To find the linearization of the function f(x) = sin(x) at the point a = π/3, we first need to determine the value of the function and its derivative at this point.
1. **Calculate f(a)**: Substitute a = π/3 into the function.
f(π/3) = sin(π/3) = √3/2
2. **Calculate f'(x)**: Now, we find the derivative of f(x).
f'(x) = cos(x)
3. **Calculate f'(a)**: Substitute a = π/3 into the derivative.
f'(π/3) = cos(π/3) = 1/2
4. **Write the linearization formula**: The linearization of the function at x = a is given by:
L(x) = f(a) + f'(a)(x – a)
5. **Substitute the values**:
L(x) = (√3/2) + (1/2)(x – π/3)
6. **Final Linearization**: Thus, the linearization of f(x) = sin(x) at a = π/3 is:
L(x) = (√3/2) + (1/2)(x – π/3)
In summary, linearization allows us to approximate the function near a given point with a linear function, making it easier to analyze and compute values close to that point.