C. Logistic Equations and Regression Models
Tải bản đầy đủ - 0trang
cob19545_ch05_553-566.qxd
11/30/10
5:03 PM
Page 556
College Algebra G&M—
556
5–78
CHAPTER 5 Exponential and Logarithmic Functions
EXAMPLE 4
ᮣ
Calculating a Logistic Regression Model
Yeast cultures have a number of applications that are
a great benefit to civilization and have been an object
of study for centuries. A certain strain of yeast is
grown in a lab, with its population checked at 2-hr
intervals, and the data gathered are given in the
table. Use the data and a graphing calculator to draw
a scatterplot, and decide on an appropriate form of
regression. If a logistic regression is the best model,
attempt to estimate the capacity coefficient c prior to
using your calculator to find the regression equation.
How close were you to the actual value?
Solution
ᮣ
WORTHY OF NOTE
Notice that calculating a logistic
regression model takes the
calculator a few seconds longer
than for other forms.
C. You’ve just seen how
we can determine when a
logistic model is appropriate,
and how to apply a logistic
model to a set of data
Elapsed Time
(hours)
Population
(100s)
2
20
4
50
6
122
After clearing the data lists, enter the input
values (elapsed time) in L1 and the output
values (population) in L2. For the viewing
window, scale the t-axis from Ϫ1 to 20 and the
P-axis from Ϫ100 to 700 to comfortably fit the
data. From the context and scatterplot, it’s
Ϫ1
apparent the data are best modeled by a logistic
function. Noting that Ymax ϭ 700 and the data
seem to level off near the top of the window, a
good estimate for c would be about 675. Using
logistic regression on the home screen (option B:Logistic),
663
1rounded2.
we obtain the equation Y1 ϭ
1 ϩ 123.9eϪ0.553X
8
260
10
450
12
570
14
630
16
650
700
20
Ϫ100
Now try Exercises 25 and 26 ᮣ
When a regression equation is used to gather information, many of the equation
solving skills from prior sections are employed. Exercises 27 through 34 offer a variety of these equations for practice and warm-up.
D. Applications of Regression
Once the equation model for a data set has been obtained, it can be used to interpolate
or approximate values that might occur between those given in the data set. It can also
be used to extrapolate or predict future values. In this case, the investigation extends
beyond the values from the data set, and is based on the assumption that projected
trends will continue for an extended period of time.
Regardless of the regression applied, interpolation and extrapolation involve substituting a given or known value, then solving for the remaining unknown. We’ll
demonstrate here using the regression model from Example 3. The exercise set offers
a large variety of regression applications, including some power regressions and additional applications of linear and quadratic regression.
EXAMPLE 5
ᮣ
Using a Regression Equation to Interpolate or Extrapolate Information
Use the regression equation from Example 3 to answer the following questions:
a. What is the average circumference of a female child’s head, if the child is
21 months old?
b. According to the equation model, what will the average circumference be
when the child turns 312 years old?
c. If the circumference of the child’s head is 44 cm, about how old is the child?
cob19545_ch05_553-566.qxd
11/27/10
12:47 AM
Page 557
College Algebra G&M—
5–79
Section 5.7 Exponential, Logarithmic, and Logistic Equation Models
Solution
557
a. Using function notation we have C1a2 Ϸ 39.8171 ϩ 2.3344 ln1a2. Substituting
21 for a gives:
ᮣ
Figure 5.63
C1212 Ϸ 39.8171 ϩ 2.3344 ln1212
Ϸ 46.9
Y1 ϭ 39.8171 ϩ 2.3344 ln x
substitute 21 for a
result
The circumference is approximately 46.9 cm. See Figure 5.63.
b. Substituting 3.5 yr ϫ 12 ϭ 42 months for a gives:
C1422 Ϸ 39.8171 ϩ 2.3344 ln1422
Ϸ 48.5
Figure 5.64
50
0
40
40
substitute 42 for a
result
The circumference will be approximately 48.5 cm. See Figure 5.63.
c. For part (c) we’re given the circumference C and are asked to find the age a in
which this circumference (44) occurs. Substituting 44 for C(a) gives the equation
44 ϭ 39.8171 ϩ 2.3344 ln X, so we set Y1 ϭ 39.8171 ϩ 2.3344 ln X and
Y2 ϭ 44. For the window size, we know the formula is valid for female infants
from 0 to 36 months, and from parts (a) and (b) a good range for the circumference
will be from 40 to 50 cm. This indicates an appropriate window might be [0, 40]
for x and [40, 50] for y. Using this window, we find the graphs intersect at about
(6, 44), showing that a female child with a cranial circumference of 44 cm
must be about 6 months old. See Figure 5.64.
