To find the linearization, or the linear approximation, of the function at a point, we follow a few simple steps. Let’s consider the function f(x) and the point where we want to linearize it.
1. **Identify the function and point**: First, we need to know the function f(x) and the point a at which we want to find the linearization. In your case, you’re mentioning ‘at a fx x12 a 4’, which I interpret as needing to linearize around x = 4.
2. **Calculate the derivative**: The next step involves finding the derivative of the function, f'(x). This derivative gives us the slope of the tangent line to the function at the given point.
3. **Evaluate the function and its derivative at point ‘a’**: You then need to find the values of the function and the derivative at the specified point:
- f(a) = f(4)
- f'(a) = f'(4)
4. **Form the linearization formula**: The linearization L(x) of the function at the point x = a can be expressed as:
L(x) = f(a) + f'(a)(x – a)
5. **Substitute the values into the formula**: Replace f(a) and f'(a) with the calculated values to get the final linearization.
Example: If our function was f(x) = x^2, then:
- f(4) = 16
- f'(x) = 2x, hence f'(4) = 8
So, the linearization at x = 4 would be:
L(x) = 16 + 8(x – 4)
This is the process to find the linearization of the function at the given point.