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Understanding Natural Growth and Decline: Exponential Functions in Population Dynamics

This section explores the principles of natural growth and decline in populations using exponential functions. It explains how a population, starting at an initial value, can grow at a constant annual percentage rate and introduces mathematical models to describe these changes. The natural growth model employs multiplication by a base constant, while the decline model is discussed with a focus on how negative growth rates affect population size. Real-world applications illustrate these concepts, including examples from U.S. history and depreciation of assets like cars.

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Understanding Natural Growth and Decline: Exponential Functions in Population Dynamics

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  1. Section 3.3 Natural Growth and Decline in the World

  2. EXAMPLE Census for early U.S. history.

  3. If a population starts at time t = 0 with initial population P0 and thereafter grows “naturally” at an annual rate of r = p%, then the number of individuals in the population after t years is given by 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.

  4. EXPONENTIAL FUNCTION Definition: An exponential function is one of the form f (x) = ax, with base a and exponent x (its independent variable).

  5. 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.

  6. 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.

  7. 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.

  8. EXAMPLE • 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? Suppose a car cost $18,000 when new. When the car is 6 years old, it has depreciated to $9,500.

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