LOGISTIC GROWTH. Section 6.3A Calculus BC AP/Dual, Revised /30/ :40 AM 6.3A: Logistic Growth 1
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1 LOGISTIC GROWTH Section 6.3A Calculus BC A/Dual, Revised /30/ :40 AM 6.3A: Logistic Growth 1
2 RECALL Solve the logistic differential equation d k 1 L. d 1 1 k 1 L 7/30/ :40 AM 6.3A: Logistic Growth 2 d L d L k k
3 RECALL Solve the logistic differential equation dy k 1. L ln + ln L kt + C ln L kt C L e 7/30/ :40 AM 6.3A: Logistic Growth 3 kt C L e C e kt L Be kt L 1 + Be kt
4 LOGISTIC CURVES A. Logistics Differential Equation: d k 1 d or k L where L k will be multiplied and the inside is divided B. The k can be different depending on the use of the equation C. Logical Growth Model: 1+Be kt 1. L Carrying Capacity (Upper Horizontal LIMIT) 2. K roportionality Constant 3. B Beginning Amount (arbitrary), use the equation: B L 0 population C. oint of Inflection: y L 2 L 0 where 0 is the initial 7/30/ :40 AM 6.3A: Logistic Growth 4
5 LOGISTIC GROWTH 7/30/ :40 AM 6.3A: Logistic Growth 5
6 EXAMLE 1 Given d identify the k and L of the equation. dy ky 1 y L d k 1 L d /30/ :40 AM 6.3A: Logistic Growth 6
7 Given d d EXAMLE 1 identify the k and L of the equation. d d 2 ( 2) /30/ :40 AM 6.3A: Logistic Growth 7 k 2 L 10, 000
8 YOUR TURN Given d identify the k and L of the equation. k L /30/ :40 AM 6.3A: Logistic Growth 8
9 EXAMLE 2 Using the equation, y sketch a graph e 3t y identify the point of inflection and L 4, B 2, and k 3 ( 3t e ) ( )( 3t ) 2 ( 3 t) y ' e 6e y 4 2e + e + e 3t ' t 1 2 3t 1 y' 3y t e + using Long Division 7/30/ :40 AM 6.3A: Logistic Growth 9
10 EXAMLE 2 Using the equation, y sketch a graph e 3t identify the point of inflection and 1 y' 3y t e + using Long Division y 4 4 ; y 30 ( ) ( 0) 1+ 2e 3 ' y y t ( 3 + e ) y y' 3y 1 4 7/30/ :40 AM 6.3A: Logistic Growth 10
11 Using the equation, y sketch a graph. y y L y 2 EXAMLE e 3t identify the point of inflection and 4 y /30/ :40 AM 6.3A: Logistic Growth 11 3t e 3t e ( 3 e t ) e t 2 OI 3t 2e 1 ln 2 :,2 3 e t e 2 3t e 2 3t ln e ln 2 3t 1 3t ln 2 ln 2 t 3
12 EXAMLE 2 Using the equation, y sketch a graph e 3t identify the point of inflection and OI ln 2 :,2 3 7/30/ :40 AM 6.3A: Logistic Growth 12
13 YOUR TURN Using the equation, y sketch a graph e 2t identify the point of inflection and OI ln 2 :,4 2 7/30/ :40 AM 6.3A: Logistic Growth 13
14 EXAMLE 3 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 elk. The growth rate of the elk population, p, is: d k 1, 40 p where t is the number of years. dt A. Write a model for the elk population in terms of t. B. Estimate the elk population in 15 years. C. Find the limit of the model as t. 7/30/ :40 AM 6.3A: Logistic Growth 14
15 EXAMLE 3A 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 elk. The growth rate of the elk population, p, is: d dt k 1, 40 p where t is the number of years. 40 ( 0) 1 + Be k A. Write a model for the elk population in terms of t. L ( ( 0) ) d kp y Be k k 1,40 kt 1 + Be L, k??,b B 99?? B kt 1 + Be 40B e kt B 99 7/30/ :40 AM 6.3A: Logistic Growth 15
16 EXAMLE 3A 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 elk. The growth rate of the elk population, p, is: d k 1, 40 p where t is the number of years e kt dt A. Write a model for the elk population in terms of t. d kp k 1,40 L, k ,??, B B t y e kt e k 7/30/ :40 AM 6.3A: Logistic Growth 16 ( 5) k ( e k ) k 1+ 99e 104 5k 99e k e k 37.5 e k e 99 ( ) 5 k 37.5 e 99 5k e k ln e ln k ln
