Download exponential growth

Quiz 3-1
1. This data can be modeled using an
Find ‘a’
exponential equation  f ( x)  ab x and ‘b’
2. Where does
g ( x)  3(0.5)
x
cross the y-axis ?
3. Is g(x) an exponential growth or decay function?
4. Convert to exponential notation:
5. Convert to logarithmic notation:
log x 243  5
x5
3
3.2
Exponential and Logistic Modeling
What you’ll learn about
•
•
•
•
Constant Percentage Rate and Exponential Functions
Exponential Growth and Decay Models
Using Regression to Model Population
Other Logistic Models
… and why
Exponential functions model unrestricted growth (money)
and decay (radioactive material);
Logistic functions model restricted growth,
(spread of disease, populations and rumors)
Factoring
Your turn: Factor the following
1. f(x) = 3 + 3x
2. g(y) = 5 + 5y
Constant Percentage Rate
A population is changing at a constant percentage
rate r, where r is the percent rate (in decimal form).
Time (years)
0
1
Your turn:
Population
P(0)  P0  “initial population”
P(1)  P1  P0  P0 * r
3.
Factor P(1)
P1  P0 (1  r )
P(2)  P2  P1  P1 * r
2
P2  P1 (1  r )
Your turn:
4.
Factor P(2)
Your turn:
5.
Write P(2) in terms of P(0) only.
P2  P0 (1  r )(1  r )
 P0 (1  r )
2
Constant Percentage Rate
Time (years)
0
Population
P0  “initial population”
1
P1  P0 (1  r )
2
P2  P0 (1  r )
Your turn:
2
6. What do you think P(3) will be?
3
P3  P0 (1  r )
4
P4  P0 (1  r )
t
P(t )  P0 (1  r )
3
4
t
Exponential Population Model
If a population is changing at a constant
percentage rate ‘r’ each year, then:
P(t )  P0 (1  r )
t
is the population as a function of time.
Finding Growth and Decay
Rates
Is the following population model an
exponential growth or decay function? Find
the constant percentage growth (decay) rate.
P(t )  782,248(1.0136) t
P(t )  P0 (1  r )
t
P(t )  782,248(1  0.0136)
t
‘r’ > 0, therefore this is exponential growth.
‘r’ = 0.0136
or 1.36%
Finding an Exponential Function
Determine the exponential function with initial
value = 10, increasing at a rate of 5% per year.
P(t )  P0 (1  r )
t
P0  10
P(t )  10(1.05)
t
or
‘r’ = 0.05
f ( x)  10(1.05)
x
Your Turn:
The population of “Smallville” in the year
1890 was 6250. Assume the population
increased at a rate of 2.75% per year.
•
What is the population in 1915 ?
Modeling Bacteria Growth
Suppose a culture of 100 bacteria cells are put into a
petri dish and the culture doubles every hour.
Predict when the number of bacteria will be 350,000.
P(t )  P0 (1  r ) t
Doubles with
Every time interval
P(t )  P0 (2)t
P(0) = 100
P(t) = 350000
P(1) = 2*P(0)
350000  100(2)t
5
3
.
5

10
3
2t 

3
.
5

10
 3500
2
110
2t  3500
Solving an Exponential
Equation
t
x
2  3500 log 2 2  log 2 3500
x  log 2 3500
Your calculator doesn’t have base 2 (it might
in some of the catalog of functions)
Change of Base Formula:
log 3500
x
 11.77
log 2
log b a
log c a 
log b c
t = 11 hours, 46 minutes
Your Turn:
The population of “Smallville” in the year
1890 was 6250. Assume the population
increased at a rate of 2.75% per year.
8. When did the population reach 50,000 ?
Exponential Regression
Stat p/b  gives lists
Enter the data:
Let L1 be years since initial value
Let L2 be population
Stat p/b  calc p/b
scroll down to exponential regression
“ExpReg” displayed:
enter the lists: “L1,L2”
f ( x)  ab
x
The calculator will display the
values for ‘a’ and ‘b’.
Modeling U.S. Population Using
Exponential Regression
Use the 1900-2000 data and
exponential regression to
predict the U.S. population for
2003. (Don’t enter the 2003
value).
Let P(t) = population,
“t” years after 1900.
Enter the data into your
calculator and use
exponential regression
to determine the model (equation).
Modeling U.S. Population Using
Exponential Regression
Your turn:
9.
What is your equation?
10. What is your predicted population
in 2003 ?
11. Why isn’t your predicted value
the same as the actual value of
290.8 million?
Maximum Sustainable
Population
Exponential growth is unrestricted, but
population growth often is not. For many
populations, the growth begins
exponentially, but eventually slows and
approaches a limit to growth called the
maximum sustainable population.
We must use Logistic function if the growth
is limited !!!
Modeling a Rumor
Roy High School has about 1500 students.
5 students start a rumor, which spreads
logistically so that
1500
S (t ) 
(1  29e 0.9t )
Models the number of students who have
heard the rumor by the end of ‘t’ days, where
‘t’ = 0 is the day the rumor began to spread.
How many students have heard the rumor
by the end of day ‘0’ ?
How long does it take for 1000 students to
have heard the rumor ?
1500
Rumors at
RHS
S (t ) 
 0.9 t
(1  29e
)
How many students have heard the rumor
by the end of day ‘0’ ?
1500
1500
S (0) 

 50
 0.9 ( 0 )
(1  29e
) (1  29)
How long does it take for 1000 students to
have heard the rumor ?
Your turn:
1500
1000 
(1  29e 0.9(t ) )
12. “t” = ? (days)
HOMEWORK