Image 1 Caption: Exponential growth is when the rate of increase of a quantity is proportional to the amount of the quantity.

Features:

1. The plot illustrates:
   1. Linear growth: red curve: y = a*x.

   2. Cubic growth: blue curve: y = a* x**3.

   3. Exponential growth: green curve:

            y = a*e**x       ,

      where e = 2.71828 18284 59045 23536 ... is a mathematical constant just called e: it's an irrational number---it's mad, bad, and dangerous to know. Actually, irrational number are real numbers that CANNOT be written as the ratio of two integers: i.e., they are NOT rational numbers. Their trailing numerical digits go on forever without terminating in 0 and without becoming a repeating decimal sequence.

      The function e**x is called exponential function.

2. Exponential growth occurs whenever the rate of change is proportional to the amount: i.e., in calculus notation

         dA/dt=kA       ,

   where t is an independent variable (e.g., time), A is amount, dA/dt is rate of change of A (i.e., the derivative of A with respect to t), and k is the rate constant.

   The mathematical solution of this 1st order linear differential equation is

         A = A_0*e**(kt) = A_0*e**(t/t_e)       ,

   where A_0 is the amount of A at t = 0 and the e-folding constant t_e = 1/k is multiplicative inverse (AKA reciprocal) of k. The e-folding constant is the change in t needed to change A by a factor of e.

3. Exponential growth is actually very common.

   Biotic populations without some controling factor (i.e., death) exhibit exponential growth.

   There was a time long ago when bank accounts exhibited exponential growth. In the early 1990s, banks discovered they did NOT have to pay interest on bank accounts---a wonderful discovery in banking.

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Images

1. Credit/Permission: User:Lunkwill, 2005 / Public domain.
   Image link: Wikipedia: File:Exponential.png.
   
2. Credit/Permission: Our World in Data, Current year / No permission: click on placeholder image alien_click_to_see_image.html.
   Image link: Our World in Data: Coronavirus (COVID-19) Cases.
   

Local file: local link: exponential_function_plot.html.
Extended file: Mathematics file: exponential_function_plot_4.html.
File: Mathematics file: exponential_function_plot.html.

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