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Your explanation is a nice restatement or interpretation for what the equation f(x)=f(0)+f'(0)x+.... is saying, but it doesn't give any insight for why the equation is true.

     and as there is no additional change that comes "out of nowhere", so to speak,

The whole explanation rests on this key point, and in fact this is the unintuitive part. In fact, it's not even always true. It's a "miracle" that it sometimes is true.

That's an interesting reaction... imho, the idea that change can't come out of nowhere is the most intuitive part. "How can something arise from nothing?" is a puzzling philosophical question precisely because our intuition is that it's impossible, that everything has to come from somewhere (and conversely, nothing can simply disappear). Functions where this expectation is violated -- e.g. non-constant continuous functions where all derivatives vanish to zero at a certain point -- feel like bizarre exceptions to me.

Fair enough. I do see how you could think that it's intuitive. I've even used your explanation when I'm teaching. Although I can't help but think there's some sort of circularity going on here, at least for me. My intuition on the subject is shaped by my knowledge of the theorem, and the functions that I tend to work with.

To know everything in derivatives you need to find the one where it's constant or a closed form series and its limit at infinity, otherwise you know nothing about how it's changing.

More than not always true, it's mostly not true. It's just true for a lot of functions we're interested in.

This is how I think of it too, and it makes some part of me want to believe that means I have an intuitive grasp do what’s going on. Except I don’t think that’s how math works. Rigor is necessary because an intuitive hand-wavey understanding is often wrong. I think there’s just as convincing a hand-wave available that dismisses the idea.

1) We have one point of the function.

2) We know one point of its derivative. We still don’t know any other points of the function.

3) We know one point of its second derivative. We still don’t know any other points of the function or its derivatives.

...

N) Same deal.

N+1) Same deal.

...

Inf) Now we know all the points in the function and also all the points in all the derivatives.

It looks absurd like this. Not to mention that we start with a countable set of points and derive a continuum from each.

It's more like at each step from n -> n+1, our understanding grows by some amount. The way you write it seems to imply that no information is gained until some arbitrary point when n is extremely large.

The moment we start taking derivatives we leave the world of countable sets and move to limits and epsilon neighborhoods. Evaluating a derivative at a single point by definition tells us something about the neighborhood of that point. Mentally expanding the series terms to their limit forms helps see that it's not just a random polynomial. I guess the real "miracle" is that for a large class of useful functions those ugly limits evaluate to simple forms.

1) We have one point of the function.

2) We know one point of its derivative. This means we know 2 more points of the function (one step in either direction).

3) We know one point of its second derivative. This means we know 2 more points of the derivative, which in turn gives us 2 more points of the function.

etc.

Or at least, that's the impression I get from this thread, since I wasn't familiar with the series before that.

But the problem is that the definitions of those derivatives are for infonitesimal change. What about far away points?

The definition works for any point. However often for higher functions you need to break the function into ranges to find closed form derivative series and get a useful result.

Far away points have properties resembling infinitesimally close points, as long as the function is continuous.

Now when there's a lack of continuity, then we run into a problem.

I'm not a mathematician but I assume there are functions with infinite number of derivatives. For such you cannot know all the values of the function by knowing the values of all derivatives because you would have to know infinitely many numbers.

Whereas if there are a fixed number of derivatives then at some level the derivative is constant and thus you can intuitively think that based on those derivatives you can then "draw" the function.

In-phony-tesimal, I like that word.

I love math.

Thank you for the beautiful explanation.