Fining the local linearization of f(x) and using it to estimate.

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) ln(1 + x) ≈ x from the series ln(1 + x) = x - x²/2 + x³/3 - .... for |x| < 1.

 

So, ln(1 + 3x) ≈ 3x

 

 

b) sin(x) ≈ x from the series sin(x) = x - x³/3! + x⁵/5! - .... for all x.

 

So, sin(0.2) ≈ 0.2