If, as in this article, we assume that real functions are infinitely differentiable, then their Taylor expansions are not… limited, since they can be carried out to any order (indeed, it would be better to call them "polynomial expansions"). At any point x0, we can therefore write f (x0 + u) = a + bu + *cu 2 + *du 3 + o(u 3 ), where a = f (x0 ), b = f ′(x0 ), c = f ′ ′ (x0 ) / 2 and d = f ′ ′ ′ (x0 ) / 6; and we could continue further.
The first two terms show that the line y = a + b (x ‒ x0) is tangent to the curve y = f (x).
If the coefficient c is nonzero, the point is said to be ordinary; the curve is concave "upward" if the coefficient is positive and "downward" otherwise.
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