Posted by E-Reverance 10 hours ago
If you're looking for a more formal approach, ie the infinitesimal analogue to the usual real analysis, it's called nonstandard analysis and you could probably start with the original, eponymous book written by the creator, Abraham Robinson, for which I unfortunately don't have a link.
If this stuff interests you btw I would also check out Knuth's book on surreal numbers[3], which I believe, in some sense, are the fullest possible extension of what we think of as numbers? But it's been a while since I read into those.
[0] https://www.bravernewmath.com/ [1] https://intellectualmathematics.com/calculus/ [2] https://people.math.wisc.edu/~hkeisler/keislercalc-06-03-26.... [3] https://people.math.harvard.edu/~knill/teaching/mathe320_201...
Generally most of these 'handwavy' notations are rigidly provable, but only under general assumptions, that might not be true in special cases.
First, limits are harder than derivatives. Historically, humans figured out the derivative in the late 1600s, but the modern rigorous definition of the limit didn't exist until the 1800s. Slow-walking the definition of the derivative doesn't fix the problem of understanding limits.
The limit of a function f at a point x is defined as the value y, if it exists, such that for all d > 0 there exists an e > 0 such that for all x' in [x - e, x + e] we have |y - f(x')| < d. That's an earful. But for essentially all limits in introductory calculus we evaluate using two rules: the limit of a continuous function f at a point x is f(x), and the squeeze theorem. So my suggestion is to elevate these to the status of axioms. Introducing another number system does not help when students will not do anything nontrivial with it anyway.
The second problem is that "introductory" calculus includes too much material and is consequently pushed too late in the curriculum and seen as a weed-out course. Students spend too much time on "preparation" that doesn't prepare them for calculus. Studying logarithms and trigonometry is orthogonal, so basically all of "precalculus" is not actually pre-calculus. To me a four-year high school curriculum could be written up just fine with two years of algebra and geometry (not separated), one year of calculus and then statistics, which provides an ideal application for the theory of derivatives when you learn regression. But the author has included multivariable calculus and fiddly techniques for taking derivatives of ugly functions into "introductory" calculus. I think this is a step in the wrong direction. Laborious algebra calculations can be moved into an optional methods course for engineering students; we should be ensuring the core ideas are as accessible as possible so that doctors don't write papers about the trapezoid rule anymore:
https://diabetesjournals.org/care/article/17/2/152/17985/A-M...
The only major change that I'd like to make is the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity.
It immediately leads to the notion of the derivative. And after that, it's just a lot of building blocks.
Strongly disagree.
Sequences(discrete) and Convergence are vital to understanding Calculus. Only then the idea of converging to a limit from left or right makes intuitive sense. Pair it with a graphical view of secants converging to a tangent(continuous) and you get the idea of instantaneous change however infinitesimal it might be.
You need both discrete and continuous ideas to build intuition before you introduce limits of functions and continuity.
Some books that i have found useful - https://news.ycombinator.com/item?id=49308281
But they are not immediately needed to understand the limits.
Try to see how far you can get just with the epsilon-delta formulation of limits of functions.
Ah yes. Algebra. The subject all students love dearly. If only we could get students to love and appreciate calculus as much they love algebra!
I have a hard time even imagining an article that is more disconnected from the reality of teaching calculus to tiny humans.