Infinitesimal calculus was discarded in late 19th century pedagogy in favor of limits as no rigorous proofs yet existed of its general correctness, whereas such proofs had been formulated for the limits version. It was suspected that while infinitesimals were a nice intuition, they lacked rigor which might lead to unknown errors.
It was eventually rigorously proven in the 1960s that the infinitesimals approach is indeed as complete and correct as the limits version, but by then standard pedagogy had become entrenched in limits.
More on this at https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini... , the modern textbook that re-introduced infinitesimals.
https://en.wikipedia.org/wiki/Nonstandard_analysis
Fun stuff.
I loved real analysis in college, for example I found cantors set kind of crazy. Does that "exist" in nonstandard analysis? Or are there other mind-bending implications?
This is a bit of a philosophical question which hinges on what the word "equivalent" means. I disagree with the other comment. It is not totally equivalent, or even at all equivalent.
I think about math in terms of set theory. You create sets, like the real numbers, the integers, or otherwise. You can add other elements to those sets. For example:
* You can add a variety of infinities (countable, uncountable, positive, negative, etc.). You have a self-consistent system.
* You can add imaginary numbers, think of the whole thing as a ball, with a zero on one end and infinity on the other. That gives you complex analysis, which is self-consistent as well.
All of those are helpful, useful, and powerful, and generally lead to the same place where they overlap, they're not very compatible with each other, either formally or intuitively.
I view the two formulations of calculus as similar. Even if they lead to the same results, that's not the same as saying they're equivalent. They're different formalisms.
If your local educators are not required to participate in this system of continuing education and improving techniques, that's a local government decision and you should seek to fix that.
Fixing it is often hard in the states because there is a not insignificant amount of Americans that see schools as undermining their authority, see education as a useless and undesirable endeavor, often as a "radicalizing" and "socialist" influence and institution, and so they fight back against any improvement of schooling. There are states in the union that allow pretty much anyone, with no qualifications or certifications, to become "educators", and other states in the union that require a masters degree and 12 credits of class work every few years to continue to be an educator. Outcomes often imply a correlation between those requirements and the result, and yet the states with much lower bars of entry do not increase them. Education in the states is a politics issue, for some stupid reason. There are people who believe for whatever reason that middle schoolers are being taught "Critical Race Theory", as if that isn't a college level course that the vast majority of humanities majors don't even experience.
Limits -> Derivatives -> Integrals -> Series.
Introduction
The Concepts of Integral Calculus
Some Applications of Integration
Continuous Functions
Differential Calculus
The Relation Between Integration and Differentiation
The Logarithm, The Exponential, and The Inverse Trigonometric Functions
Polynomial Approximations to Functions
Introduction to Differential Equations
Complex Numbers
Sequences Infinite Series, Improper Integrals
Sequences and Series of Functions
That book was widely used at Caltech, MIT, and several other top schools for decades.
Also, there's basically one way to differentiate (not really, but kind of) wheres integration is so tricky there are many, many ways to do it.
There's a reason it was developed first, historically speaking.
Learning should be like peeling onions. You peel off one entire layer, from all sides, before moving on to a deeper layer of complexity.
By the time you know about drawing lines on planes to find the integral area underneath, you've already figured out the derivative so you know what direction to move your pen in next while drawing the line
I think these are actually pretty abstract, so it's hard exactly to grasp what a derivative is until maybe Calc II or III -- imo, the main problem is that it just seems like mathematical black magic: you need to define a limit, you need to use that definition to subtract two slopes, the complexity of which magically kind of vanishes, and you end up with a "method" for spitting out derivatives. Whereas finding area under a curve seems like a common sense thing to do: you take smaller and smaller slices of a thing and add it all up.
I do agree that if you really learn anti-derivatives (which, by all accounts, is harder), you basically get derivatives for free.
Curiously his book on mathematical analysis is excellent, except that many of the exercises were downright wrong.
For many students, intuitively understanding the infinities and infinitesimals in integral calculus come easier than those in differentiation.
The best way to intuitively and comprehensively grasp this is to learn Calculus in this order:
Algebra -> Functions -> (Calculus starts here) Finite differences -> Finite sums -> Infinity/Infinitesimals -> Epsilon-Delta
Unfortunately, it would be an understatement to say that pretty much no calculus learners learn it that way. No wonder why calculus stays so foreign to people. They are mostly trained to get A's by mechanically calculating values based on memorized tables, all while completely missing the bigger picture.
That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).