First, the "indefinite integral" is really just the antiderivative (and as such, "the" indefinite integral is only unique up to a constant). This doesn't follow from the 1. FTC, but purely by definition.
OTOH, the definite integral is not "defined" as being the difference of antiderivatives; it's defined in terms of Riemann sums (at least in elementary calculus), as explained further below.
What the two parts of the FTC do is proving that those two notions, which have no a priori reason to be related, are in fact related in a particular way.
The first part of the theorem says that, if f is "nice" (in particular: continuous), the antiderivative exists and can be expressed through the definite integral with a variable upper bound.
The second part says that, if f has an antiderivative, the definite integral can be computed using that antiderivative.
It's important to keep this distinction because there are e.g. functions that are integrable but don't have an antiderivative (e.g. a function with a "jump"). In such a case, the FTC tells you nothing and you have to go back to Riemann sums to compute the definite integral.
Broadly: your criticism is a mathematician's nitpicking. I don't know that that really has much value to someone learning "Calculus" in a practical sense about understanding and working with functions. And frankly... I mean, the fundamental theorem of calculus is a historically important result and a relatively profound truth, but it's not that big a deal. If you really want to play the "you really need to know this important truth more deeply" game with amateurs, start with the incompleteness theorem or something.
I'm all for simplifying material, and I'm not saying this needs to be a mathematically rigorous treatise, but it should get the basic maths right, because there is already too much confusion out there.
FWIW, I think the FTC is deeply important and used all the time, but the first part of it - the one about writing antiderivatives as integral - could probably be omitted for non-mathematicians.
It would probably enough to establish:
- Here's what differentiation is. Here's some rules.
- The inverse of that is antidifferentiation. You compute it by going in reverse, but note that this is harder than differentiation (which is mechanical)
- Then we have the integral via Riemann sums. Maybe you can show some simple example.
- Now comes the deep result: the integral can be expressed via the antiderivative, if it exists.
You don't have to go into full mathematical detail, but this presentation wouldn't be getting it wrong anymore.
I mean, that's how it's actually taught to high school students, after all. Complexities like discontinuities are absolutely not part of the initial curriculum.
Alternatively, if you think a simple introduction to calculus requires a discussion of discontinuous functions and the fundamental theorem of calculus before it describes how to compute a definite integral, I retreat to my earlier point: that's ridiculous nitpicking.
Basically: you want a textbook. They make those. This isn't that, it's an intuitive guide for people who found the treatments in textbooks opaque. Those people exist, they want to learn this stuff too, and they aren't well served by people like you just yelling at them to read the textbooks they already have.
The fact that the theory falls apart for discontinuous functions is just a visible symptom of the deeper problem that the author confuses a definition with a theorem. I don't think this is just theoretically wrong, it also makes no pedagogical sense. To say "we define the integral as being the difference of the antiderivative at the two endpoints, but we can also define it as the Riemann sum" doesn't tell the student anything except "mathematics is magic, don't even bother to understand it". Whereas to be genuine and say "we can prove that the integral can be expressed through the antiderivative (though we choose not to do this here)" doesn't treat the reader like an idiot.
I don't think it's impossible to provide an intuitive and non-proof-based account without sacrificing getting things right conceptually. In fact, I provided a template for doing so above.
How about if there was a footnote on that point explaining that these two definitions are actually equivalent (for the continuous functions we're talking about) and that the proof is really interesting and can be found in your textbook? Would that meet your requirements?
Again, this is just nitpicking. In fact those two definitions are equivalent, as you keep pointing out. You don't have to prove everything in an introductory treatment, and in fact even high school textbooks on this subject don't even try.
In short, I'm not convinced that the author knows what he's talking about.
Or maybe even just write some mathematical overviews myself that leave out stuff that drives mathematicians insane with frustration...
It would be justice...
But I think that it's a false dichotomy to only consider "extremely rigorous" or "simplified to the point of being wrong" as the only teaching philosophies. There is a middle ground where you can be somewhat hand-wavy and inexact without getting the conceptual broad strokes wrong.
