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

• shwin
In summary, the conversation discusses using Newton's method to approximate functions f(x,y) and g(x,y), and the difference between using it for finding solutions to equations versus finding tangent plane approximations to the functions. It is noted that Newton's method is used for finding numerical solutions to equations and the tangent plane approximation is not typically referred to as Newton's method.
shwin
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)?

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
$$z= f_x(x_0,y_0)(x- x_0)+ f_y(x_0,y_0)(y- y_0)$$

## 1. What is Newton's Method for Approximating f(x,y) & g(x,y)?

Newton's Method for Approximating f(x,y) & g(x,y) is a mathematical algorithm used to find the roots of a system of equations. It is an iterative process that uses an initial guess to approximate the solutions of the equations.

## 2. How does Newton's Method work?

Newton's Method works by using the tangent line of a function at a given point to approximate the roots of the function. The process is repeated until a desired level of accuracy is achieved.

## 3. What are the advantages of using Newton's Method?

One advantage of Newton's Method is that it is a fast and efficient way to approximate the roots of a system of equations. It also allows for a high degree of accuracy, especially when the initial guess is close to the actual solution.

## 4. What are the limitations of Newton's Method?

One limitation of Newton's Method is that it may fail to converge if the initial guess is too far from the actual solution. It also requires the calculation of derivatives, which may be difficult or impossible for some functions.

## 5. When is Newton's Method used in real-world applications?

Newton's Method is commonly used in engineering, physics, and other fields to solve complex systems of equations. It is also used in optimization problems, such as finding the minimum or maximum of a function.

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