Recall the following identity, assuming continuity of $f$:
$$\lim_{\Delta x\to0} \frac{\int_{x_0}^{x_0+\Delta x}f(x)}{\Delta x}=f(x_0)$$
Can similar expressions in terms of $f(x_0)$ or its derivatives be derived for the following generalization $I_n$ assuming continuity and infinite differentiability of $f$?
$$I_n = \lim_{\Delta x\to0} \frac{\int_{x_0}^{x_0+\Delta x}(x-x_0)^nf(x)}{(\Delta x)^{n+1}}$$
COMMENT: I highly suspect that $I_n$ is proportional to $n$-th order derivatives of $f$ evaluated at $x_0$, but haven't been able to get this using typical integration by parts.