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La función exponencial

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La función exponencial

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Explora la función exponencial y sus propiedades

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Neftalí Ocampo Salazar
Neftalí Ocampo Salazar
Mexico

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La función exponencial
 

La función exponencialOnline version

Explora la función exponencial y sus propiedades

by Neftalí Ocampo Salazar
1

Introducción a la función exponencial

La función exponencial describe crecimientos o decaimientos rápidos. Se expresa como f(x) = a^x, con a > 0 y a ≠ 1. Es continua y suave.

2

Definición esencial

Una función exponencial es f(x) = a^x. Requiere una base a positiva y distinta de 1. El valor depende de la potencia de x.

3

Base natural e

La base especial e ≈ 2.718. Propiedades únicas incluyen crecimiento continuo y derivada simple: d/dx e^x = e^x.

4

Crecimiento y decaimiento

Si a > 1, la función crece a medida que x aumenta. Si 0 < a < 1, decae conforme x aumenta.

5

Derivada de la exponencial

La derivada de a^x es a^x ln(a). Para a = e, la derivada es simplemente e^x.

6

Propiedades básicas

Propiedades clave: a^{x+y} = a^x a^y; (a^x)^y = a^{xy}; a^0 = 1. Estas reglas permiten simplificar expresiones.

7

Inversa y logaritmos

La función exponencial y los logaritmos son inversas. El logaritmo base a: log_a(b) satisface a^{log_a(b)} = b.

8

Gráfica típica

La curva es positiva y suave. Crece rápido para bases mayores que 1; no cruza el eje Y y nunca es negativa.

9

Aplicaciones comunes

Interés compuesto, crecimiento poblacional, modelado de reacciones químicas y procesos de decaimiento radican en la exponencial.

10

Ejemplo práctico

Si 2^x = 8, entonces x = 3. Para e^x = 20, x = ln(20). Usa logaritmos para despejar exponentes.

11

Grafica de la exponencial

Grafica la función f(x)= (1/3)^x
 
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