Perturbing the Mean Value Theorem: Implicit Functions, the Morse Lemma, and Beyond. Issue 1 (29th December 2020)

• Record Type: Journal Article
• Authors: Lowry-Duda, David; Wheeler, Miles H.
• Is Part Of: American Mathematical Monthly. Volume 128:Issue 1(2021)
• Page Start: 50
• Page End: 61
• Publication Date: 2020-12-29
• Subjects: MSC Primary 26A24 -- Secondary 26A06 -- 26A15; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2021.1840879
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: The mean value theorem of calculus states that, given a differentiable function f on an interval [ a, b ], there exists at least one mean value abscissa c such that the slope of the tangent line at ( c, f ( c ) ) is equal to the slope of the secant line through ( a, f ( a ) ) and ( b, f ( b ) ) . In this article, we study how the choices of c relate to varying the right endpoint b . In particular, we ask: When we can write c as a continuous function of b in some interval? As we explore this question, we touch on the implicit function theorem, a simplified version of the Morse lemma, and the theory of analytic functions.