Transcript Modeling with Exponential Growth and Decay

MODELING WITH
EXPONENTIAL
GROWTH AND
DECAY Sec. 3.1c
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
Year
Population
1990
2000
782,248
894,943
Let P(t) be the population of
San Jose t years after 1990.
General Equation:
P  t   P0 b
t
Initial Population
Growth Factor
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
Year
Population
1990
2000
782,248
894,943
Let P(t) be the population of
San Jose t years after 1990.
To solve for b,
use the point (10, 894943):
P 10  782, 248b  894,943
894,943
b  10
 1.0135
782, 248
10
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
Year
Population
1990
2000
782,248
894,943
Let P(t) be the population of
San Jose t years after 1990.
P  t   782, 248 1.0135t
Now, graph this function together with the line y = 1,000,000
t  18.31
The population of San Jose will
exceed 1,000,000 in the year 2008
PRACTICE PROBLEMS
The number B of bacteria in a petri dish culture after t hours is
given by
B  100e
0.693t
1. What was the initial number of bacteria present?
B  0  100
bacteria
2. How many bacteria are present after 6 hours?
B  6  6394 bacteria
PRACTICE PROBLEMS
Populations of Two Major U.S. Cities
1990 Population
2000 Population
City
Flagstaff, AZ
Phoenix, AZ
44,545
3,455,902
67,880
4,003,365
Assuming exponential growth (and letting t = 0 represent 1990),
what will the population of Flagstaff be in the year 2030?
PF  t   44,545 1.0430
PF  40  240,199
t
PRACTICE PROBLEMS
Populations of Two Major U.S. Cities
1990 Population
2000 Population
City
Flagstaff, AZ
Phoenix, AZ
44,545
3,455,902
67,880
4,003,365
Assuming exponential growth (and letting t = 0 represent 1990),
when will the population of Phoenix exceed 4.5 million?
PP  t   3, 455,902 1.0148
t
t  17.95
 In the year 2007
PRACTICE PROBLEMS
Populations of Two Major U.S. Cities
1990 Population
2000 Population
City
Flagstaff, AZ
Phoenix, AZ
44,545
3,455,902
67,880
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 = 158.70, so
according to these models, the population of Flagstaff
will exceed that of Phoenix in the year 2148…
LOGISTIC FUNCTIONS
AND GROWTH
NOW LET’S CONSIDER THIS GRAPH…
f  x   5  0.8 
 ,  
Range:  , 0 
Domain:
Continuous
Decreasing on
x
No Local Extrema
H.A.: y = 0
V.A.: None
 ,  
No Symmetry
Bounded Above by y = 0
End Behavior:
lim f  x   0
x 
lim f  x   
x 
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???
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
c
f  x 
x
1 a b
or
c
f  x 
 kx
1 a e
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!!!
1
f  x 
x
1 e
BASIC FUNCTION: THE LOGISTIC FUNCTION
1
f  x 
x
1 e
Domain: All reals
y=1
Continuous
Range: (0, 1)
Increasing for all x
Symmetric about (0, ½), but
neither even nor odd
(0, ½)
Bounded (above and below)
No local extrema
H.A.: y = 0, y = 1
V.A.: None
End Behavior:
lim f  x   0 lim f  x   1
x 
x 
GUIDED PRACTICE
Graph the given function. Find the y-intercept, the horizontal
asymptotes, and the end behavior.
8
h  x 
x
1  3 0.7
y-intercept:
8
8

2
h 0 
0
1 3
1  3 0.7
End Behavior:
The limit to growth is 8, so:
H.A.: y = 0, y = 8
lim h  x   0 lim h  x   8
x 
x 
GUIDED PRACTICE
Graph the given function. Find the y-intercept, the horizontal
asymptotes, and the end behavior.
20
g  x 
3 x
1  2e
y-intercept:
End Behavior:
20
3
H.A.: y = 0, y = 20
lim g  x   0 lim g  x   20
x 
x 
GUIDED PRACTICE
Based on recent census data, a logistic model for the population
of Dallas, t years after 1900, is
1,301, 614
P t  
0.05055 t
1  21.603e
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???
t  84.50
The population of Dallas was 1 million by the end
of 1984
GUIDED PRACTICE
Based on recent census data, the population of New York state
can be modeled by
19.875
P t  
1  57.993e
0.035005 t
where P is the population in millions and t is the number of years
since 1800. Based on this model,
(a) What was the population of New York in 1850?
19.875
P  50  

1.794558
0.035005 50 
1  57.993e
The population was about 1,794,558 people in 1850
GUIDED PRACTICE
Based on recent census data, the population of New York state
can be modeled by
19.875
P t  
1  57.993e
0.035005 t
where P is the population in millions and t is the number of years
since 1800. Based on this model,
(b) What will New York state’s population be in 2020?
19.875
P  220  

19.366967
0.035005 220
1  57.993e
The population will be about 19,366,967 in 2020
GUIDED PRACTICE
Based on recent census data, the population of New York state
can be modeled by
19.875
P t  
1  57.993e
0.035005 t
where P is the population in millions and t is the number of years
since 1800. Based on this model,
(c) What is New York’s maximum sustainable population (limit to
growth)?
lim P  t   19.875 = 19,875,000 people
t 