Now try Exercises 37 through 44 ᮣ
D. You’ve just seen how
we can use a regression model
to answer questions and solve
applications
When extrapolating from a set of data, care and common sense must be used or results
can be very misleading. For example, while the Olympic record for the 100-m dash
has been steadily declining since the first Olympic Games, it would be foolish to think
it will ever be run in 0 sec. There is a large variety of additional applications in the
Exercise Set. See Exercises 45 through 62.
5.7 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
1. The type of regression used often depends on
(a) whether a particular graph appears to fit the
________ and (b) the ________ or ________ that
generated the data.
2. The final choice of regression can rarely be based
on the ________ alone. Relying on the basic
_________ and ________ of certain graphs can be
helpful.
3. To extrapolate means to use the data to predict
values ________ the given data.
4. To interpolate means to use the data to predict
values ________ the given data.
5. List the five steps used to find a regression equation
using a calculator. Discuss possible errors that can
occur if the first step is skipped. After the new data
have been entered, what precautionary step should
always be included?
6. Consider the eight toolbox functions and the
exponential and logarithmic functions. How many
of these satisfy the condition as x S q, y S q?
For those that satisfy this condition, discuss/explain
how you would choose between them judging from
the scatterplot alone.
cob19545_ch05_553-566.qxd
11/30/10
5:19 PM
Page 558
College Algebra G&M—
558
ᮣ
5–80
CHAPTER 5 Exponential and Logarithmic Functions
DEVELOPING YOUR SKILLS
Match each scatterplot given with one of the following: (a) likely linear, (b) likely quadratic, (c) likely exponential,
(d) likely logarithmic, (e) likely logistic, or (f) none of these.
7.
8.
y
9.
y
y
20
20
20
15
15
15
10
10
10
5
5
5
0
10.
5
10
x
0
11.
y
5
10
x
0
12.
y
20
20
15
15
15
10
10
10
5
5
5
5
10
15
x
0
5
For Exercises 13 to 20, suppose a set of data is generated
from the context indicated. Use common sense, previous
experience, or your own knowledge base to state
whether a linear, quadratic, logarithmic, exponential,
power, or logistic regression might be most appropriate.
Justify your answers.
13. total revenue and number of units sold
10
x
10
15
x
y
20
0
5
0
5
10
x
Graph the data sets, then discuss why a logarithmic
model could be an appropriate form of regression. Then
find the regression equation.
23. Total number of
sales compared to
the amount spent
on advertising
Advertising Total Number
Costs ($1000s)
of Sales
14. page count in a book and total number of words
15. years on the job and annual salary
16. time and population growth with unlimited resources
1
125
5
437
10
652
15
710
20
770
17. time and population growth with limited resources
25
848
18. elapsed time and the height of a projectile
30
858
19. the cost of a gallon of milk over time
35
864
20. elapsed time and radioactive decay
Graph the data sets, then discuss why an exponential
model could be an appropriate form of regression. Then
find the regression equation.
21. Radioactive Studies
22. Rabbit Population
24. Cumulative weight of
diamonds extracted from
a diamond mine
Time
(months)
Weight
(carats)
1
500
3
1748
6
2263
9
2610
Time in
Hours
Grams of
Material
Month
Population
(in hundreds)
12
3158
0.1
1.0
0
2.5
15
3501
1
0.6
3
5.0
18
3689
2
0.3
6
6.1
21
3810
3
0.2
9
12.3
4
0.1
12
17.8
5
0.06
15
30.2
cob19545_ch05_553-566.qxd
9/2/10
9:51 PM
Page 559
College Algebra G&M—
5–81
25. Spread of disease:
Days After Cumulative
Estimates of the
Outbreak
Total
cumulative number of
0
100
SARS (sudden acute
respiratory syndrome)
14
560
cases reported in
21
870
Hong-Kong during the
35
1390
spring of 2003 are
56
1660
shown in the table,
with day 0
70
1710
corresponding to
84
1750
February 20. (a) Use
the data to draw a scatterplot, then use the context
and scatterplot to decide on the best form of
regression. (b) If a logistic model seems best, attempt
to estimate the carrying capacity c, then (c) use your
calculator to find the regression equation.
Source: Center for Disease Control @ www.cdc.gov/ncidod/EID/vol9no12.
