Microscopes ``within'' Microscopes

This type of diagram was originally suggested and used by Keisler (1976a, 1976b), but not formalized by constructing optical lenses.

Let $f$ be a real function with continuous second derivative ($f \in C^2$). If we magnify an infinitesimal neighborhood by a more powerful tool than an optical $\Delta x$-lens, we can see other interesting properties of the curve. This is what we call a microscope ``within'' a microscope pointed in $(a+\Delta x, f(a +\Delta x))$ in the non-optical $\Delta x$-lens (because the optical lenses lose every infinitesimal details). By an optical $\Delta x$-lens pointed in $(a,f(a))$, both the curve $y=f(x)$ and the tangent $y=f'(a)(x-a)+f(a)$ are mapped in the line $(t,f'(a)t)$, where $t=\st (\lambda/\Delta x)$ and $\lambda$ is an infinitesimal of the same order as $\Delta x$. Now we can put $\lambda=\Delta x$ and point a $\Delta x^2$-lens in $(a+\Delta x, f(a +\Delta x))$. In order to visualize more details, we need to have more information about the function: our idea is to use the non-standard Taylor's second order formula for $f$ (see (Stroyan and Luxemburg, 1976)), i.e.

$$f(a+\Delta x) = f(a) + f'(a)\Delta x + \frac{1}{2}f''(a)\Delta x^2 + \varepsilon _1 \Delta x^2$$

where $\varepsilon _1$ is infinitesimal.

Thus the $\Delta x^2$-lens maps as follows

$$(x,y) \mapsto \left( \frac{x-(a+\Delta x)}{\Delta x^2} , \frac{y-f(a+\Delta x)}{\Delta x^2} \right)$$

and the point $(a+\Delta x, f(a +\Delta x))$ is mapped onto $(0,0)$. Let $\lambda$ be an infinitesimal of the same order as $\Delta x^2$. The Taylor's second order formula gives

$$f(a+\Delta x +\lambda)=f(a)+f'(a)(\Delta x + \lambda) +\frac{1}{2}f''(a)(\Delta x + \lambda)^2 + \varepsilon _2 (\Delta x + \lambda)^2.$$

Therefore, we have

$$\begin{multline*} (a+\Delta x + \lambda, f(a+\Delta x +\lambda)) \mapsto \left(\... ...Delta x \lambda - \varepsilon _1 \Delta x^2}{\Delta x^2} \right) \end{multline*}$$

and by taking the standard parts

$$\left(\st \left(\frac{\lambda}{\Delta x^2}\right), f'(a)\st \left(\frac{\lambda}{\Delta x^2}\right)\right)$$

as the other terms are all infinitesimals.

The point $(a+\Delta x +\lambda,f'(a)(\Delta x + \lambda)+f(a))$ on the graph of the tangent line is mapped in the point

$$\begin{multline*} \left(\frac{\lambda}{\Delta x^2} , \frac{f'(a)(\Delta x + \... ...ambda}{\Delta x^2} - \frac{1}{2}f''(a) - \varepsilon _1 \right) \end{multline*}$$

and then the optical lens gives

$$\left( \st \left(\frac{\lambda}{\Delta x^2}\right), f'(a)\st \left(\frac{\lambda}{\Delta x^2}\right) - \frac{1}{2}f''(a)\right).$$

Figure 3: A microscope ``within'' a microscope.

This suggests nice new (and mathematically justified, of course) mental representations of the concept of tangent line: through the optical $\Delta x^2$-lens, the tangent line can be seen as the line $(t,f'(a)t - \frac{1}{2}f''(a))$ which means that the graph of the function and the graph of the tangent are distinct, straight, and parallel lines in a $\Delta x^2$-neighborhood of $(a+\Delta x, f(a +\Delta x))$. The fact that one line is either below or above the other, depends on the sign of $f''(a)$, in accordance with the standard real theory: if $f''(x)$ is positive (or negative) in a neighborhood, then $f$ is convex (or concave) here and the tangent line is below (or above) the graph of the function.