Applied Calculus - 7e - c 05.pdf
(
1495 KB
)
Pobierz
05-W3979 11/2/06 11:35 AM Page 331
5
Exponential and
Logarithmic Functions
H
ow many bacteria will there
be in a culture at the end of a
certain period of time? How fast
will the bacteria population be
growing at the end of that time?
Example 1, page 380, answers
these questions.
T
HE EXPONENTIAL FUNCTION is, without doubt, the most important
function in mathematics and its applications. After a brief introduction
to the exponential function and its
inverse
, the logarithmic function, we
learn how to differentiate such functions. This lays the foundation for
exploring the many applications involving exponential functions. For exam-
ple, we look at the role played by exponential functions in computing
earned interest on a bank account and in studying the growth of a bacte-
ria population in the laboratory, the way radioactive matter decays, the
rate at which a factory worker learns a certain process, and the rate at
which a communicable disease is spread over time.
331
05-W3979 11/2/06 11:35 AM Page 332
332
5
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
5.1
Exponential Functions
Exponential Functions and Their Graphs
Suppose you deposit a sum of $1000 in an account earning interest at the rate of 10%
per year
compounded continuously
(the way most financial institutions compute
interest). Then, the accumulated amount at the end of
t
years (0
y
20) is
described by the function
f
, whose graph appears in Figure 1.* This function is called
an
exponential function
. Observe that the graph of
f
rises rather slowly at first but
very rapidly as time goes by. For purposes of comparison, we have also shown the
graph of the function
y
t
7000
6000
5000
4000
3000
2000
1000
y
=
f
(
t
)
0.10
t
), giving the accumulated amount
for the same principal ($1000) but earning
simple
interest at the rate of 10% per year.
The moral of the story: It is never too early to save.
Exponential functions play an important role in many real-world applications,
as you will see throughout this chapter.
Observe that whenever
b
is a positive number and
n
is any real number, the
expression
b
n
is a real number. This enables us to define an exponential function as
follows:
g
(
t
)
1000(1
y
=
g
(
t
)
t
5
10
15
20
Years
FIGURE
1
Under continuous compounding, a sum
of money grows exponentially.
Exponential Function
The function defined by
b
x
f
(
x
)
(
b
0,
b
1)
is called an
exponential function with base
b
and exponent
x
. The domain
of
f
is the set of all real numbers.
For example, the exponential function with base 2 is the function
f
(
x
)
2
x
with domain (
,
). The values of
f
(
x
) for selected values of
x
follow:
a
3
2
2
#
2
1/2
2
3
2
3/2
2
0
f
1
3
2
8
f
b
2
1
2
f
1
0
2
1
1
2
2
3
1
2
2/3
1
2
f
1
1
2
2
1
f
a
b
2
2/3
3
4
Computations involving exponentials are facilitated by the laws of exponents.
These laws were stated in Section 1.1, and you might want to review the material
there. For convenience, however, we will restate these laws.
Laws of Exponents
Let
a
and
b
be positive numbers and let
x
and
y
be real numbers. Then,
1.
b
x
b
y
b
x
y
4.
(
ab
)
x
a
x
b
x
b
x
b
y
a
a
b
b
x
a
x
b
x
2.
b
x
y
5.
3.
(
b
x
)
y
b
xy
*We will derive the rule for
f
in Section 5.3.
05-W3979 11/2/06 11:35 AM Page 333
5.1
333
EXPONENTIAL FUNCTIONS
The use of the laws of exponents is illustrated in the next example.
EXAMPLE 1
a.
16
7/4
16
1/2
16
7/4
1/2
16
5/4
2
5
32
Law 1
8
5/3
8
1/3
8
5/3
1
1/3
2
8
2
b.
64
Law 2
64
4/3
1/2
1
4/3
21
1/2
2
64
2/3
c.
1
2
64
1
64
2/3
1
1
4
2
1
16
Law 3
64
1/3
2
1
2
16
#
81
#
81
1/4
16
1/4
#
1
1
1
2
#
1
1
6
1/4
16
1/4
d.
1
2
3
Law 4
81
1/4
a
3
1/2
2
1/3
b
4
3
4/2
2
4/3
9
2
4/3
e.
