# Taylor expansion

For $LaTeX: \textstyle f \in C^n,$ Taylor's Theorem is used by dropping the remainder term. The first-order expansion is $LaTeX: \textstyle f(x) = f(y) + \nabla f(x) * (x-y),$ and the second-order expansion is
$LaTeX: \textstyle f(x) = f(y) + \nabla f(x) * (x-y) + \frac{1}{2}\, (x-y)' H_{f(x)} * (x-y).$