Transcript Logistic Functions

SECTION
7.4
Logistic Functions
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Learning Objectives
1 Graph logistic functions from
equations and tables
2 Use logistic models to predict
and interpret unknown results
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Logistic Growth
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Logistic Growth
Figure 7.23 represents the total number of people who
have had the flu as a combination of very slow exponential
growth (labeled a), rapid exponential growth (labeled b),
followed by a slower increase (labeled c), and then a
leveling off (labeled d).
Figure 7.23
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Logistic Growth
At the inflection point the rate at which the
flu is spreading is the greatest.
The horizontal asymptote (red dashed
horizontal line) represents the limiting
value for the number of people who will
contract the flu.
The mathematical model for such behavior
is called a logistic function.
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Logistic Growth
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Example 1 – Exploring Logistic Functions in a Real-World Context
The Centers for Disease Control monitor flu infections
annually, paying particular attention to flu-like symptoms in
children from birth to 4 years old.
The cumulative number of children
0–4 years old who visited the CDC’s
sentinel providers with flu-like
symptoms during the 2006–2007
flu season are displayed in
Figure 7.24, together with a
logistic model.
Children (0–4) with Flu-Like Symptoms
Figure 7.24
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Example 1 – Exploring Logistic Functions in a Real-World Context
cont’d
a. Explain why a logistic function may better model the
2006–2007 flu infection rate than an exponential
function.
Figure 7.24
Children (0–4) with Flu-Like Symptoms
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Example 1(a) – Solution
A logistic function models the growth in the
cumulative number of children with flu-like
symptoms better than an exponential function
because a logistic function grows slowly at first,
then more quickly, and finally levels off at a limiting
value.
An exponential function, on the other hand, may
grow slowly at first but then will increase at an ever
increasing rate, ultimately exceeding the available
number of children who could conceivably become
infected with flu-like illnesses.
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Example 1 – Exploring Logistic Functions in a Real-World Context
cont’d
b. Using the language of rate of change, describe the
behavior of the graph and relate this to what it tells about
the real-world context.
Children (0–4) with Flu-Like Symptoms
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Example 1(b) – Solution
cont’d
The graph of the function model is increasing
throughout the interval; however, the rate at which
the graph is increasing varies.
Initially, the graph is concave up, indicating that
the cumulative number of reported cases is
increasing at an ever increasing rate.
Around week 17, the graph changes to concave
down, indicating that the cumulative number of
reported cases are increasing at a lesser and
lesser rate.
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Example 1(b) – Solution
cont’d
Around week 32, the graph is increasing at such a
slow rate that it appears to level off.
This indicates that the number of newly reported
cases in weeks 32 and beyond is so small that it
has a negligible effect on the cumulative number of
cases.
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Example 1 – Exploring Logistic Functions in a Real-World Context
cont’d
c. The formula for the logistic function that models the
cumulative number of reported children with flu-like
symptoms is
where w is the week
of the flu season.
What does the limiting value mean in the real-world
context? How is the limiting value represented in the
formula for the function H(w) and its graph?
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Example 1(c) – Solution
cont’d
The limiting value is approximately 75,700 children.
It appears that no more than 75,700 children were
reported to have flu-like symptoms during the season.
This value is represented in the
formula for H(w) by the value 75,700
found in the numerator and on the
graph by the horizontal asymptote
at 75,700.
Children (0–4) with Flu-Like Symptoms
Figure 7.24
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Example 1 – Exploring Logistic Functions in a Real-World Context
cont’d
d. Estimate the coordinates for the point of inflection and
explain what each coordinate means in the real-world
context.
Children (0–4) with Flu-Like Symptoms
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Example 1(d) – Solution
cont’d
From the graph of H(w), an estimate for the
coordinates of the inflection point is
approximately (17, 40,000).
This means at week 17 of the 2006–2007 flu
season (late January), the number of children
with flu-like symptoms was increasing most
rapidly.
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Example – Exploring Logistic Functions From a Table
Between what intervals is the inflection point? What is the
meaning of the inflection point within the context of the
problem?
