Linearization in Single-/Multi-Variable Calculus, Etc.

Suppose we have the graph of the function $f(a,b)$. As we take the limit of this function and get exceedingly closer to a point $(a,b,f(a,b))$ on the graph of the function, the surface of the graph begins to look like a plane; this tangent plane is approximated by a differentiable linear function. If we want to find the approximation for a plane that is tangent to a single point $(a,b,f(a,b))$ on the three-dimensional function we use linearization, or the formula for the tangent plane in multi-variable calculus, which is defined as:

$f(a,b)+f_{x}(a,b)(x-a)+f_{y}(a,b)(y-b)$

where

$f(a,b) = z_{0}$ = value of the function;

$f_{x}(a,b)$ = partial derivative at point (a,b) in terms of x;

$(x-a)$ = x - given #a;

$(y-b)$ = y - given #b;

$f_{y}(a,b)$ = partial derivative at point (a,b) in terms of y