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Convex Functions and Taylor’s Theorem

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More Calculus of a Single Variable

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Abstract

In this chapter we consider the higher derivatives of a function f. These are \(f^{{\prime\prime}} = (f^{{\prime}})^{{\prime}},\) \(f^{(3)} = (f^{{\prime\prime}})^{{\prime}},\) etc. We extend the Mean Value Theorem to an analogous statement about the second derivative, and this takes us naturally to the notion of convexity. Once there, we meet the very important Jensen’s Inequality. Then we extend the Mean Value Theorem to the (n+1)st derivative—this is Taylor’s Theorem. We prove that \(\mathrm{e}\) is irrational and we take a brief look at Taylor series.

A smile is a curve that sets everything straight.

– Phyllis Diller

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Authors and Affiliations

  1. Mathematics Department, SUNY Buffalo State, Buffalo, NY, USA

    Peter R. Mercer

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  1. Peter R. Mercer
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Mercer, P.R. (2014). Convex Functions and Taylor’s Theorem. In: More Calculus of a Single Variable. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1926-0_8

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