Find the local linearization

Question

(a) Find the local linearization of f(x) = ln(1 + x3) at x = 0.

(b) Use the local linearization to estimate sin(0.2).

Answer 1

(a) f(x) = ln(1 + x³); x = 0; f(0) = ln(1) = 0; f'(x) = 3x²/(1 + x³); f'(0) = 0; L(x) = 0 + 0(x - 0) = 0; L(x) = 0

(b) f(x) = sinx, x = 0; f(0) = sin0 = 0; f'(x) = cosx; f'(0) = cos0 = 1; L(x) = 0 + 1(x - 0); L(x) = x

sin0.2 ≈ L(0.2) = 0.2