Linear (1st Order) Taylor Approximation - Expii

The 1st Taylor approximation of \(f(x)\) at a point \(x=a\) is just a linear (degree 1) polynomial, namely \[ P(x) = f(a) + f'(a)(x-a)^1.\] This make sense, at least, if \(f\) is differentiable at \(x=a\): it's just another way to phrase the tangent line approximation at a point! The intuition is that \(f(a) = P(a)\) and \(f'(a) = P'(a)\): the "zeroth" and first derivatives match.