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Example – Exploring Logistic Functions From a Table
The percentage of households with a telephone increase throughout
the table; however, the rate of increase between 10 and 15 years since
1935 begins to decrease indicating an inflection point. So between
1945 and 1950 the percentage of households with telephones
increased at it’s highest rate.
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Logistic Decay
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Logistic Decay
Many real-world data sets are modeled with
decreasing rather than increasing logistic
functions.
As was the case with logistic growth
functions, logistic decay functions have an
upper limiting value, L, and a lower limiting
value of y = 0.
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Example 2 – Recognizing a Logistic Decay Function
As shown in Table 7.19, the infant mortality rate in the
United States has been falling since 1950. (An infant is a
child under 1 year of age.)
Table 7.19
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Example 2 – Recognizing a Logistic Decay Function
cont’d
a. Using Table 7.19, show how the rate of change records
the decline in the infant death mortality rate and discuss
why this suggests that a logistic model may fit the data.
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Example 2(a) – Solution
If we calculate the decrease in the infant mortality rate (M)
and the yearly rate of change of the infant mortality rate
over each interval of time given in Table 7.20, we can
determine the behavior of the function M(y).
Table 7.20
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Example 2(a) – Solution
cont’d
Note: We must exercise caution because the
intervals between the years provided are not the
same.
From the rate of change, we can see that the
decline in the infant mortality rate tends to drop
slowly at first, then more dramatically, and then
levels off.
The change in the infant mortality rate suggests a
logistic model.
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Example 2 – Recognizing a Logistic Decay Function
cont’d
b. From the table, predict the future lower limiting value and
explain what the numerical value means in the real-world
context. Does this value seem reasonable?
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Example 2(b) – Solution
cont’d
We estimate the lower limiting value to be approximately
6.0 deaths per 1000 live births because we expect the
values to level off near 6.0.
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Example 2 – Recognizing a Logistic Decay Function
cont’d
c. Create a scatter plot of the data and estimate the upper
and lower limiting values.
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Example 2(c) – Solution
cont’d
The scatter plot is shown in Figure 7.25.
Figure 7.25
From the scatter plot, we predict the upper limiting value
will be around 35 and the lower limiting value will be around
6.
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Logistic Decay
We said earlier that all logistic functions have a
lower limiting value at y = 0.
We can use logistic regression to model data sets
with this lower limiting value.
But when a data set appears to be logistic but has
a different lower limiting value, as in Example 2,
we need to align the data before using logistic
regression to model the function.
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Example: Lower Limiting Value Alignment
a.) Use logistic regression to find the logistic model
for the data. (Remember to align your data for the
lower limiting value.)
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Example: Lower Limiting Value Alignment
a.) Use logistic regression to find the logistic model
for the data. (Remember to align your data for the
lower limiting value.)
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Example 4 – Extrapolating Exponential and Logistic Growth
cont’d
The sales of DVD hardware from 1997 to 2006 are given in
Table 7.22.
Table 7.22
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Example 4 – Extrapolating Exponential and Logistic Growth
cont’d
a. Use logistic regression to determine the function, D(y), of
the form
that best models the growth in
DVD sales from 1997 to 2004. Determine the value of L
and explain what this number means in the real-world
context.
b. Based on the data from 2005 and 2006, does it appear
the model can be used to accurately forecast future
sales? Explain why or why not.
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Example 4(a) – Solution
Using the Technology Tip and logistic regression on the
data from 1997 to 2004, we find
million dollars is the best-fit logistic function for DVD sales
(see Figure 7.29).
Figure 7.29
The limiting value, L, is 40.69, which means DVD sales
will level off at $40,690,000 per year.
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Example 4(b) – Solution
cont’d
For the years 2005 and 2006, we evaluate the function at
t = 8 and t = 9.
The model forecasts 39.3 million dollars in sales in 2005
and 40.1 million dollars in sales in 2006. In actuality, there
were 36.7 and 32.7 million dollars in sales, respectively.
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Example 4(b) – Solution
cont’d
The model predicts a leveling off of sales whereas the
actual data shows a decline in sales in 2005 and 2006
Consequently, the model does not appear to accurately
model future sales.
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