Question

Find the linearization of f(x)=1/(1/x)

a) L(x) = 1 - x
b) L(x) = 1/x
c) L(x) = 1/2x
d) L(x) = 1 + x

Answer 1

To find the linearization of f(x) = 1/(1/x), we can use the concept of linear approximation. The linearization of a function f(x) at a point a is given by the equation L(x) = f(a) + f'(a)(x - a). In this case, the linearization is L(x) = 1 - x.

Step-by-step explanation:

To find the linearization of f(x) = 1/(1/x), we can use the concept of linear approximation. The linearization of a function f(x) at a point a is given by the equation L(x) = f(a) + f'(a)(x - a), where f'(a) represents the derivative of f(x) at the point a. In this case, f(x) = 1/(1/x), so we need to find the derivative of f(x) and evaluate it at a certain point a.

First, we find the derivative of f(x) using the quotient rule. Let g(x) = 1 and h(x) = 1/x. Using the quotient rule, we have f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2 = (-1/x^2)/(1/x^2) = -1.

Since the derivative of f(x) is -1, we can use any point as a. Let's choose a = 1. Plugging into the linear approximation formula, we have L(x) = f(1) + f'(1)(x - 1) = 1 + (-1)(x - 1) = 1 - x.