Question

What is the linear approximation of `g(x)=sqrt(1+x)^(1/5)` at a =0?

Answer 1

(I suppose that you mean x=0)

The function, using the power properties, becomes: `y=((1+x)^(1/2))^(1/5)=(1+x)^((1/2)(1/5))=(1+x)^(1/10)`

To make a linear approximation of this function it is useful to remember the MacLaurin series, that is the Taylor's polinomial centered in zero.

This series, interrupted to the second power, is:

`(1+x)^alpha=1+alpha/(1!)x+(alpha(alpha-1))/(2!)x^2...`

so the linear approximation of this function is:

`g(x)=1+1/10x`