26. Cable television subscribers: The percentage of
American households having cable television is
given in the table for select years from 1976 to
2004. (a) Use the data to draw a scatterplot, then
use the context and scatterplot to decide on the best
ᮣ
form of regression.
(b) If a logistic model
seems best, attempt to
estimate the carrying
capacity c, then (c) use
your calculator to find
the regression equation
(use 1976 S 0).
Year
Percentage
1976 S 0 with Cable TV
Source: Data pooled from the 2001
New York Times Almanac, p. 393;
2004 Statistical Abstract of the
United States, Table 1120; various
other years.
0
16
4
22.6
8
43.7
12
53.8
16
61.5
20
66.7
24
68
28
70
The applications in this section require solving
equations similar to those that follow. Solve each
equation algebraically and graphically.
27. 96.35 ϭ 19.421.6x
29. 4.8x2.5 ϭ 468.75
28. 13.722.9x ϭ 1253.93
30. 4375 ϭ 1.4xϪ1.25
31. 52 ϭ 63.9 Ϫ 6.8 ln x
32. 498.53 ϩ 18.2 ln x ϭ 595.9
33. 52 ϭ
67
1 ϩ 20eϪ0.62x
34.
975
ϭ 890
1 ϩ 82.3eϪ0.423x
WORKING WITH FORMULAS
35. Learning curve: C1t2 ؍4.1 ؉ 9.5 ln t
The number of circuit boards a newly hired
employee can assemble from its component parts,
depends on the experience of the employee as
measured by the length of employment. This
relationship is modeled by the formula shown,
where C(t) represents the number of circuit boards
assembled per day, t days after employment.
(a) How many boards are being assembled after 5
days on the job? (b) How many days until the
employee is able to assemble 30 boards per day?
ᮣ
559
Section 5.7 Exponential, Logarithmic, and Logistic Equation Models
36. Bicycle sales since 1920: N1t2 ؍0.32511.0572 t
Despite the common use of automobiles and
motorcycles, bicycle sales have continued to grow
as a means of transportation as well as a form of
recreation. The number of bicycles sold each year
(in millions) can be approximated by the formula
shown, where t is the number of years after 1920
11920 S 02. According to this model, in what year
did bicycle sales exceed 10 million?
Source: 1976/1992 Statistical Abstract of the United States, Tables 406/395;
various other years
APPLICATIONS
Answer the questions using the given data and the related regression equation. All extrapolations
assume the mathematical model will continue to represent future trends.
37. Weight loss: Harold needed to lose weight and started on a new diet and exercise regimen.
The number of pounds he’s lost since the diet began is given in the table. Draw the
scatterplot, decide on an appropriate form of regression, and find an equation that models
the data.
a. What was Harold’s total weight loss after 15 days?
b. Approximately how many days did it take to lose a total of 18 pounds?
c. According to the model, what is the projected weight loss for 100 days?
Time
(days)
Pounds
Lost
10
2
20
14
30
20
40
23
50
25.5
60
27.6
70
29.2
80
30.7
cob19545_ch05_553-566.qxd
11/27/10
12:48 AM
Page 560
College Algebra G&M—
560
5–82
CHAPTER 5 Exponential and Logarithmic Functions
38. Depletion of resources: The
Time
Ounces
longer an area is mined for
(months) Mined
gold, the more difficult and
5
275
expensive it gets to obtain.
The cumulative total of the
10
1890
ounces produced by a
15
2610
particular mine is shown in
20
3158
the table. Draw the
25
3501
scatterplot, use the scatterplot
and context to determine
30
3789
whether an exponential or
35
4109
logarithmic model is more
40
4309
appropriate, then find an
equation that models the data.
a. What was the total number of ounces mined
after 18 months?
b. About how many months did it take to mine a
total of 4000 oz?
c. According to the model, what is the projected
total after 50 months?
39. Number of U.S. post
Year
Offices
offices: Due in large
(1000s)
(1900 S 0)
part to the ease of travel
1
77
and increased use of
telephones, e-mail and
20
52
instant messaging, the
40
43
number of post offices
60
37
in the United States has
80
32
been on the decline
since the twentieth
100
28
century. The data given
show number of post offices (in thousands) for
selected years. Use the data to draw a scatterplot,
then use the context and scatterplot to find the
regression equation (use 1900 S 0).
Source: Statistical Abstract of the United States; The First Measured Century
a. Approximately how many post offices were
there in 1915?
b. In what year did the number of post offices
drop below 34,000?
c. According to the model, how many post offices
will there be in the year 2015?
