Fundamentals of Mathematics - Taylor Theorem

Mathematical basis - Lagrange mean value theorem

Lagrange's mean value theorem can be understood through this diagram

Univariate Taylor Theorem

Taylor formula refers to Baidu Encyclopedia ：

For ease of understanding , You can see the following figure

Taylor's theorem extended to multivariate

Here is the upper form （2.2） It can be understood as $p$ Is the vector of the direction , function $f$ from $x$ To $x+p$ The change of is a function $f$ Gradient value and direction vector at a certain point $p$ Multiply （ Inner product ）. This point is between $(x,x+tp),t\in (0,1)$ Between .
Dan style (2.3)(2.4) I don't know how to get here . Then ask the teacher