Linearization is the process of approximating a non-linear function with a linear function. This is done by finding the tangent line of the function at a specific point, which becomes the linear function.
Linearization helps us to approximate the behavior of a non-linear function at a specific point, making it easier to analyze and understand. It also allows us to use linear equations and techniques to solve problems involving non-linear functions.
The formula for linearization of a function f(x) at a point a is given by: L(x) = f(a) + f'(a)(x-a), where f'(a) represents the derivative of f(x) at x=a.
To linearize a function, we need to follow these steps:
To linearize f(x)=x^1/3 at a=-64, we need to first find the derivative of the function, which is f'(x) = (1/3)x^(-2/3). Then, we can substitute a=-64 to find f'(a) = (1/3)(-64)^(-2/3) = -1/192. Finally, we can plug these values into the linearization formula L(x) = f(a) + f'(a)(x-a) to get L(x) = (-64)^(1/3) - (1/192)(x-(-64)), which simplifies to L(x) = x/4 + 16.