Question

Say how to linearize a function f[x, y] at a point {a, b}, and discuss how well the linearized version of f[x, y] at {a, b} approximates f[x, y] near {a, b}. Does the quality of the approximation improve or deteriorate as you go closer and closer to {a, b}

Answer 1

Explanation:

Say how to linearize a function f[x, y] at a point {a, b}, and discuss how well the linearized version of f[x, y] at {a, b} approximates f[x, y] near {a, b}. Does the quality of the approximation improve or deteriorate as you go closer and closer to {a, b}

Linear function f[x, y] at a point {a, b}

using taylor series at a point {a, b}

`f(x,y)=f(a,b)+f_x(a,b)[x-a]+(y-b)f_y(a,b)\\\\f(x,y)=f(a,b)+(x-a)f_x(a,b)+(y-b)f_y(a,b)`

The quality of linearization improves as we goes close to point {a, b}

Since,

`f(a,b)=f(a,b)+(a-a)f_x(a,b)+(b-b)f_y(a,b)\\\\f(a,b)=f(a,b) \ \ \texttt{exactly equal}`