Question

Can somebody show me how to solve this [10e^(-x^2 -y^2)(2x^3 -3x +y -(2x^2 y))]*[10e^(-x^2 -y^2) (2y^3 -3y +x -(2xy^2)]-[10e^(-x^2 -y^2)(-x^2 + x -y -(2x^2 y) -(2xy^2))]^2 with this point (1/2, -1/2)
I know I need to shrink the equation down to a manageable size so that I can pluck in the point (1/2,-1/2) and solve but I keep getting lost with this long one and end with nothing. Maybe somebody know a quick way.

Answer 1

-x^2-y^2 = -1/4-1/4 = -1/2
and thus e^(-x^2-y^2) = 1/√e
So, now you have
[5/√e(2x^3 -3x +y -(2x^2 y))]*[5/√e (2y^3 -3y +x -(2xy^2)]-[5/√e(-x^2 + x -y -(2x^2 y) -(2xy^2))]^2

2x^3 -3x +y -(2x^2 y) = 2/8 - 3/2 - 1/2 + (2 * 1/4 * 1/2) = -3/2
2y^3 -3y +x -(2xy^2) = -2/8 + 3/2 + 1/2 - (2 * 1/2 * 1/4) = 3/2
-x^2 + x -y -(2x^2 y) = -1/4 + 1/2 + 1/2 + (2 * 1/4 * 1/2) = 1
2xy^2 = 2 * 1/2 * 1/4 = 1/4

So now you have
[5/√e(-3/2)]*[5/√e (3/2)]-[5/√e(1 - 1/4)]^2
= -25/e * 9/4 - 25/e * 9/16
= -25/e * 45/16

check my arithmetic