Tangent Planes

A collection of all possible tangent lines to the surface of a multivariable function at a point $(a, b, f (a, b))$

\nabla f (x, y, z) (a, b, c) \cdot < x - a, y - b, z - c >

The normal vector $\hat{n} = (A, B, C)$ is orthogonal ???

\begin{array}{r} < x - a, y - b, z - f (a, b) > \cdot < A, B, C > = 0 \\ A (x - a) + B (y - b) + C (z - f (a, b)) = 0 \end{array}

So the equation of the tangent plane must be

z = f (a, b) - \frac{A}{C} (x - a) - \frac{B}{C} (y - b), c \neq 0

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

Taylor Polynomials

Taking the taylor expansion of a multivariable function

\begin{aligned} f (x, y) = & f (a, b) + \frac{f_{x} (a, b)}{1!} (x - a) + \frac{f_{y} (a, b)}{1!} (y - b) + \\ + & \frac{f_{x x} (a, b)}{2!} (x - a)^{2} + \frac{2}{2!} f_{x y} (a, b) (x - a) (y - b) + \frac{f_{y y} (a, b)}{2!} (y - b)^{2} + \dots \end{aligned}

Linear approximation of $f$ at $(a, b)$ is the tangent plane approximation/taylor expansion up to only one derivative

L_{a, b} (\tilde{x}) = T_{1, \tilde{a}} (\tilde{x}) = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)