![]() Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach. Keisler’s experiment teaching calculus with infinitesimals An approach to teaching introductory calculus with infinitesimals was developed by Keisler 2 based on a teaching experiment during 19734. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. In this treatment, calculus is a collection of techniques for manipulating certain limits. ![]() ![]() They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system. Limits describe the value of a function at a certain input in terms of its values at nearby input. In the 19th century, infinitesimals were replaced by limits. Leibniz gave a philosophical explanation for this fact: In nature everything bears the signature of an author with an infinite nature therefore infinitesimal. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals. This approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. ![]() From this point of view, calculus is a collection of techniques for manipulating infinitesimals. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the Archimedean property. An infinitesimal number dx could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". Historically, the first method of doing so was by infinitesimals. Our educational experience and the student reactions to our approach are detailed in this recent publication.Calculus is usually developed by manipulating very small quantities. Thus the students receive a significant exposure to both approaches. Section 2 of this article completes the proofs of Section 1 using Keisler's approach to the logic of infinitesimals from. The experimental and first edition of his book were used widely in the 1970's. To respond to the recent comment, a difference between our approach and Keisler's is that we spend at least two weeks detailing the epsilon-delta approach (once the students already understand the basic concepts via their infinitesimal definitions). Jerome Keisler developed simpler approaches to Robinson's logic and began using infinitesimals in beginning U. In fact, I did a quick straw poll in my calculus class yesterday, by presenting (A) an epsilon, delta definition and (B) an infinitesimal definition at least two-thirds of the students found definition (B) more understandable. Infinitesimals provide an alternative approach that is more accessible to the students and does not require excursions into logical complications necessitated by the epsilon, delta approach. The algebraic expressions concerned include power series in and allow the use of infinitesimal techniques for the calculus of analytic functions. The epsilon, delta techniques involve logical complications related to alternation of quantifiers numerous education studies suggest that they are often a formidable obstacle to learning calculus. This final step was taken independently by both Newton and Leibniz. To answer your question about the applications of infinitesimals: they are numerous (see Keisler's text) but as far as pedagogy is concerned, they are a helful alternative to the complications of the epsilon, delta techniques often used in introducing calculus concepts such as continuity. The quadrature problem eventually gave rise to integral calculus, which is the assimilation of the geometric and analytic methods and the understanding of the calculus of infinitesimals or infinitesimal calculus. The real numbers $\mathbb$ is algebraically simplified to $2x \Delta x$ and one is puzzled by the disappearance of the infinitesimal $\Delta x$ term that produces the final answer $2x$ this is formalized mathematically in terms of the standard part function.
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