5. Alometría y Construcción de Ecuaciones

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5. Alometría y Construcción de Ecuaciones

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Not useful  Useful • Mostrarles que muchas ecuaciones deregresión con

las cuales trabajarán durante su maestría y vida profesional expresión matemática

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Objetivos Sign up to vote on this title

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Not useful  Useful • Mostrarles que muchas ecuaciones deregresión con

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Contenido Sign up to vote on this title

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Useful  Not useful de aula • Definición de alometría y unejercicio

(analogía con partes del cuerpo y desarrollo a

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Alometría Sign up to vote on this title

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• Es el estudio del tamaño de los organismos 

vivos de

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consecuencias biológicas (Nikla

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Ejercicio de aula Sign up to vote on this title

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  • ¿Usted piensa que existe proporcionalidad Useful

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entre la altura total de una persona y el largo

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Ahora vamos a ver el concepto de  función y tasa de cambio; trabajo Sign up to vote on this title Useful

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Describiendo la evolución tempora Sign up to vote on this title

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Uno de los problemas más comunes en las ciencias naturales el de las leyes

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En otras ocasiones interesa comparar la maner como crecen las distintas partes de un  Sign up to vote on this title  Not useful individuo. Por ejemplo, se haUsefulobservado que en los seres humanos, la tasa de crecimiento

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Una de las leyes que describe la relación que existe entre las tasas de crecimiento de

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Y nació el concepto de función: y = f(x) Sign up to vote on this title

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Useful  Not useful La búsqueda de expresiones matemáticas que describan la evolución temporal de

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Una función describe la relación entre una o varia variables x 1 , x 2 , x 3 ,… (variables independientes  Sign up to vote on this title y una variable “y ”, (variable dependiente), cuya  Useful  Not useful conducta deseamos estudiar.

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Maneras de describir una función matemática. Sign up to vote on this title

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Describir una función es especificar la relación existe entre la variable independiente “

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Jerarquía de las tres formas posibles de describir una función matemática: Sign up to vote on this title

Una fórmula

En efecto:

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una tabla

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una gráfica

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Comparación de los incrementos de ambas variables. Sign up to vote on this title

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Para tener una idea de la rapidez con que varía fenómeno “ ” (“ ” podría la altura de

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La razón de cambio  La mejor manera de hacer estaSigncomparación y up to vote on this title obtener una idea del ritmo con que y está useful  Useful  Not respondiendo a los cambios en t es dividiendo ambos incrementos: y/

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Podemos expresar con más detalle la razón de cambio entre un momento inicial i y un  to vote on this title momento final f del tiempo Sign enupla forma: ∆y/ ∆t=

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(y f – y ) / (t i f ‐ t ) i

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Nótese la similitud entre la fórmula que expresa la razón de cambio y la que expresa  up to votepasa on this titlepor do pendiente “m” de una rectaSignque  Useful  Not useful puntos P1(x1, y1) y P2(x2, y2):

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La derivada Sign up to vote on this title

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Useful  Not useful La noción de derivada de una función no es otr cosa que un perfeccionamiento de la noción

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La derivada como razón de cambio puntual. Sign up to vote on this title

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 Useful  Not useful La derivada de una función no es otra cosa que una razón de cambio “puntual”, es decir, en

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Definición de derivada Sign up to vote on this title

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Es decir, la definición de la derivada de una función tomada el valor t

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Una conclusión importante Sign up to vote on this title

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 Useful  Not useful Tanto la razón de cambio como la derivada son formas de expresar la velocidad con que

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Encontrar funciones que expresen con exactitu la conducta de fenómenos Sign naturales, afecta  up to vote on this title usefuluna  Useful  Not dos por muchos factores no suele ser tarea fácil.