40. Telephone use: The
number of telephone
calls per capita has
been rising
dramatically since the
invention of the
telephone in 1876. The
table shows the
number of phone calls
per capita per year for
selected years. Use the
Year
(1900 S 0)
Number
(per capita/
per year)
0
38
20
180
40
260
60
590
80
1250
97
2325
data to draw a scatterplot, then use the context and
scatterplot to find the regression equation.
Source: The First Measured Century by Theodore Caplow, Louis Hicks, and
Ben J. Wattenberg, The AEI Press, Washington, D.C., 2001.
a. What was the approximate number of calls per
capita in 1970?
b. Approximately how many calls per capita will
there be in 2015?
c. In what year did the number of calls per capita
exceed 4000?
41. Milk production: Since
1980, the number of
family farms with milk
cows for commercial
production has been
decreasing. Use the data
from the table given to
draw a scatterplot, then
use the context and
scatterplot to find the
regression equation.
Year
(1980 S 0)
Number
(in 1000s)
0
334
5
269
10
193
15
140
17
124
18
117
19
111
Source: Statistical Abstract of the
United States, 2000.
a. What was the approximate number of farms
with milk cows in 1993?
b. Approximately how many farms will have
milk cows in 2010?
c. In what year will this number of farms drop
below 45 thousand?
42. Froth height—
Time
Height of
carbonated beverages:
(seconds)
Froth (in.)
The height of the froth
0
0.90
on carbonated drinks
and other beverages can
2
0.65
be manipulated by the
4
0.40
ingredients used in
6
0.21
making the beverage
8
0.15
and lends itself very
well to the modeling
10
0.12
process. The data in the
12
0.08
table given show the
froth height of a certain
beverage as a function of time, after the froth has
reached a maximum height. Use the data to draw a
scatterplot, then use the context and scatterplot to
find the regression equation.
a. What was the approximate height of the froth
after 6.5 sec?
b. How long does it take for the height of the
froth to reach one-half of its maximum height?
c. According to the model, how many seconds
until the froth height is 0.02 in.?
cob19545_ch05_553-566.qxd
9/2/10
9:51 PM
Page 561
College Algebra G&M—
5–83
Section 5.7 Exponential, Logarithmic, and Logistic Equation Models
43. Chicken production:
Year
Number
In 1980, the production
(1980 S 0)
(millions)
of chickens in the
0
392
United States was
about 392 million. In
5
370
the next decade, the
9
356
demand for chicken
14
386
first dropped, then rose
16
393
dramatically. The
number of chickens
17
410
produced is given in
18
424
the table to the right for
selected years. Use the
data to draw a scatterplot, then use the context and
scatterplot to find the regression equation.
Source: Statistical Abstract of the United States, 2000.
a. What was the approximate number of chickens
produced in 1987?
b. Approximately how many chickens will be
produced in 2004?
c. According to the model, for what years was the
production of chickens below 365 million?
44. Veterans in civilian
Year
Number
life: The number of
(millions)
(1950
S
0)
military veterans in
civilian life fluctuates
0
19.1
with the number of
10
22.5
persons inducted into
20
27.6
the military (higher in
30
28.6
times of war) and the
passing of time. The
40
27
number of living
48
25.1
veterans is given in the
49
24.6
table for selected years
from 1950 to 1999. Use
the data to draw a scatterplot, then use the context
and scatterplot to find the regression equation.
Source: Statistical Abstract of the United States, 2000.
a. What was the approximate number of living
military veterans in 1995?
b. Approximately how many living veterans will
there be in 2015?
c. According to the model, in what years did the
number of veterans exceed 26 million?
45. Use of debit cards:
Since 1990, the use of
debit cards to obtain
cash and pay for
purchases has become
very common. The
number of debit cards
nationwide is given in
the table for selected
Year
(1990 S 0)
Number
of Cards
(millions)
0
164
5
201
8
217
10
230
561
years. Use the data to draw a scatterplot, then use
the context and scatterplot to find the regression
equation.
Source: Statistical Abstract of the United States, 2000.
a. Approximately how many debit cards were
there in 1999?
b. Approximately how many debit cards will
there be in 2015?
c. In what year did the number of debit cards
exceed 300 million?
46. Quiz grade versus study
x
y
time: To determine the
(min study) (score)
value of doing homework,
45
70
a student in college
algebra records the time
30
63
spent by classmates in
10
59
preparation for a quiz the
20
67
next day. Then she
60
73
records their scores,
which are shown in the
70
85
table. Use the data to
90
82
draw a scatterplot, then
75
90
use the context and
scatterplot to find the
regression equation. According to the model, what
grade can I expect if I study for 120 min?
