Question

For the function f(x) = sqrt(1 + x), what is its linearization at x = 0?

a) l(x) = 1 + x
b) l(x) = x
c) l(x) = 1
d) l(x) = 0

Answer 1

The linearization of the given function f(x) = sqrt(1 + x) at x = 0 is l(x) = 1/2*x + 1. The answer does not match with the options provided .

Step-by-step explanation:

The linearization of a function at a given point is an approximation of the function using a tangent line. In this case, we want to find the linearization of the function f(x) = sqrt(1 + x) at x = 0.

To find the linearization, we need to find the slope of the tangent line at x = 0 and the y-intercept of the line. The slope can be found by taking the derivative of f(x) and evaluating it at x = 0, and the y-intercept is f(0).

First, let's find the derivative of f(x). The derivative of sqrt(1 + x) is 1/(2*sqrt(1 + x)). Evaluating this derivative at x = 0, we get 1/2.

Next, let's find f(0). Substituting x = 0 into the original function, we get sqrt(1 + 0) = sqrt(1) = 1.

Therefore, the linearization of f(x) at x = 0 is given by l(x) = 1/2*x + 1.