exponential growth

Examples of exponential growth in the following topics:

• Exponential Growth and Decay

  • Exponential decay occurs in the same way, providing the growth rate is negative.
  • In the long run, exponential growth of any kind will overtake linear growth of any kind as well as any polynomial growth.
  • If $\tau > 0$ and $b > 1$, then $x$ has exponential growth.
  • This graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
  • Apply the exponential growth and decay formulas to real world examples

• Essential Functions for Mathematical Modeling

  • For example, a simple model of population growth is the Malthusian growth model.
  • This is essentially exponential growth based on a constant rate of compound interest: $P(t)=P_0e^{rt}$ where $P_0=P(0)=\text{initial population}$, $r$ is the growth rate, and $t$ is the time.
  • A slightly more realistic and largely used population growth model is the logistic function that may be defined by the formula: $P(t) = \frac{1}{1 + e^{-t}}$.
  • The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.

• Logistic Equations and Population Grown

  • A logistic equation is a differential equation which can be used to model population growth.
  • More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative $t$, which slows to linear growth of slope $\frac{1}{4}$ near $t = 0$, then approaches $y = 1$ with an exponentially decaying gap.
  • The equation describes the self-limiting growth of a biological population.
  • where the constant $r$ defines the growth rate and $K$ is the carrying capacity.
  • It can be used to model population growth because of the limiting effect scarcity has on the growth rate.

• Predator-Prey Systems

  • The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term $\alpha x$.
  • With these two terms, the equation above can be interpreted as follows: the change in the prey's number is given by its own growth minus the rate at which it is preyed upon.
  • In the predator equation, $\delta xy$ represents the growth of the predator population.
  • $\gamma y$ represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.
  • Hence, the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.

• Derivatives of Exponential Functions

  • The derivative of the exponential function is equal to the value of the function.
  • The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative.
  • If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
  • Graph of the exponential function illustrating that its derivative is equal to the value of the function.

• The Natural Exponential Function: Differentiation and Integration

  • Now that we have derived a specific case, let us extend things to the general case of exponential function.
  • Here we consider integration of natural exponential function.
  • Note that the exponential function $y = e^{x}$ is defined as the inverse of $\ln(x)$.

• Exponential and Logarithmic Functions

  • Both exponential and logarithmic functions are widely used in scientific and engineering applications.
  • The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
  • The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.
  • The exponential function $e^x$ can be characterized in a variety of equivalent ways.
  • The derivative (or slope of a tangential line) of the exponential function is equal to the value of the function.

• Taylor Polynomials

  • The exponential function (in blue) and the sum of the first 9 terms of its Taylor series at 0 (in red).

• Taylor and Maclaurin Series

  • The exponential function (in blue) and the sum of the first $n+1$ terms of its Taylor series at $0$ (in red) up to $n=8$.

• Power Series

  • The exponential function (in blue), and the sum of the first $n+1$ terms of its Maclaurin power series (in red).