Expii

What Is Quadratic (2nd Order) Taylor Approximation? - Expii

The 2nd Taylor approximation of $$f(x)$$ at a point $$x=a$$ is a quadratic (degree 2) polynomial, namely $P(x) = f(a) + f'(a)(x-a)^1 + \frac12 f''(a)(x-a)^2.$ This make sense, at least, if $$f$$ is twice-differentiable at $$x=a$$. The intuition is that $$f(a) = P(a)$$, $$f'(a) = P'(a)$$, and $$f''(a) = P''(a)$$: the "zeroth", first, and second derivatives match.