Let f(x,y) = 6x^2 + 3xy + 6y^2 + 3x - 3y + 6.

Question

Find an upper bound for the magnitude |E| of the error in the approximation f(x,y) ~= L(x,y) over the rectangle |x+1| <= 0.4, |y+2| <= 0.4.

Answer 1

|𝐸(𝑥,𝑦)| ≤ 1/2𝑀 (|x−x₀| + |y−y₀|)² where M is the upper bound of |f_xx| , |f_yy| , |f_xy| on the rectangle.

f_x = 12x + 3y +3

f_xx = 12

f_xy = 3

f_y = 3x + 12y - 3

f_yy = 12

M = 12

Thus, for any pt (x,y) such that -.6 ≤ x ≤ 1.4 and -1.6 ≤ y ≤ 2.4 , the linearization of f(x,y), L(x,y), will have a maximum error of 6d⁴, where d = the distance between (x,y) and (.4 , .4).