To find the linearization of the function f(x) = sin(x) at the point a = π/3, we need to follow a few steps.
Step 1: Determine the function value at a
First, we evaluate the function at the point a = π/3:
f(π/3) = sin(π/3) = √3/2
Step 2: Compute the derivative of f(x)
Next, we find the derivative of the function f(x). The derivative of sin(x) is cos(x):
f'(x) = cos(x)
Step 3: Evaluate the derivative at a
Now we need to evaluate the derivative at the point a = π/3:
f'(π/3) = cos(π/3) = 1/2
Step 4: Use the linearization formula
The linearization L(x) of the function f(x) at the point a is given by the formula:
L(x) = f(a) + f'(a)(x – a)
Step 5: Substitute the values
Now we can plug in the values we computed:
L(x) = (√3/2) + (1/2)(x – π/3)
Final form of the linearization
Thus, the linearization of the function f(x) = sin(x) at the point a = π/3 is:
L(x) = (√3/2) + (1/2)(x – π/3)
This linear function approximates the behavior of the sine function near the point x = π/3.