# Exponential Growth Model

The model of exponential growth is based on the fact that the larger the stock y (e.g. a population), the more strongly it multiplies the larger the stock itself. This means in the model that the growth rate increases with the size of the population.

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

## Exponential Growth Formula

Differential equation of exponential growth:

$y′t= λyt$

$λ:Exponential growth rate$

With the growth function for the general inital values t0 and y0 = y(t0)

$y\left(t\right)={y}_{0}{e}^{\lambda \left(t-{t}_{0}\right)}$

The half-life or doubling time T describes the time in which the stock doubles or halves.

Exponential growth:

$T = ln 2 λ$

Exponential decay:

$T = ln 12 λ$

Application Examples:

• Growth of populations

• Absorption of light

• Compound interest

## Differential equation of the exponential growth

The continuous exponential growth is described by a linear homogeneous differential equation with the constant factor λ

$y′t= d y d t =λyt$

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

$y′y=λ$

Division by y

$lny′=λ$

Applying the chain rule

$lny=λ∫dt=λt+C$

Integration

$y=y0eλt-t0$

Dissolving and replacing the initial condition t0, y0 yields the solution for exponential growth. For λ > 0 there is an exponential growth process and for λ < 0 an exponential decay process.

## Calculator for the exponential growth function

Scale screen:
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Curve:
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Initial values

t0=
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Parameter value

λ=

Axes ranges

t-min=
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λ-min=
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### Screenshot of the exponential growth graph

Print or save the image via right mouse click.

## Releated sites

Here is a list of of further useful sites:

Index Logistic growth function Function Plot Normal Distribution Plot Trigonometry Curve fit calculator Damped Vibration Plot Derivative calculus Partial derivatives and gradient Tangent (tan) Plot Beat frequencies Plot