Do Now: #1-8, p.346 Let 1. Continuous for all real numbers Check the graph first? 2. 3. H.A.: y = 0, y = 50 4. In both the first and second derivatives,

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

50

1 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

50

1 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,

dPkP

dt

Notice thatdP dt

kP

is constant,

and is called the relative growth rate.

Solution (from Sec. 6.4):0ktP P e

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

1dP dt P

kP M

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

dP kP M P

dt 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 kP M P

dt M

Differential Equation:

0.1100

100P 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

0.001100

dP

P P dt

Partial Fractions1 1 1

0.001100 100

dP

P 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 10.001

100 100

dP

P P dt

Integrate ln ln 100 0.1P P t C

Prop. of Logs ln 0.1100

Pt C

P

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 Logs100

ln 0.1P

t CP

ln 0.1100

Pt C

P

Exponentiate 0.1100 t CPe

P

Rewrite 0.1100 C tPe e

P

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 0.11001 tAe

P

0.1100 C tPe e

P

–c –

Solve for P0.1

100

1 tP

Ae

Initial Condition 0

10010

1 Ae

9A

The Model:

0.1

100

1 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

10050

1 9 te

Solve:

0.11 9 2te 0.1 1 9te 0.1 9te

ln 921.972yr

0.1t

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

dP kP M P

dt M

is always

1 kt

MP

Ae

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

500

1 11.5 te

How does thegraph look???

dP kP M P

dt M

1 kt

MP

Ae

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

5.3

200

1 tP t

e

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

200

1 tP t

e 5.3

200

1 te e 1 kt

M

Ae

This is a logistic growth modelwith k = 1 and M = 200.

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

5.3

200

1 tP t

e

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

2000

1P

e

0.993 1

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