Question

Find the linear approximation of f(x) = √(x+1) at a = 1.

A. L(x) = 1 + (1/2)(x - 1)
B. L(x) = √(2x)
C. L(x) = x
D. L(x) = 1 + (1/2)(x - 2)

Answer 1

The linear approximation of f(x) at a = 1 should be L(x) = √2 + 1/2(x - 1), but the options provided do not match this result, implying an error in the question or options.

Step-by-step explanation:

The question asks to find the linear approximation of the function f(x) = √(x+1) at a = 1. The linear approximation, also known as the tangent line approximation, is given by the formula L(x) = f(a) + f'(a)(x - a). First, calculate f(1) = √(1+1) = √2. Then, find the derivative f'(x) = (1/2)(x+1)^(-1/2), and then calculate f'(1) = (1/2)(2)^(-1/2) = 1/2. Now, apply these values to the linear approximation formula to obtain L(x) = √2 + (1/2)(x - 1). However, no option matches this result, suggesting a possible mistake in the question or the provided options.