Local Linear Approximation

Question

G‎iven the funct‎ion f(x, y) = x^{2 3}√y. (Cube root y)

(a) Fin‎d the lo‎cal linear appr‎oximation of f at the p‎oint (3, 8).

(b) Use ans‎wer in (a) to approx‎imate (2.99)^{2 3}√8‎.04 (Cube root 8.04)

Answer 1

(a) L(x,y) = f(x₀,y₀) + ∂f/∂x(x - x₀) + ∂f/∂y(y - y₀), where (x₀, y₀) = (3,8) and partial derivatives evaluated at this point so.∂f/dx = 2xy^1/3 = 2·3·8^1/3 = 12; ∂f/∂y = 1/3x²y^-2/3 = 1/3·9/4 = 3/4; f(x₀, y₀) = 3²·8^1/3 = 18

L(x,y) = 18 +12(x - 3) + 3/4(y - 8); L(x,y) = 12x + 3/4y - 24;

(b) 2.99²·8.04^1/3 ≈ 12·2.99 + 3/4·8.04 - 24 = 17.91