**APPENDIX A**

**PROOFS OF SELECTED THEOREMS, AND ADDITIONAL MATERIAL**

The interpolation error theorems all depend on applying a generalization of Rolle’s Theorem to a very carefully constructed function. First, we will state and prove the general version of Rolle’s Theorem.

**Theorem A.1 (Generalized Rolle’s Theorem)** *Let f* *C*^{n}([*a, b*]) *be given, and assume that there are n points, z*_{k}, 1 ≤ *k* ≤ *n in* [*a, b*] *such that f*(*z*_{k}) = 0. *Then there exists at least one point* ξ [*a, b*] *such that f*^{(n-1)}(ξ) = 0.

**Proof:** By Rolle’s Theorem, there exists at least one point η_{k} between each *z*_{k} and *z*_{k+1} such that *f*′(η_{k}) = 0. Thus there are *n* − 1 points where *g*_{1}(*x*) = *f*′(*x*) is zero. By the same argument, then, there are *n* − 2 points where *g*_{2}(*x*) = *f*″(*x*) is zero. Continuing onward, then, we end up with a single point where *g*_{n−1}(*x*) = *f*^{(n−1)}(*x*) is zero.

This result allows us to prove, rather easily, both of Theorems 4.3 and 4.5.

**Theorem A.2 (Lagrange Interpolation Error Theorem)** *Let f* *C*^{n+1}([*a, b*]) *and let the nodes x*_{k} [*a, b*] *for* 0 ≤ *k* ≤ *n. Then, for each x* [*a, b*], *there is ...*

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