OBJECTIVE Find functions that satisfy dP dt kP Convert between growth rate and doubling time Solve application problems using exponential growth and limited growth models 33 Applications Uninhibited and Limited Growth Models ID: 603886
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Slide1
Applications: Uninhibited and Limited Growth Models
OBJECTIVE
Find functions that satisfy
dP
/
dt
=
kP
.
Convert between growth rate and doubling time.
Solve application problems using exponential growth and limited growth models.Slide2
3.3 Applications: Uninhibited and Limited Growth Models
Quick Check 1
Differentiate . Then express in terms of .
Notice that Slide3
THEOREM 8
A function
y
=
f
(
x
) satisfies the equation
if and only if
for some constant
c
.
3.3 Applications: Uninhibited and Limited Growth ModelsSlide4
Example 1:
Find the general form of the function
that satisfies the equation
By Theorem 8, the function must be
3.3 Applications: Uninhibited and Limited Growth ModelsSlide5
3.3 Applications: Uninhibited and Limited Growth Models
Quick Check 2
Find the general form of the function that satisfies the equation:
The function is , or where is an arbitrary constant.
As a check, note thatSlide6
Uninhibited Population Growth
The equation
is the basic model of uninhibited (unrestrained) population growth, whether the population is comprised of humans, bacteria in a culture, or dollars invested with interest compounded continuously. In the absence of inhibiting or stimulation factors, a population normally reproduces at a rate proportional to its size, and this is exactly what
dP
/
dt
=
kP
says.
3.3 Applications: Uninhibited and Limited Growth ModelsSlide7
Uninhibited Population Growth
The only function that satisfies this differential equation is given by
where
t
is time and
k
is the rate expressed in decimal notation. Note that
so
c
represents the initial population, which we denoted
P
0:
3.3 Applications: Uninhibited and Limited Growth ModelsSlide8
Example 2:
Suppose that an amount
P
0
, in dollars, is
invested in a savings account where the interest is
compounded continuously at 7% per year. That is, the
balance
P
grows at the rate given by
a) Find the function that satisfies the equation. Write it in terms of
P
0
and 0.07.
b) Suppose that $100 is invested. What is the balance after 1
yr
?
c) In what period of time will an investment of $100 double itself
?
3.3 Applications: Uninhibited and Limited Growth ModelsSlide9
2012 Pearson Education, Inc. All rights reserved
Slide 3.3-
9
Example 2 (concluded):
a)
b)
c)
3.3 Applications: Uninhibited and Limited Growth ModelsSlide10
THEOREM 9
The
exponential
growth rate
k
and the
doubling time
T
are related by
3.3 Applications: Uninhibited and Limited Growth Models
Slide11
Quick Check 3
Worldwide use of the Internet is increasing at an exponential rate, with traffic doubling every 100 days. What is the exponential growth rate?
The exponential growth rate is approximately 0.69% per day.
3.3 Applications: Uninhibited and Limited Growth ModelsSlide12
Example 3:
The world population was
approximately 6.0400 billion at the beginning of 2000.
It has been estimated that the population is growing
exponentially at the rate of 0.016, or 1.6%, per year.
Thus,
where
t
is the time, in years, after 2000.
3.3 Applications: Uninhibited and Limited Growth ModelsSlide13
Example 3 (continued):
a) Find the function that satisfies the equation. Assume that
P
0
= 6.0400 and
k
= 0.016.
b) Estimate the world population at the beginning of 2020 (
t
= 20).
c) After what period of time will the population be double that in 2000?
3.3 Applications: Uninhibited and Limited Growth ModelsSlide14
Example 3 (concluded):
3.3 Applications: Uninhibited and Limited Growth ModelsSlide15
Models of Limited Growth
The
logistic equation
, or
logistic function
is one model for population growth, in which there are factors preventing the population from exceeding some limiting value
L
, perhaps a limitation on food, living space, or other natural resources.
3.3 Applications: Uninhibited and Limited Growth ModelsSlide16
3.3 Applications: Uninhibited and Limited Growth Models
Models of Limited GrowthSlide17
Example 4:
Spread by skin-to-skin contact or via
shared towels or clothing, methicillin-resistant
staphylococcus aureus (MRSA) can easily spread a
staph infection throughout a university. Left unchecked, the number of cases of MRSA on a university campus t weeks after the first cases occur
can be modeled by
3.3 Applications: Uninhibited and Limited Growth ModelsSlide18
Example 4 (continued):
a) Find the number of infected students after 3 weeks; 40 weeks; 80 weeks.
b) Find the rate at which the disease is spreading after 20 weeks.
c) Explain why an uninhibited growth model is inappropriate but a logistic equation is appropriate for this situation. Then use a calculator to graph the equation.
3.3 Applications: Uninhibited and Limited Growth ModelsSlide19
Example 4 (continued):
a)
N
(3) = 11.8. So, approximately 12 students are infected after 3 weeks.
N(40) = 221.6. So, approximately 222 students are infected after 40 weeks. N(80) = 547.2. So, approximately 547 students are infected after 80 weeks.
3.3 Applications: Uninhibited and Limited Growth ModelsSlide20
Example 4 (continued):
b) Find
N
(
t) =
After 20 weeks, the disease is spreading about 4 new
cases per week.
3.3 Applications: Uninhibited and Limited Growth ModelsSlide21
Example 4 (continued):
c) Unrestricted growth is inappropriate for modeling
this situation because as more students become
infected, fewer are left to be newly infected. The
logistic equation displays the rapid spread of the
disease initially, as well as the slower growth in later weeks when there are fewer students left to be newly infected.3.3 Applications: Uninhibited and Limited Growth ModelsSlide22
3.3 Applications: Uninhibited and Limited Growth ModelsSlide23
Models of Limited Growth
Another model of limited growth is provided by
3.3 Applications: Uninhibited and Limited Growth ModelsSlide24
3.3 Applications: Uninhibited and Limited Growth Models
Section Summary
Uninhibited growth can be modeled by a
differential equation of the form , whose solutions are . The
exponential growth rate k and the doubling time T are related by the equation , or Certain kinds of limited growth can be modeled by equations such as and