Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense.
Can someone elaborate on this concept and whether there are any other things that are useful to know for a student in introductory calculus?
Ed Nelson had demonstrated in his book Radically Elementary Probability Theory that a good part of probability can be made clear to freshmen without appealing to measure theory by using the Mises frequency approach. There are other gains for various courses. But above all nonstandard analysis allows students to understand better the triumphs and tragedies along the path of mathematics. Our ancestors Wallis, Gregory, Barrow, Newton, Leibniz, Euler, Cauchy, and many others were geniuses and to try and understand their ways of thinking is much better than to arrogantly accuse the great masters of being second rate thinkers in any style similar to that of Kline, for instance: "The net effect of the century’s efforts to rigorize the calculus, particularly those of giants such as Euler and Lagrange, was to confound and mislead their contemporaries and successors. They were, on the whole, so blatantly wrong that one could despair of mathematicians’ ever clarifying the logic involved." We learn math to fit life, and infinitesimals are part of the life of humankind.
One useful concept is the Leibnizian distinction between assignable and inassignable number (according to historian Eberhard Knobloch, the distinction originates with Cusanus; Galileo's distinction between quanta and non-quanta is also traceable to Cusanus). In Robinson's framework this is implemented in terms of a distinction between a standard and a nonstandard number. Thus, ordinary real numbers are standard, whereas infinitesimals and infinite numbers are nonstandard. The sum $\pi+\epsilon$ where $\epsilon$ is infinitesimal is also nonstandard. The two domains are related by the standard part function, also known as the shadow. This is defined for any finite hyperreal. The standard part rounds off each finite hyperreal to its nearest real number.
To illustrate how this is useful in calculus, note that the derivative of $y=f(x)$ can be computed as the shadow of $\frac{\Delta y}{\Delta x}$ where $\Delta x$ is an infinitesimal $x$-increment and $\Delta y$ the corresponding change in $y$.
Short opinion answer.
I think the fact that the use of infinitesimals can be made rigorous is the most important contribution of nonstandard analysis at the elementary calculus level. That should free students and instructors to work confidently with the intuition infinitesimals provide. I don't think it's useful or necessary to work with formal nonstandard analysis, any more than it's necessary to provide a definition of the reals sufficient to work rigorously with ordinary analysis.
