1. Calculus BC AP/Dual, Revised ©2017
Logistic Growth
Section 6.3A
Calculus BC AP/Dual, Revised ©2017
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§6.3A: Logistic Growth

2. Recall Solve the logistic differential equation 𝒅𝑷 𝒅𝒕 =𝒌𝑷 𝟏− 𝑷 𝑳 .
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§6.3A: Logistic Growth

3. Recall Solve the logistic differential equation 𝒅𝒚 𝒅𝒕 =𝒌𝑷 𝟏− 𝑷 𝑳 .
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§6.3A: Logistic Growth

4. Logistic Curves
Logistics Differential Equation: 𝒅𝑷 𝒅𝒕 =𝒌𝑷 𝟏− 𝑷 𝑳 or 𝒅𝑷 𝒅𝒕 =𝒌𝑷 𝑳−𝑷 where 𝒌 will be multiplied and the inside is divided
The 𝒌 can be different depending on the use of the equation
Logical Growth Model: 𝑷= 𝑳 𝟏+𝑩 𝒆 −𝒌𝒕
𝑳 = Carrying Capacity (Upper Horizontal LIMIT)
𝑲 = Proportionality Constant
𝑩 = Beginning Amount (arbitrary), use the equation: 𝑩= 𝑳− 𝑷 𝟎 𝑷 𝟎 where P0 is the initial population
Point of Inflection: 𝒚= 𝑳 𝟐
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§6.3A: Logistic Growth

5. Logistic Growth
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§6.3A: Logistic Growth

6. Example 1
Given 𝒅𝑷 𝒅𝒕 =𝑷 𝟐− 𝑷 𝟓𝟎𝟎𝟎 identify the 𝒌 and 𝑳 of the equation.
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§6.3A: Logistic Growth

7. Example 1
Given 𝒅𝑷 𝒅𝒕 =𝑷 𝟐− 𝑷 𝟓𝟎𝟎𝟎 identify the 𝒌 and 𝑳 of the equation.
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§6.3A: Logistic Growth

8. Your Turn
Given 𝒅𝑷 𝒅𝒕 =𝟎.𝟎𝟏𝑷 𝟏𝟎𝟎−𝑷 identify the 𝒌 and 𝑳 of the equation.
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§6.3A: Logistic Growth

9. Example 2
Using the equation, 𝒚= 𝟒 𝟏+𝟐 𝒆 −𝟑𝒕 identify the point of inflection and sketch a graph.
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§6.3A: Logistic Growth

10. Example 2
Using the equation, 𝒚= 𝟒 𝟏+𝟐 𝒆 −𝟑𝒕 identify the point of inflection and sketch a graph.
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§6.3A: Logistic Growth

11. Example 2
Using the equation, 𝒚= 𝟒 𝟏+𝟐 𝒆 −𝟑𝒕 identify the point of inflection and sketch a graph.
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§6.3A: Logistic Growth

12. Example 2
Using the equation, 𝒚= 𝟒 𝟏+𝟐 𝒆 −𝟑𝒕 identify the point of inflection and sketch a graph.
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§6.3A: Logistic Growth

13. Your Turn
Using the equation, 𝒚= 𝟖 𝟏+𝟐 𝒆 −𝟐𝒕 identify the point of inflection and sketch a graph.
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§6.3A: Logistic Growth

14. Example 3
A state game commission releases 40 elk into a game refuge. After 5 years, the elk population is The commission believes that the environment can support no more than 4000 elk. The growth rate of the elk population, 𝒑, is: 𝒅𝑷 𝒅𝑻 =𝒌𝑷 𝟏− 𝑷 𝟒𝟎𝟎𝟎 ,𝟒𝟎≤𝒑≤𝟒𝟎𝟎𝟎 where 𝒕 is the number of years.
Write a model for the elk population in terms of 𝒕.
Estimate the elk population in 15 years.
Find the limit of the model as 𝒕 → ∞.
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§6.3A: Logistic Growth

15. Example 3a
A state game commission releases 40 elk into a game refuge. After 5 years, the elk population is The commission believes that the environment can support no more than 4000 elk. The growth rate of the elk population, 𝒑, is: 𝒅𝑷 𝒅𝑻 =𝒌𝑷 𝟏− 𝑷 𝟒𝟎𝟎𝟎 ,𝟒𝟎≤𝒑≤𝟒𝟎𝟎𝟎 where 𝒕 is the number of years.
Write a model for the elk population in terms of 𝒕.
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§6.3A: Logistic Growth

16. Example 3a
A state game commission releases 40 elk into a game refuge. After 5 years, the elk population is The commission believes that the environment can support no more than 4000 elk. The growth rate of the elk population, p, is: 𝒅𝑷 𝒅𝑻 =𝒌𝑷 𝟏− 𝑷 𝟒𝟎𝟎𝟎 ,𝟒𝟎≤𝒑≤𝟒𝟎𝟎𝟎 where t is the number of years.
Write a model for the elk population in terms of 𝒕.
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§6.3A: Logistic Growth

17. Example 3b
A state game commission releases 40 elk into a game refuge. After 5 years, the elk population is 104. The commission believes that the environment can support no more than 4000 elk. The growth rate of the elk population, p, is: 𝒅𝑷 𝒅𝑻 =𝒌𝑷 𝟏− 𝑷 𝟒𝟎𝟎𝟎 ,𝟒𝟎≤𝒑≤𝟒𝟎𝟎𝟎 where t is the number of years. B. Estimate the elk population in 15 years.
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§6.3A: Logistic Growth

