# Approximation of a function of two variables

• I
AAMAIK
TL;DR Summary
Why the assumptions, that the tangent plane must pass through the point we wish to approximate the function nearby and the plane has the same slopes in the i and j directions as the surface does will make the tangent plane a good approximation to the surface?
In a manner analogues to the linearization of functions of a single variable to approximate the value of a function of two variables in the neighbourhood of a given point (x0,y0,z0) where z=f(x,y) using a tangent plane. The tangent plane must pass through the point we wish to approximate z nearby and the second condition which I don't quite understand why it makes the tangent plane a good approximate is that it must have the same slopes as the surface in the i and j directions. Therefore Z≈ z0+fx(x-x0)+fy(y-y0)
I can understand that the equation of the tangent plane will serve as a good approximation if we are changing x and keeping y=y0 or vice versa. Namely, if we are varying y and keeping x=x0. Then the lines z=z0+fx(x-x0) and z=z0+fy(y-y0) of the line will serve as a good approximation to the curve traced in the vertical plane y=0 and x=x0. But since x and y are free to take any values what if we decided to change both x and y how can I prove that the tangent plane with these set of conditions will be a good approximation to the curve.

Mentor
Here's a discussion on it:

http://tutorial.math.lamar.edu/Classes/CalcIII/TangentPlanes.aspx

An aside: The Khan video is by Grant Sanderson aka 3blue1brown on Youtube. His youtube channel has some great math videos on the Essence of Calculus.

The way I look at it is the tangent plane at some point x,y is the plane that holds the tangent line of the curve in the ##XZ## plane and holds the tangent line of the curve in the ##YZ## plane at a given x and y value respectively for the function ##z=f(x,y)##

• AAMAIK
In matrix notation: $\Delta z(x_{0}, y_{0})\approx \begin{pmatrix} \frac{\partial z}{\partial x}\ & \frac{\partial z}{\partial y} \ \end{pmatrix}_{x_{0},y_{0}}\begin{pmatrix} \Delta x \\ \Delta y \ \end{pmatrix}$

• jedishrfu
AAMAIK
Here's a discussion on it:

http://tutorial.math.lamar.edu/Classes/CalcIII/TangentPlanes.aspx