R Hansens says: >To whom Joe? I have seen NO class of students get anything like the MVT from a picture, UNLESS it is explained to them.
Joe N says: >Lots of people. Probably most. We've already established you're not one. But once again your entire argument doesn't scan. Yes, one must be taught how to read the information, whether its words or curves.
Perhaps, too glib again. I suspect most people, even non mathematical types, could be taught to see the theorem, if its explained what one is looking for. Given an ordinary continuous everywhere differentiable curve, the tangent can be intuitively and visually conveyed. Once the chord between two points is illustrated, the notion that the mean value theorem could be violated would be seen to be impossible. Naturally, a guide is needed until one becomes a scout on their own. Its even more visually compelling with the intermediate value theorem.
Your assertion about classes you've experienced means little. Classes where the mean value theorem are introduced and proved without ever mentioning the visual intuition behind them would be few and far between, probably only classes where it was assumed the visual was already understood from prior exposure.
By today's standards these things were only analytically "proved" within the last couple hundred years, or even last hundred, all depending on your standard. They were known for a much longer time. The intuition was visual, not some other "sense".
By your argument, these things were not really be known before they could be proved by some modern standard. That's not the case.