47. Population of coastal
Year
areas: The percentage
(1970 S 0) Percentage
of the U.S. population
that can be categorized
0
22.8
as living in Pacific
10
27.0
coastal areas
20
33.2
(minimum of 15% of
25
35.2
the state’s land area is a
coastal watershed) has
30
37.8
been growing steadily
31
38.5
for decades, as
32
38.9
indicated by the data
33
39.4
given for selected
years. Use the data to
draw a scatterplot, then use the context and
scatterplot to find the regression equation.
According to the model, what is the predicted
percentage of the population living in Pacific
coastal areas in 2005, 2010 and 2015?
Source: 2004 Statistical Abstract of the United States, Table 23.
cob19545_ch05_553-566.qxd
9/2/10
9:51 PM
Page 562
College Algebra G&M—
562
5–84
CHAPTER 5 Exponential and Logarithmic Functions
48. Water depth and pressure:
Depth Pressure
As anyone who’s been
(ft)
(psi)
swimming knows, the deeper
15
6.94
you dive, the more pressure
you feel on your body and
25
11.85
eardrums. This pressure (in
35
15.64
pounds per square inch or
45
19.58
psi) is shown in the table for
55
24.35
selected depths. Use the data
to draw a scatterplot, then use
65
28.27
the context and scatterplot to
75
32.68
find the regression equation.
According to the model, what
pressure can be expected at a depth of 100 ft?
49. Musical notes: The
# Note Frequency
table shown gives the
1
A
110.00
frequency (vibrations
per second for each of
2
A#
116.54
the twelve notes in a
3
B
123.48
selected octave) from
4
C
130.82
the standard chromatic
5
C#
138.60
scale. Use the data to
draw a scatterplot, then
6
D
146.84
use the context and
7
D#
155.56
scatterplot to find the
8
E
164.82
regression equation.
9
F
174.62
a. What is the
10
F#
185.00
frequency of the
“A” note that is an
11
G
196.00
octave higher than
12
G#
207.66
the one shown?
[Hint: The names
repeat every 12 notes (one octave), so this
would be the 13th note in this sequence.]
b. If the frequency is 370.00 what note is being
played?
c. What pattern do you notice for the F#’s in each
octave (the 10th, 22nd, 34th, and 46th notes in
sequence)? Does the pattern hold for all notes?
50. Basketball salaries: In
1970, the average player
salary for a professional
basketball player was
about $43,000. Since that
time player salaries have
risen dramatically. The
average player salary for a
professional player is
given in the table shown
for selected years. Use the
data to draw a scatterplot,
Year
(1970 S 0)
Salary
($1000s)
0
43
10
260
15
325
20
750
25
1900
27
2200
28
2600
then use the context and scatterplot to find the
regression equation.
Source: Wall Street Journal Almanac.
a. What was the approximate salary for a player
in 1993?
b. Approximately how much will the average
salary be in 2005?
c. In what year did the average salary exceed
$5,000,000?
51. Cost of cable service:
Year
Monthly
The average monthly cost
(1980 S 0) Charge
of cable TV has been
0
$7.69
rising steadily since it
became very popular in
5
$9.73
the early 1980s. The data
10
$16.78
given shows the average
20
$23.07
monthly rate for selected
25
$30.70
years (1980 S 0). Use the
data to draw a scatterplot,
then use the context and
scatterplot to find the regression equation.
According to the model, what will be the cost of
cable service in 2010? 2015?
Source: 2004–2005 Statistical Abstract of the United States, page 725,
Table 1138.
52. Research and
Year
R&D
development
(1960 S 0) (billion $)
expenditures: The
0
13.7
development of new
products, improved health
5
20.3
care, greater scientific
10
26.3
achievement, and other
15
35.7
advances is fueled by
20
63.3
huge investments in
research and development
25
114.7
(R & D). Since 1960, total
30
152.0
R & D expenditures in the
35
183.2
United States have
39
247.0
shown a distinct pattern
of growth, and the data
are given in the table for selected years from 1960
to 1999. Use the data to draw a scatterplot, then use
the context and scatterplot to find the regression
equation. According to the model, what was spent
on R & D in 1992? In what year did expenditures
for R & D exceed 450 billion?
cob19545_ch05_553-566.qxd
11/27/10
12:48 AM
Page 563
College Algebra G&M—
5–85
563
Section 5.7 Exponential, Logarithmic, and Logistic Equation Models
53. Business start-up costs: As
Profit
many new businesses open,
Month ($1000s)
they experience a period
Ϫ5
1
where little or no profit is
realized due to start-up
Ϫ13
2
expenses, equipment
Ϫ18
3
purchases, and so on. The data
Ϫ20
4
given shows the profit of a
5
Ϫ21
new company for the first 6
months of business. Use the
6
Ϫ19
data to draw a scatterplot, then
use the context and scatterplot
to find the regression equation. According to the
model, what is the first month that a profit will be
earned?