17 EXAMLE 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 elk. The growth rate of the elk population, p, is: d k 1, 40 p where t is the number of years. dt B. Estimate the elk population in 15 years. d kp 1,40 L, k??, B?? ( ) e t e elk 7/30/ :40 AM 6.3A: Logistic Growth 17
18 EXAMLE 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 elk. The growth rate of the elk population, p, is: d k 1, 40 p where t is the number of years. dt C. Find the limit of the model as t. d k kp 1,40 L, k??,b B 99?? lim ( t 1 99 ) t t e e 7/30/ :40 AM 6.3A: Logistic Growth 18
19 EXAMLE 4 Suppose the population of bears in a national park grows according to the logistic differential equation, d 5 0. dt 0022, where is the number of bears at time t in years. Find the limit of the model as t where d 51 d 2500 k 1 L 2500 d ( t) 1+ 24e d B lim t 1 24 x ( ) e 7/30/ :40 AM 6.3A: Logistic Growth 19
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? L 1 + Be Be kt ( ) B ( 0,10) ( 10,23) B B 90 B e kt 7/30/ :40 AM 6.3A: Logistic Growth 20
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? 10k 0,10 ( 10,23) e ( ) k ln k( ) e k e k 77 9e ( ) t 1 + 9e 23 7/30/ :40 AM 6.3A: Logistic Growth 21
22 YOUR TURN e ( ) t Bears Y 50 at 22 years Y 75 at 33 years Y 100 at 75 years Years 7/30/ :40 AM 6.3A: Logistic Growth 22
23 EXAMLE 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? L 100,000 L kt 1 + Be 100,000 20,000 B 20,000 B 4 ( ) t 100, e kt ( 0, 20000) ( 1,50000) 7/30/ :40 AM 6.3A: Logistic Growth 23 B?
24 EXAMLE 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? L 100, ,000 ( t) B? 1 + 4e kt ( 0, 20000) 100,000 50,000 ( 1,50000) ( 1) 1 + 4e k 50, e k 100,000 ( ) 1+ 4e k 2 7/30/ :40 AM 6.3A: Logistic Growth 24
25 EXAMLE 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? L 100, e k 2 B? k 4e 1 ( 0, 20000) k 1 e ( 1,50000) k e 4 k e 4 7/30/ :40 AM 6.3A: Logistic Growth 25
26 EXAMLE 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? L 100,000 k e 4 B? k ( 0, 20000) ( 1,50000) ln 100,000 ( t) ( ln 4) t e 100,000 80,000 ln e 7/30/ :40 AM 6.3A: Logistic Growth 26 t
27 EXAMLE 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? L 100, ,000 B? 80,000 ln 4t 1 + 4e ( 0, 20000) ln 4 80, e t ( 1,50000) 100,000 ( ) 1+ 4 ln 4t 5 e 7/30/ :40 AM 6.3A: Logistic Growth 27 4
28 EXAMLE 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? L 100,000 ln 4t 5 ln 4t e e B? 4e ln 4t 1 e 4 ln 4t ( ln 4)( t) 1 ln ln e 16 ( 0, 20000) 16 1 ln 4t 1 ln ( 1,50000) e ln 4t ln 4 t 2 hours 7/30/ :40 AM 6.3A: Logistic Growth 28
29 A MULTILE CHOICE RACTICE QUESTION 1 (NON-CALCULATOR) Which of the following differential equations for a population could model the logistic growth equation for the graph below? (A) d (B) d (C) d (D) d /30/ :40 AM 6.3A: Logistic Growth 29
30 A MULTILE CHOICE RACTICE QUESTION 1 (NON-CALCULATOR) Which of the following differential equations for a population could model the logistic growth equation for the graph below? Vocabulary Logistic Growth Equation A) Connections and rocess d d k 1 ; k d k 0.2 d ( ) d k 7/30/ :40 AM 6.3A: Logistic Growth 30 B) d k k 200 d
31 A MULTILE CHOICE RACTICE 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 rocess Answer d d k 1 ; k ( ) d 2 k k 200 C) D) d d k k d 1 d A 7/30/ :40 AM 6.3A: Logistic Growth 31
32 ASSIGNMENT Worksheet 7/30/ :40 AM 6.3A: Logistic Growth 32
CALCULUS BC., where P is the number of bears at time t in years. dt (a) Given P (i) Find lim Pt.
CALCULUS BC WORKSHEET 1 ON LOGISTIC GROWTH NAME Do not use your calculator. 1. Suppose the population of bears in a national park grows according to the logistic differential equation 5P 0.00P, where P
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