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Prologue The Exponential Function

Keith Wright, Walter Rudin · 2003

Assuming only basic trigonometry, we define the Modifed Discrete Cosine Transform (MDCT), and prove its basic properties. The dread words “it can be shown ” do not occur. This document is not complete, but I am releasing the draft because there is enough here that it may be useful to some-one. Comments and suggestions for improvement are wel-come, especially if you know the answers to any of the ques-tions at the end.

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Exponential function and derivability

Željka Trajbar · Repository of the Faculty of Science, University of Zagreb · 2020

Ukratko, u ovom diplomskom radu proučavali smo neka svojstva eksponencijalne funkcije s naglaskom na njenu derivabilnost. Najprije smo definirali eksponencijalnu funkciju s bazom većom od 1, prvo smo za a > 0 definirali \(a^x\) za sve \(x \in \mathbb{N}\), potom za sve \(x \in \mathbb{Z}\), onda i za \(x \in \mathbb{Q}\), pri čemu smo dokazali da za svaki a > 0 i \(n \in \mathbb{N}\), postoji jedinstveni \(\sqrt[n]{a} \). Nadalje, definirali smo za a > 1 \(a^x\) za sve \(x \in \mathbb{R}\), proučavali smo konvergenciju nizova realnih brojeva te omeđene nizove u \(\mathbb{R}\). Potom smo proučavali neprekidnost eksponencijalne funkcije te posljedice koje ona ima na svojstva eksponencijalne funkcije. Također, dokazali smo da je eksponencijalna funkcija neprekidna, zatim smo došli do međuvrijednosti neprekidnih funkcija te koristeći dobivene rezultate dokazali smo da je eksponencijalna funkcija bijekcija, proučavali smo i pojmove limesa i derivacije funkcija te smo dokazali da vrijedi tzv. eksponencijalni zakon i definirali broj \(e\). Definirali smo logaritamsku funkciju s bazom većom od 1, proučavali smo derivabilnost eksponencijalne funkcije pa smo potom definirali eksponencijalnu funkciju s bazom manjom od 1, zatim logaritamsku funkciju s bazom manjom od 1 te smo dokazali neka svojstva tih funkcija. Na kraju smo dokazali da je funkcija ln derivabilna u točki 1 te, pomoću toga, došli smo do glavnog rezultata, a to je da je eksponencijalna funkcija derivabilna.

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The exponential hash function

Abdallah, Chaouki T., Smith, Bradley J., Heileman, Gregory L. · UNM’s Digital Repository (University of New Mexico) · 2012

In this paper an efficient open address hash function called exponential hashing is developed. The motivation for this hash function resulted from our ongoing efforts to apply dynamical systems theory to the study of hashing; however, the analysis conducted in this paper is primarily based on traditional number theory. Proofs of optimal table parameter choices are provided for a number of hash functions. We also demonstrate experimentally that exponential hashing essentially matches the performance of a widely-used optimal double hash function for uniform data distributions, and performs significantly better for nonuniform data distributions. We show that exponential hashing exhibits a higher integer Lyapunov exponent and entropy than double hashing for initial data probes, which offers one explanation for its improved performance on nonuniform data distributions.

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Function exponential and logarithmic

Rodrigo Augusto Tadeu Martins Leal da Silva · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 2016

No ensino da matemática um dos assuntos mais desa adores aos alunos do Ensino Médio é o de funções exponenciais e logarítmicas, sendo que grande parte dos alunos possuem difi culdades de compreensão e resolução dos exercícios propostos. Dessa maneira o trabalho em tela tem como objetivo precípuo o ensino de funções exponenciais e logarítmicas, visando apresentar aos docentes a possibilidade de ensinar o conteúdo de maneira mais atrativa e acessível aos seus alunos. Para tanto, a pesquisa se valeu de levantamento bibliográ fico e documental acerca da temática abordada. Salienta-se que ao iniciar o ensino de funções é preciso fazer um resgate histórico do conteúdo, buscando desvelar suas origens. Posteriormente o trabalho traz as concepções do tema segundo os documentos o ciais. A m de concretizar o estudo trabalha-se com o uso das defi nições aritméticas e geométricas, pois são fundamentais para o entendimento das funções, e também com situações problema contextualizadas, buscando envolver os alunos no processo de ensino aprendizagem promovendo o levantamento de hipóteses e consolidando a aprendizagem. Essas situações problemas serão difundidas por meio de atividades propostas nas quais pretende-se explorar a caracterização da função logarítmica e exponencial, buscando relacioná-las ao contexto do educando, em situações que poderiam ocorrer em seu cotidiano, para isso propomos alguns exemplos de atividades, como: a utilização do BRO ce Calc no Ensino de potenciação com números irracionais, o jogo de xadrez, mágica do baralho e a resolução de problemas da OBMEP e do ENEM.

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CMOS exponential function generator

Weihsing Liu, Shen-Iuan Liu · Electronics Letters · 2003

A new CMOS exponential function generator is presented. The proposed circuit is compact, with low power and wide dynamic range. The proposed circuit has been fabricated in a 0.50 µm CMOS process. Experimental results show that the output range of the proposed exponential function generator can be more than 15 dB with the liner error less than ±0.5 dB. The supply voltage is ±1.5 V and the power dissipation is less than 0.4 mW. Experimental results are given to demonstrate the proposed circuit.

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An Exceptional Exponential Function

Branko Ćurgus · College Mathematics Journal · 2006

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AN EXCEPTIONAL EXPONENTIAL FUNCTION

2016

There are at least two surprising results in this article. The first is that there is actually a link between the standard calculus problem of finding the best view of a painting and graphs of exponential functions. The second is that the exponential function with the “best view ” is not the one with the base e. As a bonus we get an interesting appearance of the Lambert W function. The painting problem One of the standard calculus problems is this: A painting hangs on a wall in an art gallery. Its top edge is at distance p and its bottom edge at distance q above the eye level of an art lover. How far from the wall should the art lover stand to get the best view? (Here the “best view” means the maximum viewing angle; see Figure 1.) This statement of the painting problem is similar to Exercise 44 in Stewart [9, p. 314]. In Problem 23 of Hughes-Hallett, et al. [6, p. 203], a painting is replaced by the Statue of Liberty and the values of p and q are specific numbers. In this setting the requirement that the bottom of the observed object be above the eye level of an observer seems more natural. We leave the details of the solution of the painting problem to the reader. Before proceeding, however, we make an observation. For each distance from the gallery wall, there is a corresponding viewing angle. Moreover, for each distance v other than qp from the wall there is a second distance qp/v with the same viewing angle. Figure 1 gives a geometric proof of this fact. Ray OD in the figure represents eye level. LetOA = OA ′ = p, OB = OB ′ = q, OC = OC ′ = v, and OD = qp/v. Right triangles △OAC and △OA′C ′ are congruent since their legs are congruent. Similarly, right triangles △OBC and △OB′C ′ are congruent. Hence the marked angles at C and C ′ are congruent. Furthermore, right triangles△OA′C ′ 1 2 BRANKO ĆURGUS A p B q

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