1. Modeling with Exponential Growth and Decay
Sec. 3.1c
Homework: p odd

2. Practice Problems
Using the given data, and assuming the growth is exponential,
when will the population of San Jose surpass 1 million persons?
Population of San Jose, CA
Let P(t) be the population of
San Jose t years after 1990.
Year Population
,248
General Equation:
,943
Initial Population
Growth Factor

3. Practice Problems
Using the given data, and assuming the growth is exponential,
when will the population of San Jose surpass 1 million persons?
Population of San Jose, CA
Let P(t) be the population of
San Jose t years after 1990.
Year Population
,248
To solve for b,
use the point (10, ):
,943

4. The population of San Jose will
Practice Problems
Using the given data, and assuming the growth is exponential,
when will the population of San Jose surpass 1 million persons?
Population of San Jose, CA
Let P(t) be the population of
San Jose t years after 1990.
Year Population
,248
,943
Now, graph this function together with the line y = 1,000,000
The population of San Jose will
exceed 1,000,000 in the year 2008

5. Practice Problems bacteria bacteria
The number B of bacteria in a petri dish culture after t hours is
given by
1. What was the initial number of bacteria present?
bacteria
2. How many bacteria are present after 6 hours?
bacteria

6. Practice Problems Populations of Two Major U.S. Cities
City Population 2000 Population
Flagstaff, AZ 44, ,880
Phoenix, AZ 3,455,902 4,003,365
Assuming exponential growth (and letting t = 0 represent 1990),
what will the population of Flagstaff be in the year 2030?

7. Practice Problems  In the year 2007
Populations of Two Major U.S. Cities
City Population 2000 Population
Flagstaff, AZ 44, ,880
Phoenix, AZ 3,455,902 4,003,365
Assuming exponential growth (and letting t = 0 represent 1990),
when will the population of Phoenix exceed 4.5 million?
 In the year 2007

8. Practice Problems The two curves intersect at about t = 158.70, so
Populations of Two Major U.S. Cities
City Population 2000 Population
Flagstaff, AZ 44, ,880
Phoenix, AZ 3,455,902 4,003,365
Will the population of Flagstaff ever exceed that of Phoenix?
If so, in what year will this occur? Is this a realistic estimate
in answer to this question?
The two curves intersect at about t = , so
according to these models, the population of Flagstaff
will exceed that of Phoenix in the year 2148…

9. Logistic Functions and Growth

10. Now let’s consider this graph…
Domain:
No Local Extrema
H.A.: y = 0
Range:
V.A.: None
Continuous
End Behavior:
Decreasing on
No Symmetry
Bounded Above by y = 0

11. So far, we’ve looked primarily at exponential
growth, which is unrestricted. (meaning?)
However, in many “real world” situations, it is
more realistic to have an upper limit on growth.
(any examples?)
In such situations, growth often starts
exponentially, but then slows and eventually
levels out………………………does this remind you
of a function that we’ve previously studied???

12. Definition: Logistic Growth Functions
Let a, b, c, and k be positive constants, with b < 1. A logistic
growth function in x is a function that can be written in the form or
where the constant c is the limit to growth.
Note: If b > 1 or k < 0, these formulas yield logistic decay
functions (unless otherwise stated, the term logistic functions
will refer to logistic growth functions).
With a = c = k = 1, we
get our basic function!!!

13. Basic Function: The Logistic Function
Domain: All reals
Range: (0, 1)
Continuous
Increasing for all x
y = 1
Symmetric about (0, ½), but
neither even nor odd
(0, ½)
Bounded (above and below)
No local extrema
End Behavior:
H.A.: y = 0, y = 1
V.A.: None

14. Guided Practice
Graph the given function. Find the y-intercept, the horizontal
asymptotes, and the end behavior.
y-intercept:
The limit to growth is 8, so:
H.A.: y = 0, y = 8
End Behavior:

15. Guided Practice 20 y-intercept: H.A.: y = 0, y = 20 3 End Behavior:
Graph the given function. Find the y-intercept, the horizontal
asymptotes, and the end behavior. 20
y-intercept:
H.A.: y = 0, y = 20 3
End Behavior:

16. Guided Practice
Based on recent census data, a logistic model for the population
of Dallas, t years after 1900, is
According to this model, when was the population 1 million?
Graph this function together with the line y = 1,000,000
…where do they intersect???
The population of Dallas was 1 million by the end of 1984

17. Guided Practice The population was about 1,794,558 people in 1850
Based on recent census data, the population of New York state
can be modeled by
where P is the population in millions and t is the number of years
since Based on this model,
(a) What was the population of New York in 1850?
The population was about 1,794,558 people in 1850

18. Guided Practice The population will be about 19,366,967 in 2020
Based on recent census data, the population of New York state
can be modeled by
where P is the population in millions and t is the number of years
since Based on this model,
(b) What will New York state’s population be in 2020?
The population will be about 19,366,967 in 2020

19. Guided Practice = 19,875,000 people
Based on recent census data, the population of New York state
can be modeled by
where P is the population in millions and t is the number of years
since Based on this model,
(c) What is New York’s maximum sustainable population (limit to
growth)?
= 19,875,000 people