Chapter 4

Why Exponentials Are Magic

Is there a curve whose slope is just a copy of itself?

Multiplying, again and again

Theo: Last time, the coffee's gap to room temperature shrank at a rate proportional to the gap itself. Big gap, fast shrinking. Small gap, slow shrinking. What kind of curve does that?

June: Chapter 3 already dropped a hint. Every little step multiplied the gap by the same number. So let's start with the simplest multiplying machine there is: doubling.

Take something that doubles every second. After 1 s you have 2. After 2 s, 2 × 2 = 4. After 3 s, 2 × 2 × 2 = 8. We write that as $2^t$ : the little t up top counts how many times we multiply by 2.

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Drag the number of seconds below zero, or onto a half second. Zero seconds means no doubling yet, so $2^0 = 1$ . One second back undoes a doubling: $2^{-1} = \tfrac12$ . And half a second has to be a number that, used twice, makes one doubling: $\sqrt2 \approx 1.41$ , because 1.41 × 1.41 ≈ 2.

Mika: So the exponent doesn't have to be a whole number. It's just "how much multiplying has happened", and that can be a bit, a lot, or even negative.

Here are those doublings as a ladder of bars. Split each step into halves, then quarters, then tiny slivers, and move the magnifier around:

Chapter 4: Why Exponentials Are Magic ↗

Mika: Wherever I put the magnifier, the step rises by the same share of its own height. Tall bars rise a lot, short bars rise a little, but always the same share.

Theo: That's exactly the coffee's property: the change is proportional to how much there is. And look at the last readout. As the steps shrink, rise ÷ height per second closes in on a number, about 0.69 for doubling. Hold that thought.

Growing in tiny steps

Mika: I've seen $e^t$ in textbooks. Isn't $e$ just some random number mathematicians like? 2.718-something, whatever.

June: Let's not argue about it. Let's build a growth rule and see what number falls out.

Here's the rule: grow by 100% of what you have, per second. The lazy way to follow it is one giant step: after 1 s you have 1 + 1 = 2. But that's June's giant-step trick from Chapter 3. For the whole second we used the growth rate from the start, even though the pile kept getting bigger.

So take smaller steps. Two steps of 50%: 1.5 × 1.5 = 2.25. Four steps of 25%: 1.25 × 1.25 × 1.25 × 1.25 ≈ 2.44. Every step grows a pile that has already grown.

Mika: More steps, more growth! With infinitely many steps we'd get infinitely much. I'm going to be rich.

Predict first! Keep splitting the second: a thousand steps, a million, as many as you like. What happens to the total after one second?

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Mika: It stalls! A hundred steps, a thousand, and it never gets past 2.718. So much for my infinite fortune.

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That ceiling is the number $e$ . It's what the steps close in on when they get infinitely small:

e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{n} \approx 2.71828

Theo: Now flip the widget to shrink. With $\left(1 - \frac1n\right)^n$ it lands on $1/e \approx 0.368$ : 36.8% left, so 63.2% of the way gone. That's the 63% from Chapter 3's time-constant ruler!

June: And that's my coffee! Cut one τ into n hand steps, $\Delta t = \tau/n$ : each step multiplies the gap by $1 - \frac1n$ , so after one τ the gap is $\left(1 - \frac1n\right)^n$ of what it was. One giant step ( $n = 1$ ) lands exactly on room temperature. Tiny steps leave 36.8%.

A curve that copies itself

Now let's hunt with a ruler. We want a curve whose slope, at every point, is a fixed multiple of its height. The ladder says every exponential $b^t$ should qualify: $2^t$ , $3^t$ and friends. Put a tangent on each curve and divide its slope by its height.

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Drag the cursor along any one curve and the slope ÷ height readout doesn't budge. Every exponential copies itself, scaled by a fixed number. For base 2 that number is about 0.693 (the number the ladder's shrinking steps were closing in on). For base 3 it's about 1.099.

And for exactly one base the scaling number is 1, so the slope curve lies right on top of the curve itself. Which base is it?

\frac{d}{dt}\,e^{t} = e^{t}

Mika: It's 2.718 again! The same number the tiny steps gave us. Two completely different hunts, one number. Okay, I take it back: $e$ isn't random at all.

