A question about snap, crackle and pop [duplicate]

I understand that the terms snap, crackle and pop aren't exactly official terms but I needed a compact title about various derivatives of position with time.

A body at rest has all it's time derivatives of position equal to $0$. When it starts to move it gains velocity. So it's velocity has changed implying acceleration. Acceleration has changed implying jerk and so on. So even the $1,000,000th$ derivative is not 0. Am I right about this or is there some flaw to my method?

If I'm right about this. So once the body starts to move why is it that we only consider the derivatives up to acceleration, hardly consider jerk and rarely mention higher derivatives? Intuition tells me that one the body starts to move, any change in speed can be attributed to acceleration and that takes care of everything and we need not consider anything beyond. But I would like to have a feel of the real answer. An amateur like me told me that higher derivatives are not considered for such problems and not even for some differential equations as the derivatives are small and ignoring them will result in a negligible error. The answer feels sound for engineering but not for theoretical physics. Is she right to say it's okay to ignore tiny values?

Also does it signify anything interesting when derivatives up to the 1000th derivative are all varying?

Either this is a valid question or I'm fundamentally flawed. Forgive me for wasting your time if it's the latter.

EDIT: this is not a duplicate in any way. The question similar to mine assumes advanced knowledge of lagrangian mechanics and the category it came under was differential geometry and field theory. My question is on Newtonian mechanics and neither do I have the mathematical background to understand the answers to the other question. In fact I don't know enough Mathematics to understand the other question itself. ;-)

• Possible duplicates: physics.stackexchange.com/q/18588/2451 , physics.stackexchange.com/q/4471/2451 and links therein. – Qmechanic Nov 11 '16 at 7:34
• I agree that your question is not a duplicate of that cited. But there are a lot of very similar questions in the "related" list (including "Why F=ma?"), and yet others in their related lists (eg Is it possible to have a rate of change of acceleration?. It is not clear how exactly your question differs from them. Have you considered them? – sammy gerbil Nov 11 '16 at 14:02
• I think the problem is one of mathematical modelling. Models cannot account for everything. The Taylor Expansion is only a model. It is not reality. Even in the mass-on-a-spring the acceleration is not constant, and all of the higher derivatives are non-zero. Yet motion is more simply described by $x(t)=A\sin(\omega t)$ rather than its Taylor expansion. – sammy gerbil Nov 11 '16 at 14:11