Question

state and prove taylor the orem for a function of two variables

Answer 1

Statement:

Taylor’s Theorem in two variables

If f (x,y) is a function of two independent variables x and

y having continuous partial derivatives of nth order in

some neighborhood of the point (a,b) and if (a + h, b + a)

is any point of this neighborhood, then there exists

some Ф in (0,1) such that

f (a+h, b + k) = f (a,b) + d f (a,b) + (½!) d²f (a,b)+…

                        ....+ (1/n-1!) d^n -1 (f(a,b) + (1/n!) d^n( f(a+Фh,b+ Фk) )

For the value of d in the formula see the below figure