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Unfortunately, parts of the section about integration are wrong and contribute to some of the widespread confusion about the FTC, which often seems reduced to a

Unfortunately, parts of the section about integration are wrong and contribute to some of the widespread confusion about the FTC, which often seems reduced to a mere tautology (which it's not; it's a deep result). 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.

swimfar 7y ago
FTC=Fundamental Theorem of Calculus
ajross 7y ago
You never got to the parts of the section about integration that are wrong. 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.
- yshklarov 7y ago
  The article purports to teach calculus, so it should get the basics right. The Fundamental Theorem of Calculus is called fundamental for a reason: it's the most important concept in the entire subject. It's not just a technicality. The article defines things in a non-standard way so as to reduce FTC to a tautology, and nowhere does it correctly explain the connection between differentiation and integration (which can -- and should -- be understood visually and intuitively). In short, I'm not convinced that the author knows what he's talking about.
  - oraphalous 7y ago
    Yeah well - next time I read a proof by a mathematician that skips several steps that to mathematicians are unimportant or "obvious" yet fundamentally blocked my understanding of the proof overall - I'll think on this reaction and smile... Or maybe even just write some mathematical overviews myself that leave out stuff that drives mathematicians insane with frustration... It would be justice...
    - gcc_programmer 7y ago
      Would you write code with syntax errors and blame the compiler for not understand how smart you are, or would you just learn the language and write correct code? Your frustration shows lack of understanding and appreciation for mathematics, when in reality maths and computer science are two sides of the same coin. If you can be good at one, you can be good at the other :)
      - ajross 7y ago
        > Would you write code with syntax errors and blame the compiler for not understand how smart you are 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?
- deleted 7y ago
  [deleted]
- Tainnor 7y ago
  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.
lsc 7y ago
Do you have a good reference or example or explanation of the FTC for folks who are learning calculus?
- Tainnor 7y ago
  I think for concepts like this different people might prefer different explanations, but I did some googling and this page seems to describe the heart of the theorem pretty well IMHO: http://www.intuitive-calculus.com/fundamental-theorem-of-calculus.html http://www.intuitive-calculus.com/fundamental-theorem-of-cal... (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.)
  - lsc 7y ago
    Thanks! 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.
    - Tainnor 7y ago
      Are you better with other kinds of maths? 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.
      - lsc 7y ago
        Nope, I'm not any good at any of the maths. My own model is that it's like reading, you need a critical mass of vocabulary and/or grammar (one can make up for the other, to some extent) before you can understand much of anything. with most subjects, if you understand 80% of the paragraph, you can fill in the blanks pretty easily, and thus your learning rate goes way up. 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.

swimfar 7y ago

FTC=Fundamental Theorem of Calculus

ajross 7y ago

You never got to the parts of the section about integration that are wrong. 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.

yshklarov 7y ago
The article purports to teach calculus, so it should get the basics right. The Fundamental Theorem of Calculus is called fundamental for a reason: it's the most important concept in the entire subject. It's not just a technicality. The article defines things in a non-standard way so as to reduce FTC to a tautology, and nowhere does it correctly explain the connection between differentiation and integration (which can -- and should -- be understood visually and intuitively). In short, I'm not convinced that the author knows what he's talking about.
- oraphalous 7y ago
  Yeah well - next time I read a proof by a mathematician that skips several steps that to mathematicians are unimportant or "obvious" yet fundamentally blocked my understanding of the proof overall - I'll think on this reaction and smile... Or maybe even just write some mathematical overviews myself that leave out stuff that drives mathematicians insane with frustration... It would be justice...
  - gcc_programmer 7y ago
    Would you write code with syntax errors and blame the compiler for not understand how smart you are, or would you just learn the language and write correct code? Your frustration shows lack of understanding and appreciation for mathematics, when in reality maths and computer science are two sides of the same coin. If you can be good at one, you can be good at the other :)
    - ajross 7y ago
      > Would you write code with syntax errors and blame the compiler for not understand how smart you are 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?
deleted 7y ago
[deleted]
Tainnor 7y ago
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.

lsc 7y ago

Do you have a good reference or example or explanation of the FTC for folks who are learning calculus?

Tainnor 7y ago
I think for concepts like this different people might prefer different explanations, but I did some googling and this page seems to describe the heart of the theorem pretty well IMHO: http://www.intuitive-calculus.com/fundamental-theorem-of-calculus.html http://www.intuitive-calculus.com/fundamental-theorem-of-cal... (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.)
- lsc 7y ago
  Thanks! 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.
  - Tainnor 7y ago
    Are you better with other kinds of maths? 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.
    - lsc 7y ago
      Nope, I'm not any good at any of the maths. My own model is that it's like reading, you need a critical mass of vocabulary and/or grammar (one can make up for the other, to some extent) before you can understand much of anything. with most subjects, if you understand 80% of the paragraph, you can fill in the blanks pretty easily, and thus your learning rate goes way up. 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.