Calculs 3 Tangent

Question

Find the linear approximation to the equation f(x,y)=2*√xy/6 at the point (6,4,4), and use it to approximate f(6.29,4.24)

Answer 1

Let z=f(x,y)= 2*√(xy)/6. The partial derivatives of z with respect to x and y are:

∂z/∂x = y/(√6xy), and ∂z/∂y = x/(√6xy).

The linear approximation of f can be written as:

f(6.29,4.24) = f(6,4) + (∂z/∂x (6,4))*(x-6) + (∂z/∂y (6,4))*(y-4) = 4 + 1/3*(0.29) + 1/2 *(0.24) = 4.21.