54. Low birth weight: For many
years, the association between
low birth weight (less than
2500 g or about 5.5 lb) and a
mother’s age has been well
documented. The data given
are grouped by age and give
the percent of total births with
low birth weight.
Source: National Vital Statistics Report,
Vol. 50, No. 5, February 12, 2002.
Ages
Percent
15–19
8.5
20–24
6.5
25–29
5.2
30–34
5
35–39
6
40–44
8
45–54
10
a. Using the data and the median age of each
group, draw a scatterplot and decide on an
appropriate form of regression.
b. Find a regression equation that models the
data. According to the model, what percent of
births will have a low birth weight if the
mother was 58 years old?
55. Growth of cell phone
Year
Subscriptions
use: The tremendous
(millions)
(1990 S 0)
surge in cell phone
0
5.3
use that began in the
early nineties has
3
16.0
continued unabated
6
44.0
into the new century.
8
69.2
The total number of
12
140.0
subscriptions is shown
in the table for
13
158.7
selected years, with
1990 S 0 and the
number of subscriptions in millions. Use the data
to draw a scatterplot. Does the data seem to follow
an exponential or logistic pattern? Find the
regression equation. According to the model, how
many subscriptions were there in 1997? How many
subscriptions does your model project for 2005?
2010? In what year will the subscriptions exceed
220 million?
Source: 2000/2004 Statistical Abstracts of the United States,
Tables 919/1144.
56. Absorption rates of fabric:
Using time lapse photography,
the spread of a liquid is tracked
in one-fifth of a second intervals,
as a small amount of liquid is
dropped on a piece of fabric. Use
the data to draw a scatterplot,
then use the context and
scatterplot to find the regression
equation. To the nearest
hundredth of a second, how long
did it take the stain to reach a
size of 15 mm?
Time
(sec)
Size
(mm)
0.2
0.39
0.4
1.27
0.6
3.90
0.8
10.60
1.0
21.50
1.2
31.30
1.4
36.30
1.6
38.10
1.8
39.00
57. Planetary orbits: The table
shown gives the time required for the first five
planets to make one complete revolution around the
Sun (in years), along
Planet
Years Radius
with the average
orbital radius of the
Mercury
0.24
0.39
planet in astronomical
Venus
0.62
0.72
units (1 AU ϭ 92.96
Earth
1.00
1.00
million miles). Use a
graphing calculator to
Mars
1.88
1.52
draw the scatterplot,
Jupiter
11.86
5.20
then use the
scatterplot, the context, and any previous experience
to decide whether a polynomial, exponential,
logarithmic, or power regression is most appropriate.
Then (a) find the regression equation and use it to
estimate the average orbital radius of Saturn, given it
orbits the Sun every 29.46 yr, and (b) estimate how
many years it takes Uranus to orbit the Sun, given it
has an average orbital radius of 19.2 AU.
58. Ocean temperatures: The
Depth
Temp
temperature of ocean water
(meters)
(؇C)
depends on several factors,
125
13.0
including salinity, latitude,
depth, and density. However,
250
9.0
between depths of 125 m and
500
6.0
2000 m, ocean temperatures
750
5.0
are relatively predictable, as
1000
4.4
indicated by the data shown
for tropical oceans in the
1250
3.8
table. Use a graphing
1500
3.1
calculator to draw the
1750
2.8
scatterplot, then use the
2000
2.5
scatterplot, the context, and
any previous experience to
decide whether a polynomial,
exponential, logarithmic, or power regression is
cob19545_ch05_553-566.qxd
11/30/10
5:22 PM
Page 564
College Algebra G&M—
564
5–86
CHAPTER 5 Exponential and Logarithmic Functions
most appropriate (end-behavior rules out linear and
quadratic models as possibilities).
Source: UCLA at www.msc.ucla.oceanglobe/pdf/ thermo_plot_lab
a. Find the regression equation and use it to
estimate the water temperature at a depth of
2850 m.
b. If the model were still valid at greater depths,
what is the ocean temperature at the bottom of
the Marianas Trench, some 10,900 m below
sea level?