Law 5
EXAMPLE 2
2
2
x
1
. Find the value of
x
for which
f
(
x
)
Let
f
(
x
)
16.
Solution
We want to solve the equation
2
2
x
1
2
4
16
But this equation holds if and only if
2
x
1
4
b
m
b
n
⇒
m
n
5
2
giving
x
.
Exponential functions play an important role in mathematical analysis. Because
of their special characteristics, they are some of the most useful functions and are
found in virtually every field where mathematics is applied. To mention a few exam-
ples: Under ideal conditions the number of bacteria present at any time
t
in a culture
may be described by an exponential function of
t
; radioactive substances decay over
time in accordance with an “exponential” law of decay; money left on fixed deposit
and earning compound interest grows exponentially; and some of the most impor-
tant distribution functions encountered in statistics are exponential.
Let’s begin our investigation into the properties of exponential functions by
studying their graphs.
EXAMPLE 3
2
x
.
Sketch the graph of the exponential function
y
Solution
First, as discussed earlier, the domain of the exponential function
2
x
is the set of real numbers. Next, putting
x
2
0
y
y
1,
the
y
-intercept of
f
. There is no
x
-intercept since there is no value of
x
for which
y
f
(
x
)
0 gives
y
0. To find the range of
f
, consider the following table of values:
4
5
4
3
2
1
0
1
2
3
4
5
x
1
2
1
6
1
8
1
4
1
2
2
1248 6 2
3
1
y
We see from these computations that 2
x
decreases and approaches zero as
x
decreases without bound and that 2
x
increases without bound as
x
increases with-
out bound. Thus, the range of
f
is the interval (0,
x
–2
2
)—that is, the set of positive
FIGURE
2
The graph of
y
2
x
in Figure 2.
real numbers. Finally, we sketch the graph of
y
f
(
x
)
2
x
05-W3979 11/2/06 11:35 AM Page 334
334
5
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
EXAMPLE 4
(1/2)
x
.
Sketch the graph of the exponential function
y
y
(1/2)
x
is the set of all real
Solution
The domain of the exponential function
y
numbers. The
y
-intercept is (1/2)
0
1; there is no
x
-intercept since there is no
4
value of
x
for which
y
0. From the following table of values
5
4
3
2
10
1
2
3
4
5
x
2
1
2
1
4
1
8
1
6
1
2
32
16
8
4
2
1
1
3
y
x
1/2
x
increases without bound as
x
decreases without
bound and that (1/2)
x
decreases and approaches zero as
x
increases without
bound. Thus, the range of
f
is the interval (0,
we deduce that (1/2)
x
–2
2
FIGURE
3
The graph of
y
(1/2)
x
x
). The graph of
y
f
(
x
)
1
2
¢
≤
is sketched in Figure 3.
(1/2)
x
, whose graphs you studied in Examples 3
and 4, are special cases of the exponential function
y
2
x
and
y
The functions
y
y
b
x
, obtained by set-
f
(
x
)
b
x
with
ting
b
2 and
b
1/2, respectively. In general, the exponential function
y
2
x
, whereas the graph of
y
b
x
for 0
b
1 has a graph similar to
y
b
1 is
(1/2)
x
similar to that of
y
(Exercises 27 and 28 on page 336). When
b
1, the
b
x
reduces to the constant function
y
function
y
1. For comparison, the graphs
y
=
b
x
(0 <
b
< 1)
y
=
b
x
(
b
> 1)
of all three functions are sketched in Figure 4.
y
= 1
Properties of the Exponential Function
The exponential function
y
x
b
x
(
b
0,
b
1) has the following properties:
1.
Its domain is (
,
).
FIGURE
4
y
b
x
is an increasing function of
x
2.
Its range is (0,
).
3.
Its graph passes through the point (0, 1).
4.
It is continuous on (
if
b
1, a constant function if
b
1, and a decreasing function if
,
).
0
b
1.
5.
It is increasing on (
,
) if
b
1 and decreasing on (
,
) if
b
1.
