Newton's Method for Approximating f(x,y) & g(x,y)

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    Method Newton's method
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Discussion Overview

The discussion revolves around the application of Newton's method in relation to the functions f(x,y) = cos(x-y) - y and g(x,y) = sin(x+y) - x. Participants explore whether the method is intended for approximating roots or extrema of these functions, and clarify the nature of the problem being addressed.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant inquires if Newton's method can be applied by taking partial derivatives of the functions at a given point (a,b).
  • Another participant questions whether the goal is to approximate roots or to find extrema (minima or maxima).
  • A different participant clarifies that Newton's method is used to find approximate numerical solutions to equations, prompting a request for clarification on the specific problem being addressed.
  • The original poster expresses uncertainty about using Newton's method for both roots and extrema, indicating a potential misunderstanding of terminology.
  • Another participant emphasizes the necessity of having an equation to solve and suggests that the original poster might be referring to tangent plane approximations rather than Newton's method.

Areas of Agreement / Disagreement

Participants do not reach consensus on the specific application of Newton's method, with differing interpretations regarding whether it is for approximating roots or extrema, and whether the terminology used is appropriate.

Contextual Notes

There is ambiguity regarding the definitions and applications of Newton's method versus tangent plane approximations, and the discussion lacks clarity on the specific equations or conditions being addressed.

shwin
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lets say i have the functions f(x,y) = cos(x-y) - y and g(x,y) = sin(x+y) - x. I want to use Newton's method to approximate these functions. Do I just take the partial derivatives with respect to x and y of each function and plug in a given point (a,b)?
 
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Do you mean to approximate the roots? Or do you mean to approximate an extremum (min or max)?
 
As EnumaElish said, you don't use Newton's method to "approximate" functions- you use Newton's method to find approximate (numerical) solutions to equations. What really is the problem?
 
Both, I am not sure how to use it for one and for the other. And yes I meant approximate solutions, I assumed it was just semantics but obviously it isnt.
 
The difference is that to solve and equation, you need an equation! What exactly do you want to do? Perhaps you are just referring to "tangent plane approximations" to the functions (not normally called Newton's method).

The tangent plane to z= f(x,y) at (x_0, y_0), is
z= f_x(x_0,y_0)(x- x_0)+ f_y(x_0,y_0)(y- y_0)
 

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