Do Now: #1-8, p.346Let 0.1

501 5 xf x

e

1. Continuous for all real numbers

Check the graph first?

lim 50x

f x

2. lim 0x

f x

3. H.A.: y = 0, y = 50

4. In both the first and second derivatives, the denominator0.11 5 xewill be a power of , which is never 0. Thus, the

domains of both are all real numbers.

Do Now: #1-8, p.346Let 0.1

501 5 xf x

e

Check the graph first?

5. Graph f in [–30, 70] by [–10, 60]. f (x) has no zeros.

6. Graph the first derivative in [–30, 70] by [–0.5, 2].

, Inc. interval: Dec. interval: None

7. Graph the second derivative in [–30, 70] by [–0.08, 0.08].

,16.094 Conc. up: 16.094,Conc. down:

8. Point of inflection: 16.094,25

Section 6.5a

LOGISTIC GROWTH

Review from last section…Many populations grow at a rate proportional to the size of thepopulation. Thus, for some constant k,

dP kPdt

Notice thatdP dt kP

is constant,

and is called the relative growth rate.

Solution (from Sec. 6.4):0ktP Pe

Logistic Growth ModelsIn reality, most populations are limited in growth. The maximumpopulation (M) is the carrying capacity.

1dP dt PkP M

The relative growth rate is proportional to 1 – (P/M), withpositive proportionality constant k:

dP k P M Pdt M

or

The solution to this logistic differential equation is calledthe logistic growth model.

(What happens when P exceeds M???)

A national park is known to be capable of supporting no morethan 100 grizzly bears. Ten bears are in the park at present. Wemodel the population with a logistic differential eq. with k = 0.1.(a) Draw and describe a slope field for the differential equation.

Carrying capacity = M = 100 k = 0.1

dP k P M Pdt M

Differential Equation:

0.1 100100

P P

0.001 100P P

Use your calculator to get the slope field for this equation.(Window: [0, 150] by [0, 150])

A national park is known to be capable of supporting no morethan 100 grizzly bears. Ten bears are in the park at present. Wemodel the population with a logistic differential eq. with k = 0.1.(b) Find a logistic growth model P(t) for the population and drawits graph.

Differential Equation:

0.001 100dP P Pdt

Initial Condition:

0 10P

Rewrite 1 0.001

100dP

P P dt

Partial Fractions1 1 1 0.001100 100

dPP P dt

A national park is known to be capable of supporting no morethan 100 grizzly bears. Ten bears are in the park at present. Wemodel the population with a logistic differential eq. with k = 0.1.(b) Find a logistic growth model P(t) for the population and drawits graph.

Rewrite1 1 0.1100

dP dtP P

1 1 1 0.001100 100

dPP P dt

Integrate ln ln 100 0.1P P t C

Prop. of Logs ln 0.1100P t CP

A national park is known to be capable of supporting no morethan 100 grizzly bears. Ten bears are in the park at present. Wemodel the population with a logistic differential eq. with k = 0.1.(b) Find a logistic growth model P(t) for the population and drawits graph.

Prop. of Logs100ln 0.1P t CP

ln 0.1100P t CP

Exponentiate 0.1100 t CP eP

Rewrite 0.1100 C tP e eP

A national park is known to be capable of supporting no morethan 100 grizzly bears. Ten bears are in the park at present. Wemodel the population with a logistic differential eq. with k = 0.1.(b) Find a logistic growth model P(t) for the population and drawits graph.

Let A = + e 0.1100 1 tAeP

0.1100 C tP e eP

–c –

Solve for P0.1

1001 tPAe

Initial Condition 0

100101 Ae

9A

The Model:

0.1

1001 9 tP

e

Graph this on topof our slope field!

A national park is known to be capable of supporting no morethan 100 grizzly bears. Ten bears are in the park at present. Wemodel the population with a logistic differential eq. with k = 0.1.(c) When will the bear population reach 50?

0.1

100501 9 te

Solve:

0.11 9 2te 0.1 1 9te 0.1 9te

ln 9 21.972yr0.1

t

Note: As illustrated in this example,the solution to the general logisticdifferential equation

dP k P M Pdt M

is always

1 kt

MPAe

More Practice ProblemsFor the population described, (a) write a diff. eq. for thepopulation, (b) find a formula for the population in terms of t, and(c) superimpose the graph of the population function on a slopefield for the differential equation.

1. The relative growth rate of Flagstaff is 0.83% and its current population is 60,500.

0.0083dP Pdt

0.008360,500 tP eHow does the graph look???

More Practice ProblemsFor the population described, (a) write a diff. eq. for thepopulation, (b) find a formula for the population in terms of t, and(c) superimpose the graph of the population function on a slopefield for the differential equation.

2. A population of birds follows logistic growth with k = 0.04,carrying capacity of 500, and initial population of 40.

0.00008 500P P

0.04

5001 11.5 te

How does thegraph look???

dP k P M Pdt M

1 kt

MPAe

More Practice ProblemsThe number of students infected by measles in a certain schoolis given by the formula

5.3

2001 tP te

where t is the number of days after students are first exposedto an infected student.

(a) Show that the function is a solution of a logistic differentialequation. Identify k and the carrying capacity.

5.3

2001 tP te

5.3

2001 te e 1 kt

MAe

This is a logistic growth model

with k = 1 and M = 200.

More Practice ProblemsThe number of students infected by measles in a certain schoolis given by the formula

5.3

2001 tP te

where t is the number of days after students are first exposedto an infected student.

(b) Estimate P(0). Explain its meaning in the context of theproblem.

5.3

20001

Pe

0.993 1

Initially (t = 0), 1 student has the measles.