I studied Taylor series but I would like to have an answer to a doubt that I have. Suppose I have ##f(x)=e^{-x}##. Sometimes I've heard things like: "the exponential curve can be locally approximated by a line, furthermore in this particular region it is not very sharp so the approximation is even more good.."(adsbygoogle = window.adsbygoogle || []).push({});

Now I'm aware of the fact that any function (differentiable and so on) can be approximated by a line and further polynomial, but consider this graph of ##f(x)## and the regions ##A## and ##B##.

My question is : to what extent I can say "##f(x)## is better approximated by a line in ##B## than in ##A##"? And, if I could do this, why is that ? Is that because in region ##B## ##f(x)## is less sharp than in ##A##, or is there other reason behind this?

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# I Is a function better approximated by a line in some regions?

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