6.5 Logistic Growth

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What you’ll learn about How Populations Grow Partial Fractions The Logistic Differential Equation Logistic Growth Models Essential Question How do we use logistic growth models in Calculus to help us find growth in the real world?

Partial Fraction Decomposition with Distinct Linear Denominators If where P and Q are polynomials with the degree of P less than the degree of Q, and if Q(x) can be written as a product of distinct linear factors, then f (x) can be written as a sum of rational functions with distinct linear denominators.

Example Finding a Partial Fraction Decomposition 1.Write the function as a sum of rational functions with linear denominators.

Example Antidifferentiating with Partial Fractions 2.Find

Example Antidifferentiating with degree higher in Numerator 3.Find

Example Antidifferentiating with degree higher in Numerator 3.Find

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6.5 Logistic Growth

What you’ll learn about How Populations Grow Partial Fractions The Logistic Differential Equation Logistic Growth Models Essential Question How do we use logistic growth models in Calculus to help us find growth in the real world?

Logistic Differential Equation Exponential growth can be modeled by the differential equation for some k > 0. If we want the growth rate to approach zero as P approaches a maximal carrying capacity M, we can introduce a limiting factor of M – P : This is the logistic differential equation. Population is growing the fastest. Slope is at its steepest. Maximum Capacity

Example Logistic Differential Equation 4.The growth rate of a population P of bears in a newly established wildlife preserve is modeled by the differential equation where t is measured in years. a.What is the carrying capacity for the bears in this wildlife preserve? b.What is the bear population when the population is growing the fastest? c.What is the rate of change of the population when it is growing the fastest?

Example Logistic Differential Equation 4.The growth rate of a population P of bears in a newly established wildlife preserve is modeled by the differential equation where t is measured in years. d.Solve the differential equation with the initial condition P(0) = 25.

Example Logistic Differential Equation 4.The growth rate of a population P of bears in a newly established wildlife preserve is modeled by the differential equation where t is measured in years. d.Solve the differential equation with the initial condition P(0) = 25.

The General Logistic Formula The solution of the general logistic differential equation is where A is a constant determined by an appropriate initial condition. The carrying capacity M and the growth constant k are positive constants.

Example Logistic Differential Equation 5.The table shows the population of Aurora, CO for selected years between 1950 and a.Use logistic regression to find a logistic curve to model the data and superimpose it on a scatter plot of population against years after 1950? b.Based on the regression equation, what will the Aurora population approach in the long run? c.Based on the regression equation, when will the population of Aurora first exceed 300,000 people? d.Write a logistic differential equation in the form dP/dt = kP(M – P) that models the growth of the Aurora data in the table.

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