How do you find the linearization L(x) of the function f(x) at a = 4?

To find the linearization L(x) of the function f(x) at a specific point, we first need to calculate the derivative of the function and the value of the function itself at that point.

Here’s the general formula for the linearization of a function f(x) around a point a:

L(x) = f(a) + f'(a)(x – a)

Let’s break it down:

1. Step 1: Find f(a)
   Calculate the value of the function at x = 4, which is f(4).
2. Step 2: Find f'(a)
   Calculate the derivative of the function f(x) and then evaluate it at x = 4, giving us f'(4).
3. Step 3: Substitute into the linearization formula
   Put the values from steps 1 and 2 into the linearization formula.

For example, if f(x) = x², we would first compute:

• f(4) = 4² = 16
• f'(x) = 2x, thus f'(4) = 2(4) = 8

Substituting these into the linearization formula gives:

L(x) = 16 + 8(x – 4)

This would be the linearization of the function f(x) = x² at the point x = 4.