Find the linear approximation the function f(x) = (6th root of 1 + x) at a=0. Then use this linearization to approximate the number 6th root of 0.95) .

Question

Give your answer to 4 decimal places

clearly show which value of x you are using for your approximation.

Answer 1

f(x) = (1 + x)^1/6; f'(x) = (1 + x)^-5/6/6; f(0) = 1; f'(0) = 1/6;

So, linear approximation L(x) = 1 + 1/6(x - 0); L(x) = 1/6x + 1

Let 1+ x = 0.95;x = - 0.05; f(0.95) ≈ L(- 0.05) = - 0.05/6 + 1 = 0.9917