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Calculus For The People

geogebra.org

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Re: Calculus For The People

#51

Post by gregpetrics »

rrss wrote: post #49
gregpetrics wrote: post #39

Earlier quoted context omitted.

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'…

My pages apparently don't align with yours (I see page 46 has only a single exercise), but I don't see where Rudin says anything is "good enough." He states the definition of convergence, meaning that if a sequence satisfies the property then we choose to call it convergent. There is no question of good enough I don't see a claim about an "infinite set of inequalities" - I see an infinite set of I equalities that mus…

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.

Re: Calculus For The People

#52

Post by btilly »

newprint wrote: post #40

Can I give a very practical advise to people who are reading this and trying to learn math ? As someone, who received a very strong mathematical training in a former Soviet Union, here is my practical advise: 1. Calculus books, just like this one, are absolutely impractical in real life situation, especially, if your goal is "Industrial Mathematics". All you will learn, are basic calculus notations. You will, at best…

This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM.

I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus.

I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on?

  1. The idea of approximations.
  2. The tangent line.
  3. How the tangent line connected to approximations.
  4. The derivative.
  5. The easiest formulas for differentiation and why they are true.  (All handwavy, heuristic big-O arguments.)
  6. That all possible max/min points can be found at the boundaries, or by finding where the derivative is 0 or non-existent.
  7. The Fundamental Theorem of Calculus aka why areas are the reverse of derivatives.
  8. The advice that if he had to actually calculate a derivative or integral, he should use a program like MAPLE.
Did he master the subject? Heck no!

Did he have to review his notes a bunch of times so it stuck? Of course!

But he went on to ace the course. And my guess is that he understood what makes Calculus tick better than most who took the course. (Sanity check. If you do not understand why the tangent line and derivative are connected, then you do not understand Calculus.)

Re: Calculus For The People

#53

Post by newprint »

rrss wrote: post #50
newprint wrote: post #40

Can I give a very practical advise to people who are reading this and trying to learn math ? As someone, who received a very strong mathematical training in a former Soviet Union, here is my practical advise: 1. Calculus books, just like this one, are absolutely impractical in real life situation, especially, if your goal is "Industrial Mathematics". All you will learn, are basic calculus notations. You will, at best…

I'm genuinely curious what field you work in and what problems you work on that IMO level algebra and combinatorics skills are frequently useful (IMO = the olympiad, not my opinion). In the fields I have experience with, the usually approach when faced when something gnarly involves a lot of "to first order," or "assume X is much greater than Y," or simulation. I'm somewhat doubtful that your advice is widely applica…

To answer your question, I work in optimization.

IMO (International Mathematical Olympiad) are extremely complicated problems that even professional mathematicians often struggle with them. IMO is a level on its own - Gold Standard. Not all Mathematical Olympiads are of the same level of complexity as IMO. Good example are Olympiad caliber problems that are not overkill - Hungarian Problem Books.

MO problems are useful, because they make you think out of box, they often times involve several branches of math in one problem, such as number theory problems go in hand with combinatorics; they don't require complex mathematical machinery, and technique of solving problems, directly translates to solving complex problems in analysis/abstract algebra.

Re: Calculus For The People

#54

Post by avmich »

btilly wrote: post #52
newprint wrote: post #40

Can I give a very practical advise to people who are reading this and trying to learn math ? As someone, who received a very strong mathematical training in a former Soviet Union, here is my practical advise: 1. Calculus books, just like this one, are absolutely impractical in real life situation, especially, if your goal is "Industrial Mathematics". All you will learn, are basic calculus notations. You will, at best…

This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM. I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus. I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on? 1. The idea of approximations. 2. The tangent line. 3. How the tangent line co…

I think a critical ingredient for him was that he was motivated to get the exam passed. Otherwise new terms (unknown, not heard before, so scary and opaque) would overwhelm and attention would be lost.

How you did that introduction is very important. Every time you tell something new, it better be really small - or explained quickly and well, so the concept would stick before brain would get tired.

Thank you for listing your points.

Re: Calculus For The People

#55

Post by newprint »

Railsify wrote: post #46
newprint wrote: post #40

Can I give a very practical advise to people who are reading this and trying to learn math ? As someone, who received a very strong mathematical training in a former Soviet Union, here is my practical advise: 1. Calculus books, just like this one, are absolutely impractical in real life situation, especially, if your goal is "Industrial Mathematics". All you will learn, are basic calculus notations. You will, at best…

Can you propose a specific curriculum?

There are numerous books devoted to Olympiad preparation, there are websites completely devoted to Olympiads, there are freely available problems sets (with complete solutions). To name the few: Hungarian Problem Books, Problem-Solving Strategies, Challenging Problems in Algebra, books by Titu Andreescu - a former US IMO coach, who published a lot of books on math and IMO prep, http://www.cut-the-knot.org/

Re: Calculus For The People

#56

Post by Sniffnoy »

kstenerud wrote: post #12
Sniffnoy wrote: post #9

Earlier quoted context omitted.

It's using a naive, informal notion of those. If you were to define it formally, well, you'd have the derivative. Which is what he does quite soon after. This is how definitions frequently work in mathematics -- they're meant to take some naive informal notion and formalize it, by coming up with a formal definition that matches how it should work. So, it's assuming you already have some informal notion of growth rate…

So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? Unfortunately, I don't have an informal notion of growth rate in my head :/

> So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it?

