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.