Find the linearization L(x) of the function at a.f(x) = sin(x),    a = 𝜋3

The linearization of a function f(x) at a point a is given by the formula:

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

Here, we are given f(x) = sin(x) and a = 𝜋/3.

Step 1: Find f(a) Substitute a = 𝜋/3 into f(x) to get f(a). f(a) = sin(𝜋/3) = √3/2

Step 2: Find f'(x) The derivative of sin(x) is cos(x). So, f'(x) = cos(x).

Step 3: Find f'(a) Substitute a = 𝜋/3 into f'(x) to get f'(a). f'(a) = cos(𝜋/3) = 1/2

Step 4: Substitute f(a) and f'(a) into the formula for L(x) L(x) = f(a) + f'(a)(x - a) = √3/2 + 1/2(x - 𝜋/3)

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

The linearization of a function f(x) at a point a is given by the formula:

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

Here, we have f(x) = sin(x) and a = 𝜋/3.

Step 1: Compute f(a) f(a) = sin(𝜋/3) = √3/2

Step 2: Compute f'(x) The derivative of sin(x) is cos(x), so f'(x) = cos(x).

Step 3: Compute f'(a) f'(a) = cos(𝜋/3) = 1/2

Step 4: Substitute f(a) and f'(a) into the linearization formula L(x) = f(a) + f'(a)(x - a) = √3/2 + 1/2 * (x - 𝜋/3)

So, the linearization L(x) of the function at 𝜋/3 is L(x) = √3/2 + 1/2 * (x - 𝜋/3).