My question is coming from the method Reid and Chris suggested in solving the problem here. Help on any point is greatly appreciated!

Question 1. For a real manifold $M$, consider $C^{\infty}(M,\mathbb{R})$. For a point $p\in M$, consider the ideal $I_p=\{ f : f(p)=0 \}$. Is $I_p^n$ equal to the set of smooth maps $f$ which have $n-1$st order contact (ref. Golubitsky, Guillemin pg. 37) with the $0$ function?

Question 2. Consider $C(\mathbb{R},\mathbb{R})$ and the ideal $I_0= \{ f : f(0)=0 \}$. Is it the case that $I_0^2 = \{ f : f(0)=f'(0)=0\}$? Is it the case that $I_0^n=\{ f : f(0)=f'(0)=...=f^{(n-1)}(0)=0\}$?

To question 2, the one inclusion is immediate. However the inclusion $ \{ f : f(0)=f'(0)=0\} \subseteq I_0^2 $ doesn't seem obvious to me right now. It seems like one could perhaps find a function which doesn't agree with it's Taylor series that satisfies the derivative condition, but not be in $I^2_0$.

I hope this isn't too elementary for MO. Thanks for your help!