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

• Exponential Population Growth

  • When resources are unlimited, a population can experience exponential growth, where its size increases at a greater and greater rate.
  • This accelerating pattern of increasing population size is called exponential growth.
  • The best example of exponential growth is seen in bacteria.
  • The important concept of exponential growth is that the population growth rate, the number of organisms added in each reproductive generation, is accelerating; that is, it is increasing at a greater and greater rate.
  • When resources are unlimited, populations exhibit exponential growth, resulting in a J-shaped curve.

• Logistic Population Growth

  • In the real world, with its limited resources, exponential growth cannot continue indefinitely.
  • Thus, the exponential growth model is restricted by this factor to generate the logistic growth equation:
  • A graph of this equation yields an S-shaped curve ; it is a more-realistic model of population growth than exponential growth.
  • Initially, growth is exponential because there are few individuals and ample resources available.
  • When resources are unlimited, populations exhibit exponential growth, resulting in a J-shaped curve.

• Generation Time

  • If the number surviving exceeds unity on average, the bacterial population undergoes exponential growth.
  • The measurement of an exponential bacterial growth curve in batch culture was traditionally a part of the training of all microbiologists.
  • In autecological studies, bacterial growth in batch culture can be modeled with four different phases: lag phase, exponential or log phase, stationary phase, and death phase .
  • For this type of exponential growth, plotting the natural logarithm of cell number against time produces a straight line.
  • However, exponential growth cannot continue indefinitely because the medium is soon depleted of nutrients and enriched with wastes.

• Limited Growth

  • A realistic model of exponential growth must dampen when approaching a certain value.
  • Exponential functions can be used to model growth and decay.
  • Exponential functions are ever-increasing so saying that an exponential function models population growth exactly means that the human population will grow without bound.
  • This is the idea behind limited growth, that a population may grow exponentially but only to a point at which time the growth will taper off.
  • Thus, the model of population growth among sheep will no longer be exponential.

• Graphs of Exponential Functions, Base e

  • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.
  • The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$ where $e$ is the number (approximately 2.718281828) described previously.
  • If the change is positive, this is called exponential growth and if it is negative, it is called exponential decay.
  • For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function.
  • Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as $y=Ae^{kt}$ where $A$ is the number of bacteria present initially (at time $t=0$) and $k$ is a constant called the growth constant.

• Problem-Solving

  • The exponential function has numerous applications.
  • Economic growth is expressed in percentage terms, implying exponential growth.
  • Compound interest at a constant interest rate provides exponential growth of the capital.
  • Think about a microorganism's growth.
  • In what other circumstances would you see exponential growth?

• Population Growth

  • Population can fluctuate positively or negatively and can be modeled using an exponential function.
  • Population growth can be modeled by an exponential equation.
  • It is the Population Growth Rate ($PGR$).
  • If the current rates of births and deaths hold, the world population growth can be modeled using an exponential function.
  • The graph below shows an exponential model for the growth of the world population.

• 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.

• Introduction to Exponents

  • Exponential form, written $b^n$, represents multiplying the base $b$ times itself $n$ times.
  • Exponentiation is a mathematical operation that represents repeated multiplication.
  • Exponentiation is used frequently in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
  • Now that we understand the basic idea, let's practice simplifying some exponential expressions.
  • Let's look at an exponential expression with 2 as the base and 3 as the exponent: