Second-Order Taylor: Where the Quadratic Model Can Be Trusted

Here is the surface of f, the log of one plus e to the x zero plus e to the x one, and its second order Taylor expansion at the origin: the amber bowl. The value log three, the gradient one third, one third, and the Hessian build that bowl.

Near the origin, the two surfaces are almost the same. The teal region on the floor marks where the error is below one percent. Farther away, the bowl curves up, while f keeps growing almost linearly.

The rest of the narration is in the book's Companion.