No. We're talking about instantaneous velocity. You know, the thing the speedometer displays. How fast is the car moving at any given moment? Like, a car doesn't need to be moving at constant speed for a speedometer to give meaningful information, right? Sometimes it is moving faster and sometimes it is moving slower. Sometimes it is moving at a rate such that if it stayed at that rate it would go 60 miles in an hour, and sometimes it is moving at a rate such that if it stayed at that rate it would go 30 miles in an hour. This is the informal notion of instantaneous velocity you should already have. Now the question becomes, how do we formalize this? Which is what the page is trying to answer.

Re: Calculus For The People

#57

Post by master_yoda_1 »

vibrio wrote: post #27
master_yoda_1 wrote: post #19

For which people? Who does not go to school?

Do you believe school is sufficient?

Yes if you focus and don't waste time. Your whole life won't be sufficient if you can't focus and keep wasting your time in unrelated stuff like this shallow article.

Re: Calculus For The People

#58

Post by btilly »

avmich wrote: post #54
btilly wrote: post #52

Earlier quoted context omitted.

This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM. I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus. I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on? 1. The idea of approximations. 2. The tangent line. 3. How the tangent line co…

I think a critical ingredient for him was that he was motivated to get the exam passed. Otherwise new terms (unknown, not heard before, so scary and opaque) would overwhelm and attention would be lost. How you did that introduction is very important. Every time you tell something new, it better be really small - or explained quickly and well, so the concept would stick before brain would get tired. Thank you for list…

I think that it really helped that back in grad school I read https://www.docdroid.net/z8ki/knuth.pdf and thought hard about Knuth's ideas. I even went so far as to make a course outline for a first course in Calculus based on his ideas.

The result is that I was able to break my exposition into one piece at a time, starting with ideas that were already accessible. More specifically I started with the idea that f(x) = approx(x) + error(x) where we want approx simple and error small. This motivates a language for describing what "small error" means, which motivates little-o for polynomials.

That's enough to do tangent lines and answer questions like, "Given the equation for position vs time, how fast is the rock going when it hits the bottom?"

Only after he could calculate tangent lines did I introduce the derivative.

Contrast to the usual approach where limits are an abstract concept with no obvious application, and then the derivative is introduced. It actually combines several ideas jumbled together. That's a big mental knot that almost nobody gets. And explaining it more carefully doesn't help because people keep getting to holding too many unintegrated ideas in their head at once.

And seriously, heuristic arguments that you can reproduce whenever you forget them are good. Take the product rule.

  f(x0+h) = f(x0) + f'(x0) h + o(h)
  g(x0+h) = g(x0) + g'(x0) h + o(h)

  (f*g)(x0+h)
    = (f(x0) + f'(x0) h + o(h)) * (g(x0) + g'(x0) h + o(h))
    = f(x0) * g(x0) + f(x0) * g'(x0) h + f(x0) * o(h) +
      f'(x0) h * g(x0) + f'(x0) h * g'(x0) h + f'(x0) h * o(h) +
      o(h) * g(x0) + o(h) * g'(x0) h + o(h) * o(h)
    = f(x0) * g(x0) + f(x0) * g'(x0) h + o(h) +
      f'(x0) * g(x0) h +                o(h) +           o(h) +
              o(h) +            o(h) +        o(h)
    = f(x0) * g(x0) + (f(x0)*g'(x0) + f'(x0)*g(x0)) h + o(h)
And we recognize the form of the tangent line and so the derivative of f*g is f' g + g f'.

I had him do that calculation with only minimal prompting. And I think that this is a calculation that I'd like every Calculus student to be able to do on demand. If you forget the rule and you know that argument, you can figure it out again.

Re: Calculus For The People

#59

Post by phaedrus »

btilly wrote: post #52
newprint wrote: post #40

Can I give a very practical advise to people who are reading this and trying to learn math ? As someone, who received a very strong mathematical training in a former Soviet Union, here is my practical advise: 1. Calculus books, just like this one, are absolutely impractical in real life situation, especially, if your goal is "Industrial Mathematics". All you will learn, are basic calculus notations. You will, at best…

This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM. I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus. I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on? 1. The idea of approximations. 2. The tangent line. 3. How the tangent line co…

Counting Differential Equations, I've had 5 semesters of calculus. (Calc I,II,III,IV, Diff Equ.) Until reading your comment, item 7, I had never heard this simple explanation of the meaning of the Fundamental Theorem of calculus ("why areas are the reverse of derivatives"). I was taught the fundamental theorem algebraically, and how to apply it, but none of my professors or textbooks ever explained what it meant.

Re: Calculus For The People

#60

Post by btilly »

phaedrus wrote: post #59
btilly wrote: post #52

Earlier quoted context omitted.

This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM. I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus. I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on? 1. The idea of approximations. 2. The tangent line. 3. How the tangent line co…

Counting Differential Equations, I've had 5 semesters of calculus. (Calc I,II,III,IV, Diff Equ.) Until reading your comment, item 7, I had never heard this simple explanation of the meaning of the Fundamental Theorem of calculus ("why areas are the reverse of derivatives"). I was taught the fundamental theorem algebraically, and how to apply it, but none of my professors or textbooks ever explained what it meant .

Thank you for your honesty.

I wish it surprised me. But I suspect that your experience is more normal than not.

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