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1. Primero se busca formular una ley que describ cómo depende la tasa de crecimiento del  up to vote on this title fenómeno (la derivada) conSign elUseful fenómeno mism  Not useful Esta ley suele expresarse en forma de una ecuación diferencial, es decir, una ecuación qu

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Repertorio de funciones Sign up to vote on this title

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Existen un amplio repertorio de funciones que pueden servir “ladrillos” de base

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Potencias de exponente entero: la base es variabl y el exponente es constante: f(t) = t n donde n es  Sign up to vote on this title un entero. Ejemplo: t 5 

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Polinomios: sumas de términos formados por

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También podemos agregar a la lista:

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 Useful  Not useful En la era de las computadoras, han cobrado fuerza los llamados modelos computacionale

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La función lineal Sign up to vote on this title

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 Useful  Not useful Una de las leyes más simples para describir la relación entre dos variables es la llamada ley

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Se han empleado varios tipos de fórmulas  no matemáticas para describirSignelupcrecimiento to vote on this title useful aquí  Useful  NotHe lineal de poblaciones o individuos. algunas y a la derecha la ecuación diferencia

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La razón de cambio ∆y/ ∆t nos da una medida absoluta de la rapidez con que varía “y ” al varia  Sign up to vote “t ”. Volviendo a nuestro ejemplo, si “ony this” title denota  Useful  Not useful número de árboles en una parcela, y “t ” el tiempo medido años, ∆y/ ∆t dice

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Razón de cambio relativa Sign up to vote on this title

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Para resolver el problema de comparar razones de cambio, la solución dichas

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La ley alométrica Sign up to vote on this title

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Se conoce con el nombre de ley alométrica a ley tipo de comparación

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Si “y ” denota una parte de un ser vivo y “ x ” denota otra de sus partes, sabemos que  Sign up to vote on thistiempo title ambas partes crecen en función del  Useful  Not useful con sus respectivas tasas puntuales: dy/dt y dx/dt

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Ecuación diferencial alométrica Sign up to vote on this title

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La ley alométrica se expresa matemáticamente ecuación diferencial de la forma:

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Solución de la ecuación alométric Sign up to vote on this title

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La ecuación (1/y)(dy/dt) = k (1/x)(dx/dt)

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Solución de ecuación alométrica Sign up to vote on this title

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La ecuación (dy/dx)=k(y/x) es de un tipo muy sencillo de resolver (ecuación de variables

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Esta última ecuación Sign up to vote on this title

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nos dice que las partes “ x ” y “y ” varían con el

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Para propósitos numéricos y gráficos es práctic tomar logaritmos en ambos miembros de la  ecuación anterior, obteniéndose ecuación: Sign up to votela on this title 

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log(y) = k log(x) + log(c) (*)

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Estimación de biomasa Sign up to vote on this title

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Ejemplo de tabla de volumen. Tabla de volumen total para Cordia alliodora con corteza (m3 árbol-1) Ejemplo:

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Ahora veamos un ejemplo concreto de construcción de modelos alométricos de biomasa Sign up to vote on this title

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Location of the study Area: 8,327 ha


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Indigenous Reserve Cabécar Indigenous Reserve Bribri

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Sample and tree size Sign up to vote on this title

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Diametric distribution of sampled Theobroma cacao trees Sign up to vote on this title

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Relation between total aboveground biomass and stem diameter at 30 cm above ground for Theobroma cacao Sign up to vote on this title

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Relationship between total aboveground biomass and a) and b) total height of C. alliodora Sign up to vote on this title

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Stem diameter distribution of sampled fruit tr Sign up to vote on this title

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Relation between total aboveground biomass and dbh and b) total height of fruit trees Sign up to vote on this title

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Stem diameter distribution of sampled individuals (h ≥ 1.5 m and dbh < 10 cm young forest fallows  Sign up to vote on this title 

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Relation between total aboveground biomass and a) dbh and b) total height of individuals (h ≥ 1.5 m and dbh < 10 cm) of natural regenerat in fallows Sign up to vote on this title

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Lecturas • Obligatorias


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