59. Predater/prey model: In
Predators Rodents
the wild, some rodent
10
5100
populations vary inversely
with the number of
20
2500
predators in the area. Over
30
1600
a period of time, a
40
1200
conservation team does an
50
950
extensive study on this
relationship and gathers
60
775
the data shown. Draw a
70
660
scatterplot of the data and
80
575
(a) find a regression
90
500
equation that models the
data. According to the
100
450
model, (b) if there are
150 predators in the area,
what is the rodent population? (c) How many
predators are in the area if studies show a rodent
population of 3000 animals?
60. Children and AIDS:
Largely due to research,
education, prevention, and
better health care, estimates
of the number of AIDS
(acquired immune
deficiency syndrome) cases
diagnosed in children less
than 13 yr of age have been
declining. Data for the
years 1995 through 2002 is
given in the table.
Source: National Center for Disease
Control and Prevention.
Years
Since 1990
Cases
5
686
6
518
7
328
8
238
9
183
10
118
11
110
12
92
a. Use the data to draw a scatterplot and decide
on an appropriate form of regression.
b. Find a regression equation that models the
data. According to the model, how many cases
of AIDS in children are projected for 2010?
c. In what year did the number of cases fall
below 50?
61. Growth rates of children: After reading a report
from The National Center for Health Statistics
regarding the growth of children from age 0 to 36
months, Maryann decides to track the relationships
(length in inches, weight in pounds) and (age in
months, circumference of head in centimeters) for
her newborn child, a beautiful baby girl—Morgan.
a. Use the (length, weight) data to draw a
scatterplot, then use the context and scatterplot
to find the regression equation. According to
the model, how much will Morgan weigh when
she reaches a height (length) of 39 in.? What
will her length be when she weighs 28 lb?
b. Use the (age, circumference) data to draw a
scatterplot, then use the context and scatterplot
to find the regression equation. According to
the model, what is the circumference of
Morgan’s head when she is 27 months old?
How old will she be when the circumference
of her head is 50 cm?
Exercise 61a
Length
(in.)
17.5
Weight
(lb)
5.50
Exercise 61b
Age
(months)
Circumference
(cm)
1
38.0
21
10.75
6
44.0
25.5
16.25
12
46.5
28.5
19.00
18
48.0
33
25.25
21
48.3
62. Correlation coefficients: Although correlation
coefficients can be very helpful, other factors must
also be considered when selecting the most
appropriate equation model for a set of data. To see
why, use the data given to (a) find a linear
regression equation and note its correlation
coefficient, and (b) find an exponential regression
equation and note its correlation coefficient. What
do you notice? Without knowing the context of the
data, would you be able to tell which model might
be more suitable? (c) Use your calculator to graph
the scatterplot and both functions. Which function
appears to be a better fit?
cob19545_ch05_553-566.qxd
11/30/10
5:23 PM
Page 565
College Algebra G&M—
5–87
ᮣ
565
Making Connections
MAINTAINING YOUR SKILLS
63. (4.4) State the domain of the function, then write it
in lowest terms:
x2 Ϫ 6x ϩ 5
h1x2 ϭ 3
x Ϫ 4x2 Ϫ 7x ϩ 10
64. (2.5) Find a linear function that will make p(x)
continuous.
2
x
p1x2 ϭ • ??
2x Ϫ 4 ϩ 1
65. (2.1) For the graph of f 1x2
given, estimate max/min
values to the nearest tenth
and state intervals where
f 1x2c and f 1x2T.
y
5
4
3
2
1
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
Ϫ1
2
3
4
5
x
2
3
4
5
x
Ϫ3
Ϫ4
Ϫ5
66. (2.2) The graph of f 1x2 ϭ x3
is given. Use it to
sketch the graph
of
2
F 1x2 ϭ 1x Ϫ 22 3 ϩ 3 , and
use the graph to state the
domain and range of F.
Ϫ2 Յ x 6 2
?Յx 6 ?
xՆ4
1
Ϫ2
2
y
5
4
3
2
1
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
Ϫ1
1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
MAKING CONNECTIONS
Making Connections: Graphically, Symbollically, Numerically, and Verbally
Eight graphs (a) through (h) are given. Match the characteristics or equations shown in 1 through 16 to one of the
eight graphs.
