Natural Growth and Decline in the World

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Section 3.3

• Natural Growth and Decline in the World

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EXAMPLE
Census for early U.S. history.
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NATURAL GROWTH
The world is full of quantities that appear to
grow at a constant percentage rate per unit of
time. This is so common for populations that it
is called natural growth.
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EXPONENTIAL FUNCTION
Definition An exponential function is one of
the form f (x) ax, with base a and exponent x
(its independent variable).
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NATURAL GROWTH MODEL
Natural Growth Model Multiplication by a every
year The base constant a in P(t) P0 at is
the factor by which the population is multiplied
every year. In short, the function P(t) P0
at models a population with annual multiplier a.
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ANOTHER NATURAL GROWTH MODEL
Natural Growth Model Multiplication by b every
N years If a population with initial (time t
0) value P0 grows naturally and is multiplied by
the number b every N years, then it is described
by the function P(t) P0 bt/N.
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NATURAL DECLINE
If the growth rate r in the equation
P(t)  P0(1  r)t is negative, then the
population is declining or decaying.
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EXAMPLE
Suppose a car cost 18,000 when new. When the
car is 6 years old, it has depreciated to 9,500.

• Find a exponential function that models this
  situation.
• Find the value of the car when it is 10 years
  old.
• How long (in years and months) will it take for
  the car to be worth 5,000?