4.8
Exponential Growth and Decay
We’ve had some experience dealing
with exponential functions, but this
section takes what we know and puts
it in a real-world context.
Exponential growth and decay are rates;
that is, they represent the change in some
quantity through time. Exponential growth
is any increase in quantity over time, while
exponential decay is any decrease in
quantity over time.
N(t) = N0ekt (exponential growth)
or
N(t) = N0e-kt (exponential decay)
where:
• N0 is the initial quantity
• t is time
• N(t) is the quantity after time t
•k is a constant not equal to zero, and
•ex is the exponential function
Exponential growth is also called the Law
of uninhibited growth, and can be used
with any variable for your initial and
ending quantities.
For example :
A = A0ekt
Can you think of a familiar example?
Some examples that follow the law of
uninhibited growth:
-interest compounded continuously
(A = Pert)
-cell and bacterial growth
- population growth
Let’s go through an example:
A colony of bacteria grows according to
the law of uninhibited growth according to
the function N(t) = 100e0.045t, where N is
measured in grams and t is measured in
days.
N(t) =
0.045t
100e
a) Determine the initial amount of bacteria
N(t) =
0.045t
100e
b) What is the growth rate of the bacteria?
N(t) =
0.045t
100e
c) Graph the function using a graphing utility
N(t) =
0.045t
100e
d) What is the population after five days?
N(t) =
0.045t
100e
e) How long will it take for the population to reach
140 grams?
N(t) =
0.045t
100e
f) What is the doubling time for the population?
Ready to try some problems?
Homework: p. 334/ 1-4
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