泰勒定理与黑塞矩阵

One variable, reviewed

The Hessian matrix

A worked Taylor polynomial

What the quadratic form reveals

At a point where $f_{xx} = 2$, $f_{xy} = 2$ and $f_{yy} = 6$, what is the second-order term of the Taylor approximation in terms of the step $(u, v)$?

The second-order Taylor polynomial of $f(x, y) = x^2 y + y^3$ at $(1, 1)$ is $P_2 = 2 + 2u + 4v + u^2 + 2uv + 3v^2$ with $u = x-1$, $v = y-1$. What does it predict at $(1.1, 0.9)$?

For $f(x, y) = x^2 y + y^3$, what is the Hessian matrix at the point $(1, 1)$?

The quadratic form $u^2 + 2uv + 3v^2$ can be rewritten as a sum of squares $(u+v)^2 + c\,v^2$. Give the coefficient $c$, as a plain number.

At the step $(0.1, -0.1)$ toward $(1.1, 0.9)$, the linear approximation of $f(x,y) = x^2y + y^3$ misses the true value by about $0.018$, and the second-order polynomial misses by about $0.002$. Give the ratio of the two errors, rounded to a plain number.