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Perturbing the Mean Value Theorem: Implicit Functions, the Morse Lemma, and Beyond.
- Source :
-
American Mathematical Monthly . Jan2021, Vol. 128 Issue 1, p50-61. 12p. - Publication Year :
- 2021
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029890
- Volume :
- 128
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- American Mathematical Monthly
- Publication Type :
- Academic Journal
- Accession number :
- 148137174
- Full Text :
- https://doi.org/10.1080/00029890.2021.1840879