Find the linearization L(x,y) of the function f(x,y)=sqrt(338−9x^2−16y^2) at (−3,4).

Question

Find the linearization L(x,y) of the function f(x,y)=sqrt(338−9x^2−16y^2) at (−3,4).

L(x,y)= ?

Note: Your answer should be an expression in x and y; e.g. "3x - 5y + 9"

Answer 1

L(x,y) = f(-3,4)+ f_x(-3,4)(x + 3) + f_y(-3,4)(y − 4);

f(-3,4) = √(338 - 81 - 256) = 1; f_x(x,y) = - 9x/f(x,y); f_x(-3,4) = 27; f_y(x,y) = - 16y/f(x,y); f_y(-3,4) = - 64;

L(x,y) = 1 + 27(x + 3) - 64(y - 4) = 27x - 64y + 338