Integration question: Fundamental Theorem of Calculus?

Question

Let f be a continuous function taking positive real values, and set

G(x) = S f(z) dz

limits are from z=x to x^2

Use the chain rule to find G'(x)

Estimate the value of G(x) for values of x near 1. (You should do more than just give the value of lim x-->1 G(x).

Answer: I can do the first bit, G'(x) = 2x f(x^2) - f(x).

But I can't figure out the second bit where the answer is :

G(1) = 0 and G'(x) = f(1) > 0. Thus an approximation to G(x) near x=1 is

G(x) approximately equals to G(1) + (x-1)G'(!) = (x-1) f(1)

Why is this so?

Answer 1

If you take the Taylor series for G(x) with center the point x= 1

G(x) = G(1) +(x-1)G´(1) +(x-1)^2/2!*G´´(1)+++
so you are taking the two firts terms of the Taylor series