I am reading out of Marsdens Vector Calculus and the text gives the same equation under two different headings, namely Linear or Affine Approximations and Tangent Plane to a Surface. The equation is the following at given points $(x_0, y_0)$: $$z= f(x_0 ,y_0) +[\frac{\partial f}{\partial x} (x_0,y_0) ](x - x_0)+[\frac{\partial f}{\partial y} (x_0,y_0) ](y - y_0)$$
Question 1 Can we generalize this equation to an n-dimensional object which has a tangent (n-1) dimensional object at point $(x_1, x_2, \ldots x_{n-1})$?? Or is the case that this equation of the form
$$z= f(x_1,x_2, \ldots x_{n-1}) +[\frac{\partial f}{\partial x_1} (x_1,x_2, \ldots x_{n-1} ) ](x-x_1)+ \cdots + [\frac{\partial f}{\partial x_{n-1}} (x_1,x_2, \ldots x_{n-1})](x- x_{n-1})$$ will represent the tangent plane of an n-dimensional object at point $(x_1, x_2, \ldots x_{n-1})$ in $\mathbb R^n$?
Question2: Can someone prove either of these statements if they are true or suggest where I might find it?
Question 3: Is the equation of affine approximation and the tangent plane always the same equation as dimensions increase or does the affine approximation become expressed by a different equation than the tangent plane equation(or the tangent $n-1$ dimensional-object equation which ever is the case with the equation of this form)??
Thanks in advance, and I apologize for any errors in questions if there be, I'm working off a phone and it is incredibly frustrating.
(Condensing my comments into an answer.)
The proof of multivariate Taylor's Theorem covers this as the degree 1 case is the affine approximation by an $(n-1)$-dimensional object. As multivariate Taylor's Theorem is a generalization of the univariate one, the guess of "tangent plane formula but with more terms corresponding to more input variables" is a good one.
This affine approximation is often called the "tangent hyperplane (line/plane/space/hyperspace)". Be careful, in the field of differential geometry, there is a slightly different (related) definition of "tangent space".