Linear approximation, new volume versus volume change

Question

Would it be correct to say, supposing that we are dealing with the volume of a ball (just to exemplify), that:

$f(a+\Delta x)\approx L(a+\Delta x)=f(a)+f'(a)\Delta x$ is an approximation of the new volume

and

$\Delta f(x) \approx f'(a) \Delta x$ is an approximation of the change of volume?

I am confused because in Adam's calculus they write the following:

$L(r+\Delta r)=V(r)+4\pi r^{2}\Delta r$ Thus, $\Delta V\approx L(r+\Delta r)=4\pi r^{2}\Delta r $

I do not understand how $L(r+\Delta r)$ can equal the new volume and approximately equal the volume change at the same time. Where does my logic fail here?

Answer 1

It should say $\Delta V \approx L(r + \Delta r) - V(r) = 4 \pi r^2 \Delta r$ (assuming $L$ is the local linear approximation to $V$ at $r$).