Find the linear approximation of the function f(x) =

Question

Find the linear approximation of the function f(x) =4-x. at a = 0.

Use L(x) to approximate the numbers 3.9 and 3.99

Answer 1

Lenar approximation at x = 0 L(x) = f(0) + f'(0)x; f(0) = sqrt(4 - 0) = 2; f'(x) = - 1/2(4 - x)^-1/2; f'(0) = -1/2·1/2 =

- 1/4. L(x) = 2 - 1/4x We say (4 - x)^1/2 ≈ 2 - 1/4x.

If x = 0.1 we get 3.9^1/2 ≈ 2 - 1/4·0.1 = 1.975

If x = 0.01 we get 3.99^1/2 ≈ 2 - 1/4·0.01 = 1.9975