Find the linearization L(x) of the function at a.

Question

f(x)=sin(x) , a=  𝜋/3

Answer 1

The derivative of the function f is given by:

f'(x) = cos(x)

f(π/3) = sin(π/3) = (√3)/2

f'(π/3) = cos(π/3) = 1/2

The linearization of the function f around a = π/3 is given by:

L(x) - f(a) = f'(a)(x - a)

⇒ L(x) - f(π/3) = f'(π/3)(x - π/3)

⇒ L(x) - (√3)/2 = (1/2)(x - π/3)

⇒ L(x) = (1/2)x - π/6

⇒ L(x) - (√3)/2 + (√3)/2 = (1/2)x - π/6 + (√3)/2

⇒ L(x) = (1/2)x + (√3)/2 - π/6