Question

Determine the 1st and 2nd degree Taylor polynomials L(x,y) and Q(x,y) for

f(x, y) = tan^−1(xy) for (x, y) near the point (1,1).

Answer 1

Explanation:

`f(x,y)=arctan(xy)\\\\( \partial f)/( \partial x)=(y)/(1+x^2y^2) \\\\( \partial f)/( \partial y)=(x)/(1+x^2y^2) \\\\( \partial ^2 f)/( \partial x^2)=(-2xy^3)/((1+x^2y^2)^2) \\\\( \partial ^2 f)/( \partial y^2)=(-2x^3y)/((1+x^2y^2)^2) \\\\( \partial ^2 f)/( \partial x\ \partial y)=(1-x^2y^2)/((1+x^2y^2)^2) \\`

`f(x,y)= f(1,1)+(x-1)(\partial f)/(\partial x) (1,1)+(y-1)(\partial f)/(\partial y) (1,1)+((x-1)^2)/(2) (\partial^2 f)/(\partial x^2) (1,1)+(x-1)(y-1)(\partial^2 f)/(\partial x\ \partial y) (1,1)+((y-1)^2)/(2) (\partial^2 f)/(\partial y^2) (1,1)+...\\\\=(\pi)/(4)+((x-1))/(2) +((y-1))/(2) -((x-1)^2)/(4) +(x-1)(y-1)*(0)/(4)-((y-1)^2)/(4) \\\\`

`\boxed{f(x,y)=(\pi)/(4)+((x-1))/(2) +((y-1))/(2) -((x-1)^2)/(4) -((y-1)^2)/(4) }\\\\`