Neocalculus

What is a function?

A function is a machine: something goes in, something comes out — always the same output for the same input. Functions are far more general than "plug in a number, get a number." They are the universal language of transformation:

A function can square a number, measure the length of a word, take the derivative of another function, or compute the divergence of a vector field. In every case the contract is the same: one input, one output, deterministic. In calculus we focus on functions from numbers to numbers — like $f(x) = x^2$ — but the deeper idea is universal.

Calculus studies how functions change: how the output responds when you nudge the input. To do that cleanly, we need to be precise about what kind of functions we allow.

Smoothness

Not all functions change the same way. Some flow gently — a ball arcing through the sky traces a smooth curve. Others break. There are two ways a function can fail to be smooth:

• A corner — like $|x|$ at the origin, where the slope changes abruptly.
• A jump — like a step function that leaps from 0 to 1, with no intermediate values.

In Neocalculus, we work exclusively with smooth functions: no corners, no jumps, no breaks of any kind. This is a modeling choice that matches many physical regimes and keeps the calculus workflow algebraic.

Zooming in: the key insight

Take any smooth curve. Pick any point. Now zoom in. What happens?

The curve flattens. The bends straighten out. And at the very bottom of the zoom — at a scale we call infinitesimal — the curve is represented by its straight, first-order part.

This is microstraightness. Try it yourself:

For smooth functions in this framework, zooming to infinitesimal scale makes the curve and tangent agree to first order. That infinitesimal straight segment has a tiny width we call d, and its slope is what we call the derivative.

Meet d

Infinitesimals live on the number line alongside 1, ½, and π. They obey one special rule:

When you multiply an infinitesimal by itself, you get exactly zero. But d itself is not zero — it has direction and slope. It is the mathematical atom of change: too small to measure, but large enough to carry information.

This is microstraightness stated algebraically: over a distance of d, smooth functions restrict to an affine first-order form, because curvature requires a $d^2$ term and $d^2 = 0$ on $D$.

We write $D = \{d \in \mathcal{R} : d^2 = 0\}$ for the set of all infinitesimals — the infinitesimal neighborhood of zero. Our number line $\mathcal{R}$ contains everything you already know (integers, fractions, irrationals) plus these infinitesimals.

Arithmetic with infinitesimals

The only new rule is $d^2 = 0$. Everything else is ordinary algebra.

• Constant multiples: $(3d)^2 = 9d^2 = 0$, so any multiple of an infinitesimal is infinitesimal.
• Higher powers vanish: $d^3 = d \cdot d^2 = 0$, and $d^4 = 0$, and so on.

Expand normally, kill every $d^2$ and higher, done. On $D$, the result has the form "number + coefficient × $d$."

A first taste: the slope of $x^2$

Take $f(x) = x^2$ and nudge its input by an infinitesimal:

The slope of $x^2$ at any point $x$ is $2x$. At $x = 3$, the slope is 6. At $x = -1$, it's −2. We just computed the derivative — and all we did was algebra in the infinitesimal model. No limit computation was needed for this derivation.

The word uniquely is the key. It means there is exactly one slope that works — and that uniqueness is what lets us extract the derivative by "reading off the coefficient of $d$."

This is the workhorse of every derivation in this book. Expand $f(x+d)$, apply $d^2 = 0$, and match the coefficients of $d$ on both sides.

How can $d^2 = 0$ without $d = 0$?

In ordinary algebra, $d^2 = 0$ forces $d = 0$. The proof goes: "Suppose $d \neq 0$. Divide both sides of $d \cdot d = 0$ by $d$. Then $d = 0$. Contradiction." But this argument assumes you can always decide whether $d = 0$ or $d \neq 0$ — that is, it uses the law of excluded middle.