icon-komplexe-Zahl Math Tutorial: Exponential growth

Model

The model of exponential growth is based on that a stock y (eg a population) is all the more increased the greater the inventory itself. That in the model with the magnitude of the population continues to increase the growth rate.

Limitations of the model

The model does not account limiting factors such as only a finite available resources. A model that takes this into account leads to the model of logistic growth.

Half-life or doubling time T

The half-life or doubling time T is the period in which the stock is doubled or halved.

Exponential growth:

T = ln 2 λ

Exponential decay:

T = ln 12 λ

Application Examples

  • Growth of populations

  • Radioactive decay

  • Absorption of light

  • Compound interest

Differential equation of the exponential growth

The continuous exponential growth by a homogeneous linear differential equations with constant factor λ described.

yt= d y d t =λyt

The equation states that the change of the stock is proportional to the component itself. λ is the proportionality constant.

yy=λ

Division by y

lny=λ

Applying the chain rule

lny=λdt=λt+C

Integration

y=y0eλt

General solution of the equation for exponential growth with the constant y 0 is the initial stock. For lambda & gt; 0 exists and for lambda an exponential growth process; & lt; 0, an exponential decay process.

Calculator for the model of exponential growth

Zoom

y=y0eλt

Increment direction field
step=
Range of values ​​of the axes
t-min= t-max=
y-min= y-max=
Range of values ​​of the parameters
λ-min= λ-max=
Current value of the parameter
λ=
Current value of the initial values
t0= y0=