows, populations that explode, atoms that decay, and a drug leaving the bloodstream. By the end, “half-life” will be something you can derive, not something you look up.

One honest warning before we start. This chapter takes a small loan. We are going to use a fact about $e$ before we have fully earned it, and I am going to tell you exactly where the loan is and exactly when we pay it back (Chapter 10, once we have integrals). Borrowing a result you are confident in, using it, and settling the debt later is not cheating. It is how working mathematicians actually move. I want you to see it done in the open instead of pretended away.

The question nobody asks first

Most books hand you $e^x$ on a plate. Here is the function, here is its graph, here is the rule, memorize it. We are going to do the opposite. We are going to ask a question, and the number $e$ is going to be the answer.

Here is the question. Is there a function whose rate of change is equal to itself?

That sounds abstract, so look at where it comes from. Three completely different situations from the real world:

• Money in an account earning interest. The more money you have, the more interest you earn each year — so the rate at which the money grows is proportional to how much money is there.
• A population of rabbits, or bacteria, with plenty of food. The more rabbits there are, the more baby rabbits appear each month — the growth rate is proportional to the current number.
• A radioactive lump, or a drug in your blood. The more atoms (or drug molecules) are present, the more of them break down or get cleared each second — the rate of decrease is proportional to how much is left.

Strip the stories away and they are all the same sentence:

$$\text{rate of change of the amount} \;=\; (\text{a constant}) \times (\text{the amount}).$$

In the language we have been building, if $N(t)$ is the amount at time $t$, this says

$$\frac{dN}{dt} = k\,N.$$

This little equation is one of the most important sentences in all of science, and we will meet it again. For now, notice what it is asking of us. We want a function $N(t)$ whose derivative is a constant times itself. So before we can solve any of those three problems, we have to answer the bare question: what kind of function is proportional to its own slope?

That is the whole chapter. Let us go find the function.

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