Let $\mathcal{L}^i (i = 1,2, \cdots ,n)$ denote n linear functionals on $C^n [a,b]$, $f \in C^n [a,b]$, $p \in P_{n - 1} $ (the space of polynomials of degree $ < n$), where $\mathcal{L}^i p = \mathcal{L}^i f$. Then one has the formula $| {f(x) - p(x)} | \leqq U(x)\| {f^{(n)} } \|_\infty $, where U is the upper envelope of all functions $R \in C^n [a,b]$ such that $\left\| {R^{(n)} } \right\|_\infty \leqq 1 $ and $\mathcal{L}^i R = 0$. $U(x)$ can be determined for each x to be the total variation of a certain measure calculated directly from the $\mathcal{L}^i $. One can, however, bypass this procedure and still obtain easily calculable formulas for $V(x)$, an upper bound for $U(x)$. In addition, situations in which $f(x) - p(x) = L(x)f^{(n)} (\xi _x )$, $\xi _x \in [a,b]$, will be investigated, where L is the unique element of $C^n [a,b]$ such that $L^{(n)} \equiv 1$ and $\mathcal{L}^{(i)} L = 0$.