Theo: Of course it's the same. "Grow by 100% of what you have, per second, smoothly" means slope = height. It's one wish wearing two costumes.

Speeding it up: e^(at)

Real systems don't all change at the same pace, so let's put a knob in the exponent: $e^{at}$ . The number a stretches or squeezes time.

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Now slope ÷ height is exactly a, everywhere on the curve. Picture why: $e^{at}$ is the film of $e^{t}$ played a times faster, so every slope is a times steeper. Positive a: it grows (and grows faster and faster). Negative a: it shrinks towards zero, like the coffee's gap. a = 0: it just sits there.

\frac{d}{dt}\,e^{at} = a\,e^{at}

The knob and the base are two views of the same thing: $e^{at} = \left(e^{a}\right)^{t}$ . Every second, the curve multiplies itself by $e^{a}$ .

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Predict first! Take the slope of $e^{2t}$ , and then take the slope of that slope. What do you get?

\frac{d}{dt}e^{at} = a\,e^{at},\qquad \frac{d^2}{dt^2}e^{at} = a^2\,e^{at}

Theo: Wait, that's huge. For exponentials, taking a slope (calculus) turns into multiplying by a (just algebra).

Guess an exponential

For a rule where the slope is just fixed numbers times the thing itself (nothing squared, no numbers that change with time), like our coffee rule, we can try an exponential: plug it in and see which value of a makes the rule true. Other kinds of rules may need other shapes.

The coffee rule was $\frac{dT}{dt} = -\frac{T - 20}{\tau}$ with τ = 10 min. Our guess is $T = 20 + 70\,e^{at}$ (70 is the starting gap). Tune a until the guess's slope agrees with what the rule demands.

Chapter 4: Why Exponentials Are Magic ↗

The algebra agrees with the slider. The slope of the guess is $a \cdot 70e^{at}$ (the 20 never changes, so its slope is 0, and the 70 just comes along for the ride). The rule wants $-70e^{at}/\tau$ . The $70e^{at}$ appears on both sides and cancels:

a \cdot 70e^{at} = -\frac{70e^{at}}{\tau} \quad\Longrightarrow\quad a = -\frac{1}{\tau}

Theo: And it lands exactly on the dots we built by hand in Chapter 3. Guess-an-exponential works. The equation that pins down a is called the characteristic equation.

The spring breaks our trick

June: Let's try it on something that bounces. A mass on a spring is basically our drone under P control: pulled back harder the further it is from the target.

The spring pulls back with force $-kx$ . Force = mass × acceleration, and acceleration is the slope of the slope (Chapter 3). So the rule is:

m\,\frac{d^2x}{dt^2} = -k\,x

Guess $x = e^{at}$ . The slope of the slope is $a^2 e^{at}$ , so $m a^2 e^{at} = -k e^{at}$ . Cancel $e^{at}$ :

a^2 = -\frac{k}{m}

With k/m = 4, we need a number whose square is −4. Hunt for one:

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Theo: Squares are never negative. The blue parabola never gets down to the green line. So there's no such a. Our trick fails: springs just can't be described with exponentials.

Mika: But the spring is right there, bouncing happily. It has a motion. Maybe it's not the spring that's wrong, maybe it's our idea of what counts as a number?

Theo: …Hmm. I'll admit I said "no number" a bit fast. Every number I know about squares to something positive.

What we know so far

What we figured out

$b^t$ means "multiply by b, t times", and t can be anything: $2^0 = 1$ , $2^{-1} = \tfrac12$ , $2^{0.5} = \sqrt2$ .
Growing 100% per second in ever tinier steps levels off at $e \approx 2.718$ . Shrinking the same way leaves $1/e \approx 0.368$ : the 63% ruler from Chapter 3.
Every exponential's slope is a copy of itself: slope ÷ height is constant.
$e \approx 2.718$ is the base where that ratio is exactly 1. With $e^{at}$ the ratio is a.
Taking a slope of $e^{at}$ just multiplies by a. Calculus becomes algebra.
Guess an exponential, plug it into the rule, and solve for a. For the coffee, a = −1/τ.
For a spring the guess demands $a^2 = -k/m$ , and no ordinary number does that. Yet.

The drone and the spring wiggle. Exponentials only grow or shrink. Unless… a number could square to a negative?

Chapter 4: Why Exponentials Are Magic ↗