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Comment by boyobo | Hacker News Reader

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.

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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.

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