No, but you might tolerate example code with mild syntax errors (like using a "..." to skip unrelated material, or skipping import declarations and boilerplate to better clarify the point being made). You don't demand that every example in a programming tutorial be a fully buildable piece of software, why are you demanding that a simpified calculus tutorial exhibit full rigor?
Aren't they just suggesting a marginal amount of rigor to communicate that the "definition" is in fact based on justifications which render it non-arbitrary?
(sidenote: the first theorem is technically only valid for continuous functions, but I think it's ok to omit that detail in an intuitive treatment.)
my own strategy is to try to get it explained to me in as many different ways as possible; I think it's often easiest to learn by integrating a bunch of different viewpoints of the same thing.
on the other hand, I haven't had a lot of luck with calculus, so who knows.
IMHO many people struggle with maths because they're not used to the fact that you have to read maths differently. You can't just digest it quickly like many other subjects. You have to ponder over definitions for a while and play with them and try to come up with examples and counterexamples.
That, and I think like reading, it takes a lot of practice, and that practice is really hard before you are good enough at reading to see the story. I think I'm at the point in math where I can't read it well enough for it to tell me a story, so practice is a real slog, just like it was for reading at that point.
If you're having trouble with specific material (e.g. calculus), sometimes the reason might be that there's something else that you don't quite understand well enough yet that is a prerequisite. In such a case, the first step would be to identify what that is (it could be unfamiliarity with trigonometry or a lack of skills in algebraic manipulations) and then go back and refresh that.
> We all know that the slope of a linear equation/function has a constant value
Er... no. We don't "all" know that.
Actually, I'd say that 99.99% of the population of the planet has no idea what that means.
I find that people who are very familiar with a subject usually have a difficult time explaining it without slipping into jargon that will lose the listener.
Also Shoshin, "beginner's mind":
At most, he could have written "The slope of a linear equation or "line" is constant. This is a known fact."
If you shrink equations, then if someone wants a closer look they can zoom in/out. But if you have them scroll off the side of the view, it is impossible to see the whole equation at once.
If you really want to understand these things, start here: https://www.youtube.com/playlist?list=PL5A714C94D40392AB and watch them as you work through Analysis I by Terence Tao, skipping to chapter 2 where he begins from scratch introducing the naturals, integers, etc. Wildberger even shows you how to do algebraic calculus which will help to learn the Tao book when he goes about rigorously introducing calculus defined using limits. For me that helped anyway, seeing Naturals, Ints, Rationals, and Reals constructed from scratch with all their laws/operations defined and proved by Tao then him using analysis to introduce how these tiny changes in calculus work.
To wrap up Gelfand is one bird in the hand, Apostol chapter 1? with integration from stratch is another bird in the hand. That would be a good first two steps to get a sense of what calculus is really about.
Both are outstanding.
3D graphics is mostly vector maths and linear algebra but you can do some calculus too. Derivatives are tied to continuity and continuity is tied to smoothness, which make things easy to visualize.
Funny, I don't think I realized it at the time, but is calculus the first time we are formally introduced to the concept of a function in math? If so, that gives me a new found respect for calc class :)
And then once students are initiated into the (disappointing) mysteries of calculus, what next? They spend the next few years mechanically carrying out analytic differentiations and integrations of a bunch of ever-more-artificial one-dimensional functions. Do you know what the integral of tanh (x + x^x) is? You shouldn’t, because it doesn’t fricking matter! Most of the actually useful integrals that anyone would ever calculate in real life turn out to be only solvable numerically.
In conclusion, put an introduction to the intuitive ideas behind calculus earlier (Year 7), emphasise numerical over analytic solutions, and use the time saved to move on to things like multivariable calculus (and path integrals, which I still don’t understand properly).
Path integral may refer to:
Line integral, the integral of a function along a curve
Functional integration, the integral of a functional over a space of curves
Path integral formulation by Richard Feynman of quantum mechanics
I find integrals useful.
Also I see my computer science pals have a lack of faith in math. I don't understand why.
I never needed actual Math in my career so far.
I would be impressed if he came up with it before or after that time.
It made it click for myself and it's fun to do!
I tend to get distracted when the notation in a field is awkward to me. Statistics confuse me with notations that resemble function applications, but are not. I like how physics customize the notation to their needs, like the use of brackets in quantum mechanics.