1. Do Now: #1-8, p.346 Let Check the graph first?
1. Continuous for all real numbers 2.
3. H.A.: y = 0, y = 50
4. In both the first and second derivatives, the denominator
will be a power of , which is never 0. Thus, the
domains of both are all real numbers.

2. Do Now: #1-8, p.346 Let 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].
Conc. up:
Conc. down:
8. Point of inflection:

3. Logistic Growth
Section 6.5a

4. Review from last section…
Many populations grow at a rate proportional to the size of the
population. Thus, for some constant k,
Notice that
is constant,
and is called the relative growth rate.
Solution (from Sec. 6.4):

5. Logistic Growth Models
In reality, most populations are limited in growth. The maximum
population (M) is the carrying capacity.
The relative growth rate is proportional to 1 – (P/M), with
positive proportionality constant k: or
The solution to this logistic differential equation is called
the logistic growth model.
(What happens when P exceeds M???)

6. Use your calculator to get the slope field for this equation.
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model 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
Differential Equation:
Use your calculator to get the slope field for this equation.
(Window: [0, 150] by [0, 150])

7. Rewrite Partial Fractions
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(b) Find a logistic growth model P(t) for the population and draw
its graph.
Differential Equation:
Initial Condition:
Rewrite
Partial Fractions

8. Rewrite Integrate Prop. of Logs
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(b) Find a logistic growth model P(t) for the population and draw
its graph.
Rewrite
Integrate
Prop. of Logs

9. Prop. of Logs Exponentiate Rewrite
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(b) Find a logistic growth model P(t) for the population and draw
its graph.
Prop. of Logs
Exponentiate
Rewrite

10. Let A = + e Solve for P Initial Condition
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(b) Find a logistic growth model P(t) for the population and draw
its graph.
The Model: –c
Let A = + e –
Solve for P
Graph this on top
of our slope field!
Initial Condition

11. A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(c) When will the bear population reach 50?
Solve:
Note: As illustrated in this example,
the solution to the general logistic
differential equation
is always

12. More Practice Problems
For the population described, (a) write a diff. eq. for the
population, (b) find a formula for the population in terms of t, and
(c) superimpose the graph of the population function on a slope
field for the differential equation.
1. The relative growth rate of Flagstaff is 0.83% and its
current population is 60,500.
How does the graph look???

13. More Practice Problems
For the population described, (a) write a diff. eq. for the
population, (b) find a formula for the population in terms of t, and
(c) superimpose the graph of the population function on a slope
field 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.
How does the
graph look???

14. This is a logistic growth model
More Practice Problems
The number of students infected by measles in a certain school
is given by the formula
where t is the number of days after students are first exposed
to an infected student.
(a) Show that the function is a solution of a logistic differential
equation. Identify k and the carrying capacity.
This is a logistic growth model
with k = 1 and M = 200.

15. Initially (t = 0), 1 student has the measles.
More Practice Problems
The number of students infected by measles in a certain school
is given by the formula
where t is the number of days after students are first exposed
to an infected student.
(b) Estimate P(0). Explain its meaning in the context of the
problem.
Initially (t = 0), 1 student has the measles.