9

9. Find the linear approximation of f(x,y) = (20 - x² -7y²)^1/2 at
(2,1) and use it to approximate f(1.95, 1.08).

Solution: We have f(2,1) = 3 and

f_x = (1/2) (20 - x² -7y²)^-1/2(-2x), f_x(2,1) = -(2/3)

f_y = (1/2) (20 - x² -7y²)^-1/2(-14y), f_y(2,1) = -(7/3)

So, the linear approximation is

f(x,y) » 3 -(2/3)(x-2) -(7/3)(y-1).

Consequently,

f(1.95, 1.08) » 3 -(2/3)(-0.05) -(7/3 )(0.08) = 2.85