18. Example 3c
A state game commission releases 40 elk into a game refuge. After 5 years, the elk population is 104. The commission believes that the environment can support no more than 4000 elk. The growth rate of the elk population, p, is: 𝒅𝑷 𝒅𝑻 =𝒌𝑷 𝟏− 𝑷 𝟒𝟎𝟎𝟎 ,𝟒𝟎≤𝒑≤𝟒𝟎𝟎𝟎 where t is the number of years. C. Find the limit of the model as 𝒕 → ∞.
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§6.3A: Logistic Growth

19. Example 4
Suppose the population of bears in a national park grows according to the logistic differential equation, 𝒅𝑷 𝒅𝑻 =𝟓𝑷−𝟎.𝟎𝟎𝟐 𝑷 𝟐 , where 𝑷 is the number of bears at time 𝒕 in years. Find the limit of the model as 𝒕→∞ where 𝑷 𝟎 = 𝟏𝟎𝟎.
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§6.3A: Logistic Growth

20. Your Turn
Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears. Assuming a logistic growth model, when will the bear population reach 50? 75? 100?
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§6.3A: Logistic Growth

21. Your Turn
Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears. Assuming a logistic growth model, when will the bear population reach 50? 75? 100?
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§6.3A: Logistic Growth

22. Your Turn 𝒀=𝟓𝟎 𝒂𝒕 𝟐𝟐 𝒚𝒆𝒂𝒓𝒔 𝒀=𝟕𝟓 𝒂𝒕 𝟑𝟑 𝒚𝒆𝒂𝒓𝒔 𝒀=𝟏𝟎𝟎 𝒂𝒕 𝟕𝟓 𝒚𝒆𝒂𝒓𝒔 Bears
Years
Bears
𝒀=𝟓𝟎 𝒂𝒕 𝟐𝟐 𝒚𝒆𝒂𝒓𝒔
𝒀=𝟕𝟓 𝒂𝒕 𝟑𝟑 𝒚𝒆𝒂𝒓𝒔
𝒀=𝟏𝟎𝟎 𝒂𝒕 𝟕𝟓 𝒚𝒆𝒂𝒓𝒔
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§6.3A: Logistic Growth

23. Example 5
In a particular town of 100,000 residents, 20,000 watched a viral video on the Internet. The rate of growth of the spread of information was jointly proportional to the amount of people who had not watched it. If 50% watched it after one hour, how long does 80% of the population watched the viral video?
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§6.3A: Logistic Growth

24. Example 5
In a particular town of 100,000 residents, 20,000 watched a viral video on the Internet. The rate of growth of the spread of information was jointly proportional to the amount of people who had not watched it. If 50% watched it after one hour, how long does 80% of the population watched the viral video?
5/1/2019 1:01 AM
§6.3A: Logistic Growth

25. Example 5
In a particular town of 100,000 residents, 20,000 watched a viral video on the Internet. The rate of growth of the spread of information was jointly proportional to the amount of people who had not watched it. If 50% watched it after one hour, how long does 80% of the population watched the viral video?
5/1/2019 1:01 AM
§6.3A: Logistic Growth

26. Example 5
In a particular town of 100,000 residents, 20,000 watched a viral video on the Internet. The rate of growth of the spread of information was jointly proportional to the amount of people who had not watched it. If 50% watched it after one hour, how long does 80% of the population watched the viral video?
5/1/2019 1:01 AM
§6.3A: Logistic Growth

27. Example 5
In a particular town of 100,000 residents, 20,000 watched a viral video on the Internet. The rate of growth of the spread of information was jointly proportional to the amount of people who had not watched it. If 50% watched it after one hour, how long does 80% of the population watched the viral video?
5/1/2019 1:01 AM
§6.3A: Logistic Growth

28. Example 5
In a particular town of 100,000 residents, 20,000 watched a viral video on the Internet. The rate of growth of the spread of information was jointly proportional to the amount of people who had not watched it. If 50% watched it after one hour, how long does 80% of the population watched the viral video?
5/1/2019 1:01 AM
§6.3A: Logistic Growth

29. AP Multiple Choice Practice Question 1 (non-calculator)
Which of the following differential equations for a population could model the logistic growth equation for the graph below? (A) 𝒅𝑷 𝒅𝒕 =𝟎.𝟐𝑷−𝟎.𝟎𝟎𝟏 𝑷 𝟐 (B) 𝒅𝑷 𝒅𝒕 =𝟎.𝟏𝑷−𝟎.𝟎𝟎𝟏 𝑷 𝟐 (C) 𝒅𝑷 𝒅𝒕 =𝟎.𝟐 𝑷 𝟐 −𝟎.𝟎𝟎𝟏𝑷 (D) 𝒅𝑷 𝒅𝒕 =𝟎.𝟏 𝑷 𝟐 −𝟎.𝟎𝟎𝟏𝑷
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§6.3A: Logistic Growth

30. AP Multiple Choice Practice Question 1 (non-calculator)
Which of the following differential equations for a population could model the logistic growth equation for the graph below?
Vocabulary
Connections and Process
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§6.3A: Logistic Growth

31. AP Multiple Choice Practice Question 1 (non-calculator)
Which of the following differential equations for a population could model the logistic growth equation for the graph below?
Connections and Process
Answer
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§6.3A: Logistic Growth

32. Assignment
Worksheet
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§6.3A: Logistic Growth