Las derivadas de la vida cotidiana

Muchas veces, con la ayuda del sentido común, estamos derivando sin darnos cuenta. Naturalmente, uno no necesita derivar en la vida diaria, sin embargo las derivadas son necesarias en muchas aplicaciones prácticas en biología, mecánica, en medicina bacteriológica, etc.

Especialmente el concepto de derivada es fundamental para comprender y derivar fórmulas que luego tienen una aplicación importante en la industria y en la ciencia en general, que es la que definitivamente inspira las innovaciones industriales. Las derivadas se utilizan para optimizar sistemas que se expresan mediante funciones más o menos complejas. Otra de sus aplicaciones es hallar los valores máximos o mínimos de ciertas expresiones. También puedes hallar los intervalos de crecimiento o decrecimiento de valores de interés.

Por eso mismo utilizamos las derivadas sin darnos cuenta, por ejemplo, cuando vamos al supermercado y vemos que han subido el precio de las patatas respecto al mes pasado, nosotros podemos hallar la variación con una simple derivada. Otro caso, se puede encontrar en “casa”, es decir, cuando queremos calentar la casa encendiendo la calefacción podemos saber cuánto tiempo tardará en calentarse derivando en función del espacio, la temperatura y el tiempo. O simplemente cuando vamos en el coche y aceleramos, hallamos la velocidad derivando esa aceleración en función del tiempo.

Entrevista con un matemático

INTERVIEW WITH A MATHEMATICIAN

Charles L. Fefferman is a mathematician who wrote his first scientific work at age 15 and was Professor at 22. This mathematician, who is at the University of Princeton (USA), was in 1978 the equivalent Fields Medal to Nobel Prize in mathematics for researchers under 40 years of age.

Q. When working, think on possible applications, or only on the problem?
A. Only in the problem, but I hope to eventually take to applications. I have been involved in works that have led to very interesting applications, and am very pleased with this. Working for years in a relevant problem for applications. It is easy to state: I have, say, a million points in a room, and I'd like to draw a surface as smooth as possible quepase by all of them. It is a deep problem, probably will continue a few more years with him.

Q. How long do you work in a problem, average?
A. The most that I have devoted to a group of problems is 15 years. They were problems of quantum mechanics. Then I was interested in another area, the fluids, which I now work with, among others, Antonio Córdoba.

Q. But solved those problems of quantum mechanics.
A. Says that I made a lot of progress.
Q. In mathematics, physics... There are problems that seem actually insoluble, given its resistance. How should we deal with them? Is there some mental wall effect?
A. Happens constantly problems that seem unsolvable are actually very simple. Suddenly someone discovers a particular way of looking at them makes that easily resolved. Moreover, one forgets that once you were difficult.
Q. What has happened to you ever?
A. Yes, I think that I have had two such moments. They are the sweetest. In particular one. It was something that no one imagined that it could be true. Remember be explaining it, before I met, very distinguished mathematicians, and before reaching the show one of them told me "no, that is not ridiculous". I felt great.
Q. It is important to keep this in mind when dealing with problems?

A. Of course. My Professor, Elias Stein, taught me that optimism is important. There are problems that really are not solvable, but other, very hard, they can be solved, and it is important to not have them fears.

Q. As a child prodigy, you educated are with people who were not of age. How it felt?

A. In most aspects it was happy. It was difficult not to be part of a group of people that, on the other hand, had much in common. But it was a price that was worth paying. If you had followed the normal education, I think I would have felt very frustrated

Q. Said that it does not reject any student who wants to make the thesis with you. How many do you have?

A. Now I have two, but I have come to have five at the same time. It takes a long time, of course, but it's satisfying. I try to organise myself to continue to have time to think, although I do not always succeed.

Q. You need a special environment to think?

A. This is a very sensitive issue. To think you must be in the right state of mind, you have rested well, your work must be excited... And you need some time.

Fuente: I-Math; Mónica Salomé - Madrid - 03/07/2009

Límites de dos variables

Definición: sea f una función de dos variables cuyo dominio D incluye puntos tan cercanos como se quiera a (x0, y0). Entonces decimos que el límite de f cuando (x, y) se aproxima a   (x0, y0) es L y escribimos:


                 Lim   f(x, y) = L

                                      (x, y)→(x0, y0)

                     

 Calculo: Las dos formas más comunes del cálculo de límites en función de varias variables son:

• ·         Límites iterados

Sea f una función de dos variables. Llamaremos límites iterados de f en el punto (x0, y0) a los límites:

L1 = Lim   ( Lim  f (x, y))

           x→ x0     y→ y0

L2 = Lim    ( Lim  f (x, y))

          y→ y0       x→ x0

• ·         Límites polares

En algunos casos, la introducción de coordenadas polares puede simplificar las expresiones y facilitar el cálculo de los límites. Supongamos que tenemos una función f de dos variables y que queremos calcular el límite en un punto (x0, y0). Para hacer el límite podemos probar con el cambio de variable x = x0 + ρ cos θ   ,   y = y0 + ρ sin θ.

Se pueden hacer ejercicios en el siguiente enlace:  

http://www2.topografia.upm.es/asignaturas/matematicas/segundo/Hojas%20de%20ejercicios%20MII/F.%20de%20varias%20variables,%20limites%20y%20continuidad/soluciones/Limites%20y%20continuidad.pdf