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

[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)?

 

[EnumaElish]

Do you mean to approximate the roots? Or do you mean to approximate an extremum (min or max)?

 

[HallsofIvy]

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?

 

[shwin]

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.

 

[HallsofIvy]

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