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##\Delta L=Lk\Delta T##

Where L is length and T is temperature

I'm wondering, is this just an approximation? Because, if you were to increase the temperature by [itex]\Delta T[/itex] and then calculate the new length, and then decrease the temperature by the same [itex]\Delta T[/itex] and calculate the new length again, you would not get back to your original length.

Wouldn't the "symmetrical" definition of [itex]k[/itex] be

##L_{f}=L_{0}e^{k\Delta T}##

This leads me to the question:

Is the reason they don't define it like this because the idea of 'linear thermal expansion' is not true to that degree of accuracy?

(In other words, finding the new length (to that accuracy) is not as simple as using a single constant?)