Help With Spivak: Learning Calculus from First Principles

[E'lir Kramer]

Mod note: post split of from this thread: https://www.physicsforums.com/showthread.php?p=4187253[/color]

Hi. I know this thread hasn't been active in a year, but it still seems the most appropriate place to post.

On the advice in this thread, I've bought Michael Spivak's book, Calculus, 4th ed.

Now I am working through the book, and I'm hoping for some advice on how to use the book to teach myself mathematics. In fact, it's a question about how to learn mathematics.

I am trying to learn calculus from first principles. As a software engineer, I've used the tools of calculus without really (or only briefly) understanding the underlying principles of it. As I mature professionally and intellectually, I am more and more interested in math for math's sake. And I find it much easier now, after five years of heavy programming, to think logically and prove theorems.

So I want to learn math - calculus, set theory, and algebra. Spivak's book seemed like the best one for a person in my situation, and the first 26 pages, which have taken me almost 16 hours to get through, have been fascinating.

Some of these problems in the back of the chapter, though, are really hard. For instance, 3(d) is "Prove the binomial theorem".

He's pointed towards me the answer by previously asking me to prove that (n+1 choose k) = (n choose k-1) + (n choose k). And many of the proofs he's been asking for are inductive, and this one will probably also be inductive. n=1 or n=0 is an easy inductive case, and then I need only prove that if some the binomial theorem is true for some number k ∈ N, then it will also be true for k+1.

With (a + b)ⁿ * (a + b) = (a+b) ⁿ⁺¹, I know I'm on the right path to the solution. With a good night's sleep and another hour of work tomorrow morning, I'm sure that I'll crack it, and I'll be gratified to do so.

But if I were to solve every problem in this book, it might take me a year. Or maybe more than that. Suppose that my goal is to learn - deeply, and durably - the mathematics of calculus in the least amount of time. Do you think that working through every Spivak problem would be the best use of my time? Or is this too masochistic? Should I struggle for a while, and then give up? How long is "a while"? What light can guide my self-study approach?
Mason

 

[micromass]

In order to really understand calculus, it is of course not necessary to work through all the problems in Spivak.

I like Spivak's problems very much and I always found them to be extremely fun. So there is something fun about solving all exercises. The exercises in Spivak will show you many nice results.

But not every problem in Spivak is as useful to learning calculus. If you have little time, then I would make a certain selection of exercises. Exercises that you know how to solve immediately are useless to make, for example. If you want to know which exercises are worthwhile to make, then I can always help you with your selection.

 

[E'lir Kramer]

Thanks, micromass. I'll keep working through the problems as long as I feel like I've got a chance at them, and not feel terrible about skipping some of them. I may very well take you up on your offer to make some selections for me when I get deeper into the book.

 

[QuantumP7]

In order to really understand calculus, it is of course not necessary to work through all the problems in Spivak.

I like Spivak's problems very much and I always found them to be extremely fun. So there is something fun about solving all exercises. The exercises in Spivak will show you many nice results.

But not every problem in Spivak is as useful to learning calculus. If you have little time, then I would make a certain selection of exercises. Exercises that you know how to solve immediately are useless to make, for example. If you want to know which exercises are worthwhile to make, then I can always help you with your selection.

Which would be the best problems to solve for the chapter on Limits?

 

[E'lir Kramer]

Here's one I feel I need to ask about. Chapter 2, problem 20: Prove Binet's Formula.

Once again, I knew this would be a natural fit for an inductive proof. I got quite far into the proof before I eventually gave up and looked it up. I found the proof I was trying for here: http://fabulousfibonacci.com/portal/index.php?option=com_content&view=article&id=22&Itemid=22 .

On my own, I worked down to ø^k-1(ø+1) - τ^k-1(τ+1) step (that is the first use of the distributive prop. in the proof). At that point I got stuck, and eventually wandered off down several dead ends.

What happens next in this proof feels like magic to me. The substitution of 2/2 for 1 seems like a fine exploratory step that doesn't need any justification. But when the author gets down to multiplying (3+sqrt(5))/2 by 2, and then separates 6 into 1 + 5 - that makes my head explode. How did he know to search there? What hints did he have to go on?

Edit: now it occurs to me that if I wanted ø^k+1 - τ^k+1 = ø^k-1(ø+1) - τ^k-1(τ+1), then it follows that ø^k-1(ø+1) - τ^k-1(τ+1) = ø^k-1ø² - τ^k-1τ² and thus ø+1 = ø², etc.

It's easier to see how working for the identity ø² = ø+1 would lead to the following "magic" steps.

(The even more head-exploding thing is how this formula was conjectured in the first place - but I'm assuming that it was derived in some other, more straightforward way.)