calculus help?

Question

use a linear approximation to estimate the value of (41/10)^1/2

Answer 1

Let x = (41/10)^(1/2)

Then:
x = (4 + 0.1)^(1/2)
= 4^(1/2)(1 + 0.1/4)^(1/2)
= 2(1 + 0.1/4)^(1/2)

Expanding the root by the binomial theorem as far as the 2nd. term:
x = 2(1 + (1/2)*0.1/4 + ... )
= 2(1 + 0.0125)
= 2.025.