Approximation of a function of two variables

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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.
 

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Here's a discussion on it:

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

and Khan Academy on it:

https://www.khanacademy.org/math/mu...local-linearization/v/what-is-a-tangent-plane

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)##
 
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In matrix notation: [itex]
\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}
[/itex]
 
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Here's a discussion on it:

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

and Khan Academy on it:

https://www.khanacademy.org/math/mu...local-linearization/v/what-is-a-tangent-plane

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)##
I read the first link but they don't answer my question which is that x and y are free to take any values and that I can understand why the tangent plane approximation would serve as a good approximation to the surface at some specific point if we were to vary x or y holding the other as constant.
The second link video series also laid down the criteria to approximation and how the criteria (slopes in the i and j direction=slope of the surface) narrows down the many candidates for the tangent plane approximation and that the tangent plane passing through the point of approximation alone is not enough.
 

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