Math & Beyond

Exponential Growth And Mathematics!

July 30, 2020

Reading Time: 2 minutes

This article would give the reader a good understanding about Exponential function, Exponential growth with examples.

Exponent Description

In Mathematics, Exponent is a quantity that represents the power to be raised for a given number or an expression, usually expressed as a raised number or variable as a superscript of the number or an expression.

Exponent Description            

Special symbol, the up-arrow "↑" or “^” (in keyboard) also can be used to represent an exponent.

v↑k = vk

s^b = S

The exponent of a number is the number of times that number should be multiplied by itself. 

For example: 74 = 7 × 7 × 7 × 7 = 2401. (The exponent "4" says the number 7 to be multiplied four times by 7 itself)


Exponential Function

To the list of mathematical functions namely linear functions, quadratic functions, rational functions, and radical functions, in addition comes Exponential Functions. 

 Exponent function

Exponential functions are further classified into Exponential Growth and Exponential Decay and this blog is confined to discuss Exponential Growth.

expoential


Exponential Growth

Exponential growth is a mathematical change that increases without limit based on an exponential function and the change is in the positive direction. The important concept is that the rate of change continues to increase over time.

 Exponential Growth           

Exponential growth occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself.


Exponential Formula

The original exponential formula is y = abx. However in Exponential growth, ‘b’ value (growth factor) has been replaced by (1 + r) and hence growth "rate" (r) is determined as b = 1 + r.

Exponential Growth

where,

a = initial value (the amount before measuring growth)

r = growth (most often represented as a percentage and expressed as a decimal)

x = number of time intervals that have passed

 

Example: The population of Vatican City in 2016 was estimated to be 55,000 people with an annual rate of increase of 

  1. What is the growth factor for Vatican City?

           After one year the population would be 55,000 + 0.024(55000).

          By factoring, we have 55000(1 + 0.024) or 55000(1.024).

          The growth factor is 1.024. (Growth factor should be greater than 1)

  1.  Write an equation to model future growth.

             y = abx = a(1.014)x = 55000(1.024)x

Most naturally occurring Exponential growth phenomena grow continuously. For example, bacteria will continue to grow over a 24 hours period, producing new bacteria which will also grow. The bacteria do not wait until the end of the 24 hours, and then all reproduce at once.

The exponential ‘e’ is used when modeling continuous growth that occurs naturally such as populations, bacteria, etc.

You can think of e like a universal constant representing how fast you could possibly grow using a continuous process.

Also, the beauty of e is that not only is it used to represent continuous growth, but it can also represent growth measured periodically across time (such as the growth in previous example).

The following formula is used to illustrate continuous growth. If a quantity grows continuously by a fixed percent, the pattern can be depicted by this function.

The exponential ‘e’ is used when modeling continuous growth that occurs naturally such as populations, bacteria, etc.

You can think of e like a universal constant representing how fast you could possibly grow using a continuous process. And, the beauty of e is that not only is it used to represent continuous growth, but it can also represent growth measured periodically across time (such as the growth in previous example).

Continuous Exponential Growth

exC

A = ending value (amount after growth)

A0 = initial value (amount before measuring growth)

e = exponential e = 2.71828183...

k = continuous growth rate (also called constant of proportionality)

(k > 0, the amount is increasing (growing); 

t = time that has passed


Example: A strain of bacteria growing on your desktop doubles every 5 minutes. Assuming that you start with only one bacterium, how many bacteria could be present at the end of 96 minutes? (Hint: bacteria continuously grow)

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Now, form the equation using this k value, and solve the problem using the time of 96 minutes.

expAA


Exponential Growth Facts

REALITY EXPONENTIAL GROWTH

 

School Kid using exponential growth formula for calculating spread of virus

EXPONENTIAL GROWTH

 

Exponential growth inversely proportional to Exposure Reduction

Exposure Reduction

 

Graph depicting simulated exponential growth

simulated exponential growth


Conclusion

Exponential growth is a pattern of data that shows sharper increases over time that is versatile across various sectors like medical, finance, marketing, economy, banking, insurance, information technology etc. Surf the net to find more about Exponential growth.

Thank you for spending your time here and please do comment below if you have any queries.

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