Abstract
In this chapter we study derivatives of functions and some of their applications. Along the way, we show how to use derivatives to solve the problem of finding the monotonicity (resp. concavity) intervals of a differentiable (resp. twice differentiable) function, as well as to build an accurate sketch of the graph of such functions. In turn, the analysis of such problems will motivate several interesting applications of the concept of derivative to problems of maxima and minima.
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Notes
- 1.
Gottfried Wilhem Leibniz , German mathematician and philosopher of the XVII century. Together with Sir Isaac Newton, Leibniz is considered to be one of the creators of the Differential and Integral Calculus. Some of the notations used in Calculus up to this day go back to Leibniz, having survived the ruthless test of time.
- 2.
- 3.
Later, in Chap. 11, we shall see an example of a continuous function \(f: \mathbb{R} \rightarrow \mathbb{R}\) that is not differentiable at any point.
- 4.
Since f (k) is also used to denote the composition of f: I → I with itself, k times, we will rely on the context to clear any danger of confusion.
- 5.
We shall have more to say on Newton’s method in Problem 7, page 455.
- 6.
After Guillaume F. Antoine, Marquis de l’Hôpital , French mathematician of the XVII century. A more refined version of l’Hôpital’s rule will be presented at Proposition 9.50.
- 7.
After Michel Rôlle , French mathematician of the XVII century.
- 8.
This terminology alludes to the key role of the first derivative in this result, as well as to the fact that, classically, the first derivative of a (differentiable) function was called its first variation.
- 9.
After Jean-Gaston Darboux , French mathematician of the XIX and XX centuries.
- 10.
After Johan Jensen , Danish engineer and mathematician of the XIX and XX centuries.
Bibliography
A. Caminha, An Excursion Through Elementary Mathematics II - Euclidean Geometry (Springer, New York, 2018)
A. Caminha, An Excursion Through Elementary Mathematics III - Discrete Mathematics and Polynomial Algebra (Springer, New York, 2018)
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Caminha Muniz Neto, A. (2017). Limits and Derivatives. In: An Excursion through Elementary Mathematics, Volume I. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-53871-6_9
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