Then I did a MathSciNet search: "Anywhere=(mean value theorem)". This list was compiled as follows: first I started with the five or so articles on MVT that I remembered, mostly from having appeared in the Monthly. Koliha, Mean, Meaner, and the Meanest Mean Value Theorem. Swann, Commentary on Rethinking Rigor in Calculus: The Role of the Mean Value Theorem. Tucker, Rethinking Rigor in Calculus: The Role of the Mean Value Theorem. Boas Jr., Who needs these mean-value theorems anyway? Two-Year College Math. Sanderson, Classroom Notes: A Versatile Vector Mean Value Theorem. Boas Jr., Classroom Notes: Lhospital's Rule Without Mean-Value Theorems. Levi, Classroom Notes: Integration, Anti-Differentiation and a Converse to the Mean Value Theorem. Bers, On avoiding the Mean Value Theorem. Cohen, On being mean to the Mean Value Theorem. Barrett, Classroom Notes: Methods of Proving Mean Value Theorems. Evans, Classroom Notes: Sequences Generated by Use of the Mean Value Theorem. (In this article, the derivation of MVT from Rolle's theorem by tilting one's head is presented in horrible analytic detail.) Wang, Classroom Notes: Proof of the Mean Value Theorem. Zeitlin, Classroom Notes: An Application of the Mean Value Theorem. Spiegel, Mean Value Theorems and Taylor Series. Bush, Classroom Notes: On an Application of the Mean Value Theorem. Certainly they are written by people who have thought deeply and in novel ways about this result - i.e., by mathematicians. they discuss logical implications between different forms of MVT. Most of these articles have some real mathematical content: e.g. The purpose of this answer (which I would make CW even if the question weren't) is to collect references to scholarly articles on MVT and its role in introductory calculus courses. This is obvious from the intuitive point of view, but not from the formal point of view. This choice varies from class to class.ĮDIT: You use the MVT to show that positive derivative corresponds to increasing function. When I realized this, I decided that this point was so subtle that I either have to make a big point of explaining the question or else drop it. That is, the metamathematical content of the MVT is that the intuition definition matches the formal definition. The MVT is the basis for all proofs that geometric intuition about slopes of tangent lines holds for the limit definition. To be sure, the limit is motivated by the tangent approach, but no attempt is ever made to show that the two approaches give the same answer (indeed this can't be proved using the usual definitions, since ultimately the definition of tangent line is based on the derivative). The second is to compute a certain limit. The first is to draw the tangent line and measure its slope. This means that we are really working with two different definitions of the derivative. The second is concerned with the technicalities, showing how abstract mathematics can lead to very useful, interesting, and important results. The first is not really concerned with a rigorous presentation rather it tries to get the main ideas, their interrelations, and uses across. My view is that there are essentially two strands in a first calculus course. Note added later: "Franklin" appears to have altered the meaning of the question with his later edits. In what ways is MVT used in further studies that they have to take or that they need)?Ĭan the role of MVT be replaced by a more easy to use/easy to grasp result?Īre there other uses (exercises) more suitable for the uses that MVT has for students of this specialties? How does eliminating MVT from the curriculum affect students from specialties exemplified above (i.e. Is it possible to remove MVT from the program and get a consistent exposition of the rest of the results and techniques in Calculus? It is a fact that many times we fail in making students able to use MVT in these kind problems.Īssume that we are considering a first introduction to calculus for students that mostly will use it in application, students that will work in Biology, Engineering, Chemistry. ) I can imagine a text that doesn't require the student to understand logical rigor, but that instructs them on the role of the MVT in the theory. general properties of functions, or derivatives of functions, inequalities, Taylor. just checking it) or about proving theoretical facts (e.g. Most of the related exercises that books list are either about examples related to the necessity of the hypothesis of the theorem (i.e. Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that interval. Grand Rapids is in Kent, Allendale in Ottawa.Should the mean value theorem be taught in first-year calculus?
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