(a)
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
(e)
y
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
5
4
3
2
1
y
(b)
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
1
1. ____ y ϭ Ϫ x Ϫ 2
5
2. ____ domain: x ʦ 1Ϫq, 3 4
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
y
(f)
y
(c)
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
y
(g)
y
(d)
y
(h)
1 2 3 4 5 x
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
9. ____ range: y ʦ 1Ϫq, q 2, f 1Ϫ22 ϭ Ϫ1
3. ____ as x S q, y S 0
4
1 ϩ 1.5eϪ2x
11. ____ y ϭ 23 Ϫ x Ϫ 1
4. ____ y ϭ log2 1x ϩ 42 Ϫ 2
12. ____ y ϭ
10. ____ y ϭ
5. ____ y ϭ Ϫ1x ϩ 12 2 ϩ 4
1
1x ϩ 32 1x Ϫ 12 2 1x Ϫ 52
20
13. ____ axis of symmetry x ϭ Ϫ1
6. ____ as x S q, y S 4
14. ____ y ϭ 2Ϫx
7. ____ f 102 ϭ 1, f 1Ϫ22 ϭ 4
8. ____ f 1x2c for x ʦ 1Ϫq, q 2
15. ____ y ϭ 2xϪ2 Ϫ 3
16. ____ f 1x2 Յ 0 for x ʦ 3 Ϫ3, 54
1 2 3 4 5 x
1 2 3 4 5 x
cob19545_ch05_553-566.qxd
9/2/10
9:52 PM
Page 566
College Algebra G&M—
566
5–88
CHAPTER 5 Exponential and Logarithmic Functions
SUMMARY AND CONCEPT REVIEW
SECTION 5.1
One-to-One and Inverse Functions
KEY CONCEPTS
• A function is one-to-one if each element of the range corresponds to a unique element of the domain.
• If every horizontal line intersects the graph of a function in at most one point, the function is one-to-one.
• If f is a one-to-one function with ordered pairs (a, b), then the inverse of f exists and is that one-to-one function
f Ϫ1 with ordered pairs of the form (b, a).
• The range of f becomes the domain of f Ϫ1, and the domain of f becomes the range of f Ϫ1.
• To find f Ϫ1 using the algebraic method:
1. Use y instead of f(x).
2. Interchange x and y.
3. Solve the equation for y.
4. Substitute f Ϫ1 1x2 for y.
• If f is a one-to-one function, the inverse f Ϫ1 exists, where 1 f ؠf Ϫ1 21x2 ϭ x and 1f Ϫ1 ؠf 21x2 ϭ x.
• The graphs of f and f Ϫ1 are symmetric to the identity function y ϭ x.
EXERCISES
Determine whether the functions given are one-to-one by noting the function family to which each belongs and
mentally picturing the shape of the graph.
1. h1x2 ϭ ϪͿx Ϫ 2Ϳ ϩ 3
2. p1x2 ϭ 2x2 ϩ 7
3. s1x2 ϭ 1x Ϫ 1 ϩ 5
Find the inverse of each function given. Then show using composition that your inverse function is correct. State any
necessary restrictions.
4. f 1x2 ϭ Ϫ3x ϩ 2
5. f 1x2 ϭ x2 Ϫ 2, x Ն 0
6. f 1x2 ϭ 1x Ϫ 1
Determine the domain and range for each function whose graph is given, and use this information to state the domain
and range of the inverse function. Then use the line y ϭ x to estimate the location of three points on the graph, and use
these to graph f Ϫ1 1x2 on the same grid.
y
y
y
7.
8.
9.
f(x)
5
4
3
2
1
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
5
4
3
2
f(x) 1
1 2 3 4 5 x
Ϫ2
f(x)
Ϫ3
Ϫ4
Ϫ5
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
10. Fines for overdue material: Some libraries have set fees and penalties to discourage patrons from holding
borrowed materials for an extended period. Suppose the fine for overdue DVDs is given by the function
f 1t2 ϭ 0.15t ϩ 2, where f (t) is the amount of the fine t days after it is due. (a) What is the fine for keeping a DVD
seven (7) extra days? (b) Find f Ϫ1 1t2, then input your answer from part (a) and comment on the result. (c) If a fine
of $3.80 was assessed, how many days was the DVD overdue?
SECTION 5.2
Exponential Functions
KEY CONCEPTS
• An exponential function is defined as f 1x2 ϭ bx, where b 7 0, b 1, and b, x are real numbers.
• The natural exponential function is f 1x2 ϭ ex, where e Ϸ 2.71828182846.
• For exponential functions, we have
• one-to-one function
• y-intercept (0, 1)
• domain: x ʦ ޒ
range:
increasing
if
y
ʦ
10,
q
2
b
7
1
•
•
• decreasing if 0 6 b 6 1
• asymptotic to x-axis