gregpetrics

Good feedback.

I struggled with deciding if I should write activities that illustrate the full algorithm for derivatives and antiderivatives. At this time I left it out, but I do have the materials...

The book was written with a bit of a promise to keep the algebra out, and overdoing it on Monkey Rules (derivatives) and Lucifer's Rules (antiderivatives) breaks that promise. That said, calculating derivatives and antiderivatives is the fundamental algebraic task of a calculus student.

I'm thinking about your feedback right now... and will likely make adjustments in the near future to introduce optional tracks for extra practice on this.

That's really sad.

You might check out the sequence of activities in the Integral chapter of this book to get another perspective on the FTC.

I totally agree. It's the entire point of calculus, and it's what sets the entire field of Differential Equations in motion!

Whoops. Page 47. I paraphrase: Sequences "converge" if there exists an N such that the sequence stays within epsilon of p (the mark) for all indexes larger than N.

I know it seems formal because it adheres to a certain structure, but even this is informal at a fundamental level.

Specifically, how can we be sure we can perform a countably infinite number of distance measurements in the metric space to be sure the sequence stays close to p? (this is the infinite stack of inequalities I alluded to)

He doesn't say. Implicitly, Rudin is saying here that this definition of "converges" is good enough. And he's not wrong. It is a very good definition. To me at least this is Rudin, the towering statue of formality, being informal.

He could/should have actually gone down to a more fundamental level and whipped out mathematical induction as an axiom to assure us that we can do such things, but then that would have taken him off his narrative goal, and also probably lost even more readers. Furthermore, even if he did so, an axiom is an assertion that "you just have to trust me on this one."

Now look, I'm not bashing formality. I'm a huge fan of it, and teach upper level math classes formally. But it has it's place and it is NOT in Calculus 1. Furthermore, I think folks need to realize that even the most formal of treatises have informalities buried in them at the very least in the form of stated axioms.

I agree Linear and Combi is far more useful once you get going on a technical degree and/or career, but go to a university in the US and check the prerequisites on these courses: CALCULUS.

This was another reason I wrote this book. For people who just need to get through calc, here's some help that you can pick up and read in a couple hours.

This.

I've Seen it time and again: Great mathematicians who are awful mathematics students get turned away. They end up making great full stack engineers.

Author of book here. Nice comment, and very interesting question.

I am not worried about using "informality" to get more people studying mathematics.

The book is informal, but by the end of the book, the integral that gets presented is the correct definition of the integral. I've just collapsed as much of the technical language at possible and focused on the core idea. My thinking is: if someone is hooked, sure they'll run up against walls if they try to use my book and only my book, but that would be the time to turn to Stewart (famous Calc text) or comparable. My thinking it that at that point the student is ready for "rigor" and "formality", and they won't even think twice about. They might even appreciate it. I've seen it happen over a decade of calculus teaching. It happens more than you think.

But to take this a little further, I believe the "formality" you mention actually hides a fundamental and insidious truth about mathematics: Mathematics fundamentally is informal. Burrow down deep enough into the epsilon/delta of limit definitions, and you'll see at the bottom is what amounts to an informal "this is good enough I guess".

For instance, at the bottom of epsilon/delta definition of what it means to converge in Baby Rudin (pg. 46), he essentially says "if you can get sequence within epsilon of the target anywhere past N" that's good enough. But why?! There is no more unpacking or additional fundamentalism at that point. How can we be sure we can make a claim about an infinite set of inequalities? Do if/then statements work this way? How can we be sure we can use the natural numbers this way? That fundamental informality then persists throughout the text. It's fine of course, and this is the agreed upon way to do mathematical calculus, but it's also a fundamental informality.

From my point of view (and this is part of what got me writing this book in the first place): why bother going all the way "down there" just to say "good enough"? Why not say "good enough" a lot higher up the ladder closer to where the problem originated.

I'm hardly the final arbiter on this matter. But that's my opinion.

Good point. I've put in a request with Geogebra for the feature.

In the meantime, I'm thinking of how I might hack it with what I have now by just using URLs attached to text like "Help me! I don't understand!" to out-of-book auxiliary resources to provide another perspective on some of the slipperiest topics (like "growth rate" vis-a-vis derivatives).

I really would like to know where people stumble with these activities... and then fix it. So that feature would be really helpful. Or if someone wants to copy this and change it themselves, go for it! I put this up on Geogebra because it is a fully open platform.

Which brings me to another reason for going with Geogebra: your point.

Geogebra books are a nice balance of free hosting, wide distribution, and a reasonably friendly UX format. That said, it's hardly perfect. But it offers me what I think is my best chance to fail fast, get feedback, and make changes with the approach to teaching calculus that I present in this book. But I want to fully acknowledge your point regarding UX. I absolutely agree. The "Geogebra Book" format has its limitations and introduced new assumptions along the way that make it harder for users. If nothing else, it's A BOOK. There is an inherent "one-way-ness" to it. I don't see that being as much of an issue as you, but I agree some of it needs to be broken down. How else will revisions make it into the book? That's the whole point of it being open.

As a sidenote: I also want to point out that I have tested these activities as best I can in a variety of LMSs and in a variety of classrooms from Ivy League students gunning for top marks to adult learners who "can't do math" and gave up on Calculus 20 years ago, but due to some reason or another now need to know it. This book is sort of the "least common denominator" of all these testings, and is itself a testing.

So... you're right! Thanks. Want to work together to test it more?

I've already changed it just from reading these comments, and I'd be really interested in continued revisions!

It's a "Geogebra Book." Get a free Geogebra account at geogebra.org, and get started writing.

Hiya. I wrote a more detailed response above. Thanks for sharing your struggle.

Good point. Thanks. I do think that limit is obvious and equal to 4. The point tends to 4.

That said, I also agree the "growth rate" thing is coming in a little too quickly there. It's meant to foreshadow derivatives in the next chapter, but it seems like maybe it's introducing confusion to the reader.

I went ahead and made some revisions to try to ease that connection of the "slope of a secant line" as an estimate of "growth rate" of a function.

That said, no matter what I do, this is one of those "object equivalencies" in calculus that there's no way to really make for someone. At the end of the day "slope of secant line" and "growth rate" are two different objects that in the context of a mathematical model are equivalent, but in a mathematical vacuum, are not. I write about this a little bit at the end of the book here: https://www.geogebra.org/m/x39ys4d7#material/fxpkwpt7

Sadly, the resolution for you isn't really very concrete. To get another person to "learn" an object equivalence is a challenging thing. There's really only two options: 1. tell them. 2. put evidence in front of them and hope they make it themselves. I went for option 1 after sprinkling in a bit of option 2. I've tried to slow it down a bit more, but of course, every learner will be different on when they're ready to make this important connection.

So at some point or another, this speed-bump needs to get hit.

If you have more thoughts let me know!