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I think I did, but I'll be more precise. The first problem is that the FTC is reduced to a tautology. But the more serious problem is that the definite integral

I think I did, but I'll be more precise. The first problem is that the FTC is reduced to a tautology. But the more serious problem is that the definite integral is _defined_ via the antiderivative. This is wrong, as the existence of integrable functions without antiderivative shows. 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.

ajross 7y ago
So... if it were made clear that the discussion was specific to continuous functions, your criticism would disappear? Is that really worth the level of your dismissal? 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.
- Tainnor 7y ago
  I don't feel you've actually read and understood my criticism. 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.
  - ajross 7y ago
    > 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" is [like, really bad!] 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.

ajross 7y ago

So... if it were made clear that the discussion was specific to continuous functions, your criticism would disappear? Is that really worth the level of your dismissal? 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.

Tainnor 7y ago
I don't feel you've actually read and understood my criticism. 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.
- ajross 7y ago
  > 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" is [like, really bad!] 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.