Generalizing Rolle's Theorem in Elementary Calculus
Rodney D. Gentry · The Two-Year College Mathematics Journal · 1973
Rodney D. Gentry is Assistant Professor of Mathematics at the University of Guelph, Ontario. He received his Ph.D. from the University of California at Davis, and his research has been in the area of differential equations. He has taught for several years and has used discovery teaching methods in two-year and university level mathematics courses. In elementary calculus classes, Rolle's Theorem is frequently generalized to obtain the Mean Value Theorem. I present here some less widely noted generalizations of Rolle's Theorem which may, however, be successfully developed in elementary calculus classes. I also indicate a method of introducing Rolle's Theorem which differs from that found in most calculus texts. There are many interesting extensions of Rolle's Theorem and the Theorem of the Mean which consider several functions or functions defined on vector or norm spaces. In general, such generalization through abstraction does not come easily to freshman calculus students. Instead, I have found that students naturally seek to generalize theorems by simply removing a portion of the hypothesis or by replacing it with a weaker condition. The latter occurs most frequently when students lack a good grasp of particular definitions or conditions; consequently they subjectively attribute the properties needed to prove a theorem to a weaker hypothesis. This is particularly the case when students examine and attempt to generalize Rolle's Theorem.