by Jeffrey Clark
This article originally appeared in:
College Mathematics Journal
November, 2011
Subject classification(s):
Calculus | Single Variable Calculus | DifferentiationApplicable Course(s):
3.2 Mainstream Calculus IIMotivated by the observation that the derivatives of \(e^x\) are all positive and the derivatives of \(e^{-x}\) alternate sign, the author asks what kinds of ``sign patterns" are possible for infinitely differentiable real functions. The author shows that infinitely differentiable functions with domain \(\mathbb{R}\) can exhibit one of only four possible sign patterns. However, Taylor series are used to show that all sign patterns are possible if the domain is a bounded open interval.
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Capsule Course Topic(s):
One-Variable Calculus | Differentiation: Definition and Elementary Application