Approximation of a function of two variables

In summary, the tangent plane approximation to a curve traced in the vertical plane y=0 and x=x0 will be a good approximation if the following conditions are met: 1) the lines z=z0+fx(x-x0) and z=z0+fy(y-y0) of the line hold at a given x and y value respectively for the function ##z=f(x,y)##2) the slopes of the lines in the ##XZ## and ##YZ## planes are the same as the slope of the surface at the point of approximation.
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AAMAIK
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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.
 
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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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  • #4
jedishrfu said:
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.
 

1. What is the purpose of approximating a function of two variables?

The purpose of approximating a function of two variables is to simplify a complex mathematical equation into a more manageable form. This allows for easier analysis and interpretation of the function's behavior.

2. How is a function of two variables approximated?

A function of two variables can be approximated using various methods such as linear interpolation, polynomial interpolation, or least squares regression. These methods involve finding a simpler function that closely matches the behavior of the original function.

3. What are the benefits of approximating a function of two variables?

Approximating a function of two variables can provide a better understanding of the relationship between the two variables and can also help in making predictions or solving problems that involve the function.

4. Are there any limitations to approximating a function of two variables?

Yes, there are limitations to approximating a function of two variables. The accuracy of the approximation depends on the method used and the complexity of the original function. In some cases, the approximation may not be able to capture all the nuances of the original function.

5. How is the accuracy of an approximation of a function of two variables measured?

The accuracy of an approximation of a function of two variables can be measured by calculating the error between the approximated function and the original function. This can be done by comparing the values of the two functions at different points or by calculating the difference between the two functions over a range of values.

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