Abstract
ABSTRACT Error estimates for tangent line approximations and for numerical integration are found using special cases of the error formulas for Taylor's Theorem and the Trapezoidal Rule, respectively. Proofs of these theorems rely on a modification of Rolle's Theorem. By recasting these theorems, their proofs are more accessible to first semester calculus students. The advantages are several. First, students are able to estimate errors in approximations. Second, to take full advantage of the results, students need to use a graphing calculator or computer algebra system. Third, students are introduced to two theorems which use the same proof-technique. Fourth, students become familiar with one of calculus' big-name theorems.
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