Download Exponential growth occurs when the growth rate of the

Exponential growth occurs when the growth rate of the value of a
mathematical function is proportional to the function's current value.
LEARNING OBJECTIVE [ edit ]
Apply the exponential growth and decay formulas to real world examples
KEY POINTS [ edit ]
The formula for exponential growth of a variable x at the (positive or negative) growth rate r, as
time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is: x t = x 0 (1 + r t ) where x0 is the value of x at time 0.
Exponential decay occurs in the same way as exponential growth, 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.
TERMS [ edit ]
polynomial
an expression consisting of a sum of a finite number of terms, each term being the product of a
constant coefficient and one or more variables raised to a non­negative integer power
linear
having the form of a line; straight
exponential
any function that has an exponent as an independent variable
Give us feedback on this content: FULL TEXT [ edit ]
Exponential growth occurs when the growth rate of the value of a mathematical function is
proportional to the function's current value. Exponential decay occurs in the same way,
providing the growth rate is negative.
In the long run, exponential growth of any
kind will overtakelinear growth of any
kind as well as any polynomial growth, as
shown in .
Interactive Graph: Exponential Growth
The graph illustrates how exponential growth
(purple) surpasses both linear (red) and cubic
(blue) growth.
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The formula for exponential growth of
a variable x at the (positive or negative) growth rate r, as time t goes on in discrete intervals
(that is, at integer times 0, 1, 2, 3, ...), is: x t = x 0 (1 + r t ) where x0 is the value of x at time 0.
For example, with a growth rate of r = 5% = 0.05, going from any integer value of time to the
next integer causes x at the second time to be 1.05 times (i.e., 5% larger than) what it was at
the previous time.
t
A quantity x depends exponentially on time t if x t = ab τ where the constant a is the initial
value of x, x(0) = a , the constant b is a positive growth factor, and τ is the time constant—
the time required for x to increase by one factor of b: x(τ + t) = ab
τ+t
τ
t
τ
= ab τ b τ = x(t)b
If τ > 0 and b > 1, then x has exponential growth. If τ < 0 and b > 1, or τ > 0 and 0 < b < 1,
then x has exponential decay.
Let's assume that a species of bacteria doubles every ten minutes. Starting out with only one
bacterium, how many bacteria would be present after one hour? The question implies a = 1, b
= 2, and τ = 10 min.
t
60min
x(t) = ab τ = 1 × 2 10min
x(1hour) = 1 × 26 = 64
After one hour, or six ten­minute intervals, there would be sixty­four bacteria.