6.5

Logistic Growth

Quick Review

2

3

2

1. Use polynomial division to write the rational

( )function in the form ( ) , where the degree

( )

of is less than the degree of .

a. 13

b. 1

R xQ x

D x

R D

x x

xx

x

1

22

x

x

1

3

x

xx

Quick Review

-0.1

x

x

302. Let ( ) .

1 5a. Find where ( ) is continuous.

b. Find lim ( ).

c. Find lim ( ).

d. Find the y-intercept of the graph of .

e. Find all horizontal asymptotes of the graph of .

xf x

ef x

f x

f x

f

f

,

30

0

5

30 and 0 yy

What you’ll learn about How Populations Grow Partial Fractions The Logistic Differential Equation Logistic Growth Models

Essential QuestionHow do we use logistic growth models inCalculus to help us find growth in the realworld?

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.

,xQ

xPxf

Example Finding a Partial Fraction Decomposition

122

xxx 486221212 2 xxxxCxxBxxA

14A 10B 42Set x 1 So A

14

4862

2

xx

xxxf1. Write the function as a sum of

rationalfunctions with linear denominators.

122

486 2

xxx

xxxf

122

x

C

x

B

x

A

12 xxA 12 xxB 22 xxC

40C 30 A 34 B 362Set x 3 So B 04C 03A 01 B 61Set x 2 So C 31C

122

486 2

xxx

xxxfTherefore

2

1

x 2

3

x 1

2

x

Example Antidifferentiating with Partial Fractions

2ln x

Cxxx 2312 2 ln

122

486 2

dxxxx

xx

2. Find

dxxxx

xx

122

486 2

dxxxx

1

2

2

3

2

1

2ln3 x 1ln2 x C

Example Antidifferentiating with degree higher in Numerator

4

22

3

dxx

x

3. Find

00024 232 xxxxx2

xxx 802 23 x8

4

82

2 x

xx

dxx

x

4

22

3

dx

x

xx

4

82

2

2x dxx

x

4

82

Example Antidifferentiating with degree higher in Numerator

4

22

3

dxx

x

3. Find

2x dxx

x

4

82

22

8

xx

x

22

x

B

x

A

22

xx 2xA 2 xB xxBxA 822

4A 0B 162Set x 4 So A 0A 4 B 162Set x 4 So B

dxxx

x 2

4

2

42

2x 2ln4 x 2ln4 x

Cxxx 2 2 ln42

C

Pg. 369, 6.5 #1-22

6.5

Logistic Growth

What you’ll learn about How Populations Grow Partial Fractions The Logistic Differential Equation Logistic Growth Models

Essential QuestionHow do we use logistic growth models inCalculus to help us find growth in the realworld?

Logistic Differential Equation

Exponential growth can be modeled by the differential equation

for some k > 0.kP

dt

dP

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.

PMkPdt

dP

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

,100008.0 PPdt

dP 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?

bears 100

bears 50

008.0dt

dP 50 50100 20 yearper bears

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

,100008.0 PPdt

dP where t is measured in years.

d. Solve the differential equation with the initial

condition P(0) = 25.

dtPP

dP008.0

100

dP

dP

P

B

P

A

100 dt .0080

1100 PBPA1100 A0Set P 01.0 So A1100 B100Set P 01.0 So B

dtdPPP

008.0 100

01.001.0

dtdPPP

8.0 100

11

Pln P 100ln Ct 8.0 P100ln Pln Ct 8.0

P

P100ln Ct 8.0

1100

P

Cte 8.0

P

100 Cte 8.01

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

,100008.0 PPdt

dP where t is measured in years.

d. Solve the differential equation with the initial

condition P(0) = 25.

P

100 Cte 8.01

25

100 Ce1

Ce3

P

100 ct ee 8.01

teP

8.031100

te

P8.031

1

100

teP

8.031

100

The General Logistic FormulaThe solution of the general logistic differential equation

is

PMkPdt

dP

tkMeA

MP

1

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 2003.

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.

teP

1026.0577.231

7.316440

people 441,316

te 1026.0577.231

7.316440000,300

0548.1577.231 1026.0 te

0548.577.23 1026.0 te

00232.1026.0 te

1.59t2009in

dt

dP1026.0Mk

7.316440M71024.3 k 71024.3 P P7.316440

Pg. 369, 6.5 #23-34