Neocalculus

First: what is a function?

Before we study change, we need one simple piece of language. A function is a rule that accepts an input and gives back exactly one output.

We usually call the input $x$. If the function is named $f$, its output is written $f(x)$, read “f of x.” Thus $f(x)=x^2$ means “square the input.” For example, $f(3)=9$ and $f(-2)=4$. The parentheses in $f(x)$ do not mean multiplication; they tell us which input was given to $f$.

Watch one cycle of the machine below. A number enters from the left. The displayed rule acts on it, and the result leaves on the right. The dot on the small graph marks that same input-output pair.

Each cycle uses a different rule, but the notation keeps the same meaning. A graph is a picture of all the input-output pairs: when the input $x$ produces the output $f(x)$, the point $(x,f(x))$ lies on the graph.

Now we can ask the question that begins calculus: when $x$ changes, how does $f(x)$ change? To answer it at one particular input, we look closely at the graph near the corresponding point. Sometimes the graph settles toward one line direction. Sometimes it does not.

Which curves have a local direction?

For now, call a graph smooth at a point when magnifying near that point reveals one line direction. A parabola has one. At the corner of $|x|$, the directions from the left and right disagree. At a jump, the two nearby pieces do not even connect.

Look through a first-order lens

The purple dashed line in the demo is the tangent line: it passes through the chosen point and points in the graph’s local direction. Keep the parabola selected, turn on the tangent, and increase the zoom. Watch the gap between the curve and the line.

At ordinary scale, the parabola and the tangent are visibly different. As the window shrinks around the chosen point, the bend becomes harder to see and the graph looks more like the line. Numerical zoom reveals this pattern, but it never reaches a final smallest magnification.

Synthetic differential geometry asks an exact version of the zooming question. It represents a first-order change in the input by d and uses the rule

This d is not an ordinary real number that happens to be very small. It is a first-order displacement in the model. The term $d$ records the direct response to an input change; $d^2$ records a second-order effect such as bending. The first-order lens discards the latter. It does not discard terms such as $6d$: their coefficients still tell us how the output responds.

Now fix an input $a$. Smoothness says that, for first-order displacements, the output has one and only one form

Here $f(a)$ is the original output, and $md$ is its first-order change. The number $m$ is determined by the function and the point. We will calculate it before giving it a name.

This is microstraightness. For an ordinary finite change $\Delta x$, the line usually gives an approximation: $f(a+\Delta x)\approx f(a)+m\Delta x$. For a first-order displacement $d$, the equation $f(a+d)=f(a)+md$ is exact. The curve has not become a line everywhere; curve and line agree only at first order near the chosen point.

Finding the local coefficient

Now let us find $m$ for the squaring function. Start at the input 3 and change it to $3+d$. Expand in the ordinary way, then use $d^2=0$:

The answer has two parts. The 9 is the original output. The coefficient 6 tells us the first-order change: near 3, the squaring function changes by $6d$ when the input changes by $d$. In the microstraightness equation, this says $m=6$ at the input 3.

Now let the starting input be $x$ instead of 3. For $f(x)=x^2$:

This has the form $f(x+d)=f(x)+md$ with $m=2x$. Because the starting input can vary, the local coefficient varies with it. This coefficient has a name.

A derivative at one input is a number: the slope there. But let the input move, and each point supplies its own slope. Those slope values together form a new function, $f'$. Watch the point travel along each curve below. The purple tangent turns as the local slope changes; the function that records all those slopes emerges on the right.

The trigonometric outputs are a preview. In Chapter 2, we will derive them instead of asking you to take them on faith.

For $f(x)=x^2$, we found $f'(x)=2x$. At $x=3$, the slope is 6. At $x=-1$, it is −2. The sign tells the direction; the size tells how steeply the output responds.

Notice what we did not do: we never divided by $d$. We expanded the function and read the unique first-order coefficient.