The Base
e
Exponential functions to the base
e
, where
e
is an irrational number whose value is
2.7182818. . . , play an important role in both theoretical and applied problems. It
can be shown, although we will not do so here, that
TABLE
1
b
m
1
m
m
a
1
10
2.59374
b
m
1
m
e
lim
m
S
a
1
(1)
100
2.70481
1000
2.71692
10,000
2.71815
However, you may convince yourself of the plausibility of this definition of the
number
e
by examining Table 1, which may be constructed with the help of a cal-
culator.
100,000
2.71827
1,000,000
2.71828
EXPLORING WITH TECHNOLOGY
To obtain a visual confirmation of the fact that the expression (1
1/
m
)
m
approaches
the number
e
2.71828. . . as
m
increases without bound, plot the graph of
f
(
x
)
(1
1/
x
)
x
in a suitable viewing window and observe that
f
(
x
) approaches
2.71828. . . as
x
increases without bound. Use
ZOOM
and
TRACE
to find the value of
f
(
x
) for large values of
x.
05-W3979 11/2/06 11:35 AM Page 335
5.1
335
EXPONENTIAL FUNCTIONS
EXAMPLE 5
Sketch the graph of the function
y
e
x
.
y
5
Solution
Since
e
1, it follows from our previous discussion that the graph of
y
e
x
is similar to the graph of
y
2
x
(see Figure 2). With the aid of a calcula-
tor, we obtain the following table:
3
3
2
1
0
1
2
3
x
1
0.05
0.14
0.37
1
2.72
7.39
20.09
y
x
–3
–1
1
3
The graph of
y
e
x
is sketched in Figure 5.
FIGURE
5
The graph of
y
e
x
Next, we consider another exponential function to the base
e
that is closely
related to the previous function and is particularly useful in constructing models that
describe “exponential decay.”
y
EXAMPLE 6
Sketch the graph of the function
y
e
x
.
5
Solution
(1/
e
)
x
is an exponential function with base less than 1. Therefore, it has a graph similar
to that of the exponential function
y
Since
e
1, it follows that 0
1/
e
1, so
f
(
x
)
e
x
1/
e
x
(1/2)
x
. As before, we construct the fol-
3
lowing table of values of
y
e
x
for selected values of
x
:
1
3
2
1
0
1
2
3
x
x
20.09
7.39
2.72
1
0.37
0.14
0.05
–3
–1
1
3
y
FIGURE
6
The graph of
y
e
x
Using this table, we sketch the graph of
y
e
x
in Figure 6.
5.1
Self-Check Exercises
Solutions to Self-Check Exercises 5.1 can be found on
page 337.
1.
Solve the equation 2
2
x
1
2
3
2
x
1
.
2.
Sketch the graph of
y
e
0.4
x
.
5.1
Concept Questions
2.
For the exponential function
y
b
x
(
b
0,
b
1), state
(a) its domain and range, (b) its
y
-intercept, (c) where it is
continuous, and (d) where it is increasing and where it is
decreasing for the case
b
1 and the case
b
1.
1.
Define the exponential function
f
with base
b
and exponent
x
.
What restrictions, if any, are placed on
b
?
5.1
Exercises
In Exercises 1–8, evaluate the expression.
1. a.
4
3
3. a.
9(9)
1/2
b.
5(5)
1/2
4
5
b.
3
3
3
6
1
2
b
3
d
2
1
3
b
2
d
3
4. a.
ca
b.
ca
2. a.
(2
1
)
3
b.
(3
2
)
3
Plik z chomika:
Januszek66
Inne pliki z tego folderu:
Back Seat_ A Mumbai Tale - Aditya Kripalani.mobi
(755 KB)
Brief Wondrous Life of Oscar Wao, The - Junot Diaz.opf
(3 KB)
Don't Make Me Think, Revisited_ - Steve Krug.mobi
(9256 KB)
M. T. Anderson - Norumbegan 03 - The Empire of Gut and Bone # (v5.0).epub
(2209 KB)
M. T. Anderson - Norumbegan 02 - The Suburb Beyond the Stars # (v5.0).epub
(2105 KB)
Inne foldery tego chomika:
Dokumenty
Galeria
LUDLUM ROBERT
Midi - Kar
Mszał Rzymski PL
Zgłoś jeśli
naruszono regulamin