Find the linearization l(x) of the function at a. f(x) = sin(x), a = π/3

Solution:

Given, the function f(x) = sin(x)

We have to find the linearization L(x) of the function at a = π/3.

We know, π/3 = 60°

Using the formula,

L(x) = f(a) + f’(a)(x - a)

Now,

f(x) = sin(x)

f(a) = f(π/3)

sin(π/3) = sin 60° = √3/2

f’(x) = cos(x)

f’(a) = f’(π/3) = cos(π/3)

cos(π/3) = cos 60° = 1/2

Substituting the values of f(a) and f’(a), the function becomes

L(x) = sin(π/3) + cos(π/3)(x - (π/3))

L(x) = (√3/2) + (1/2)(x - (π/6))

Therefore, the linearization of f(x) = sin(x) at a = (π/3) is L(x) = (√3/2) + (1/2)(x - (π/3))

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Find the linearization l(x) of the function at a. f(x) = sin(x), a = π/3

Summary:

The linearization of the function f(x) = sin(x) at a = (π/3) is L(x) = (√3/2) + (1/2)(x - (π/3)).