To find the linearization of the function f(x) = sin(x) at the point x = π/6, we will use the formula for linearization, which is:
L(x) = f(a) + f'(a)(x – a)
Here, a is the point where we want to linearize the function, in this case π/6.
Step 1: Calculate f(a)
First, we need to find f(π/6):
f(π/6) = sin(π/6) = 1/2
Step 2: Calculate the derivative f'(x)
The derivative of f(x) = sin(x) is f'(x) = cos(x). Now, we compute f'(π/6):
f'(π/6) = cos(π/6) = √3/2
Step 3: Substitute a, f(a), and f'(a) into the linearization formula
Now that we have f(π/6) = 1/2 and f'(π/6) = √3/2, we can plug these values into our linearization formula:
L(x) = 1/2 + (√3/2)(x – π/6)
This is the linear approximation (linearization) of the function f(x) = sin(x) near the point x = π/6.