Final answer:
The local linear approximation of the function f(x) = √(2+x) at x₀ = 7 is used to approximate √8.9 and √9.1. The approximations are calculated as 3.3167 for √8.9 and 3.3500 for √9.1, respectively.
Step-by-step explanation:
To find the local linear approximation of the function f(x) = √(2+ x) at x₀ = 7, we use the formula for the linear approximation, which is f(x) ≈ f(a) + f'(a)(x-a), where a is the point of approximation (x₀) and f'(a) is the derivative of f(x) evaluated at a.
First, we compute the derivative f'(x):
f'(x) = (1/2)(2+x)^{-1/2}(1),
Then, evaluate it at x₀ = 7:
f'(7) = (1/2)(2+7)^{-1/2} = (1/2)(9)^{-1/2} = (1/2)(1/3).
The linear approximation at x₀ = 7 is then:
f(x) ≈ f(7) + f'(7)(x-7) = √9 + (1/6)(x-7).
We can now use this approximation to estimate √8.9 and √9.1.
- For √8.9: f(7) + f'(7)(8.9-7) = 3 + (1/6)(1.9) ≈ 3 + 0.3167 ≈ 3.3167
- For √9.1: f(7) + f'(7)(9.1-7) = 3 + (1/6)(2.1) ≈ 3 + 0.3500 ≈ 3.3500
Therefore, the approximations for √8.9 and √9.1 using the local linear approximation at x₀ = 7 are approximately 3.3167 and 3.3500, respectively.