When Are There Continuous Choices for the Mean Value Abscissa?
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866908464970727424 |
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| author | Lowry-Duda, David Wheeler, Miles H. |
| author_facet | Lowry-Duda, David Wheeler, Miles H. |
| contents | 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 $c$ 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?
Drawing inspiration from graphed examples, we first investigate this question by proving and using a simplified implicit function theorem. To handle certain edge cases, we then build on this analysis to prove and use a simplified Morse's lemma. Finally, further developing the tools proved so far, we conclude that if $f$ is analytic, then it is always possible to choose mean value abscissae so that $c$ is a continuous function of $b$, at least locally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_02026 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | When Are There Continuous Choices for the Mean Value Abscissa? Lowry-Duda, David Wheeler, Miles H. Classical Analysis and ODEs 26A06, 26B10 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 $c$ 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? Drawing inspiration from graphed examples, we first investigate this question by proving and using a simplified implicit function theorem. To handle certain edge cases, we then build on this analysis to prove and use a simplified Morse's lemma. Finally, further developing the tools proved so far, we conclude that if $f$ is analytic, then it is always possible to choose mean value abscissae so that $c$ is a continuous function of $b$, at least locally. |
| title | When Are There Continuous Choices for the Mean Value Abscissa? |
| topic | Classical Analysis and ODEs 26A06, 26B10 |
| url | https://arxiv.org/abs/1906.02026 |