Microscopes and Differentiable Functions

Now we can easily generalize Tall's example about the role of microscopes (Tall, 1982, 2001). An infinitesimal increment $\Delta x$ of a differentiable function $f$ from its point $x$ can be written as follows

$$f(x+\Delta x)=f'(x)\Delta x + f(x) + \varepsilon \Delta x$$

where $\varepsilon$ is infinitesimal. Thus, we can fix $(a,f(a))$ on the graph and point on it an optical $\Delta x$-lens to magnify infinitesimal details that are too small to see to the naked eye. We have

$$\mu(x,y)=\left(\frac{x-a}{\Delta x} , \frac{y-f(a)}{\Delta x} \right).$$

An infinitely close point $(a+\lambda,f(a+\lambda))$, when viewed through $\mu$, becomes

$$\mu(a+\lambda,f(a+\lambda))=\left(\frac{\lambda}{\Delta x} ,\frac{f'(a)\lambda + \lambda\varepsilon }{\Delta x} \right).$$

Suppose that $\lambda$ is of the same order as $\Delta x$, i.e. $\lambda/\Delta x$ is finite. This means that $\lambda\varepsilon /\Delta x$ is infinitesimal. By taking the standard parts, we have

$$\left(\st \left(\frac{\lambda}{\Delta x}\right), \st \left(\frac{... ...bda}{\Delta x}\right), f'(a) \st \left(\frac{\lambda}{\Delta x}\right)\right).$$

If $a$ is fixed, putting $\st (\lambda/\Delta x)=t$, we see that the points on the graph in the field of view are mapped in the straight line $(t,f'(a)t)$, where $t$ varies (see Figure 2). Note that the slope of the line is, in effect, the derivative of $f$ in the point $a$ and the function is really indistinguishable from its tangent in an infinitesimal neighborhood of $a$.

Figure 2: A graph of a differentiable function through an optical $\Delta x$-lens.

In the following sections we will describe some interesting new mathematical situations in which such lenses can be used to construct a suitable mental representation.