2
CHAPTER 0
.
.
Preliminaries
02
1
2
3
4
5
6
7
8
x
y
5
10
15
20
25
30
35
0
FIGURE 0.1
The Fibonacci sequence
these is the attempt to find patterns to help us better describe the world. The other theme is
the interplay between graphs and functions. By connecting the powerful equationsolving
techniques of algebra with the visual images provided by graphs, you will significantly
improve your ability to make use of your mathematical skills in solving realworld problems.
0.1
POLYNOMIALS AND RATIONAL FUNCTIONS
The Real Number System and Inequalities
Although mathematics is far more than just a study of numbers, our journey into calculus
begins with the real number system. While this may seem to be a fairly mundane starting
place, we want to give you the opportunity to brush up on those properties that are of
particular interest for calculus.
The most familiar set of numbers is the set of
integers,
consisting of the whole numbers
and their additive inverses: 0,
±
1
,
±
2
,
±
3
,
....
A
rational number
is any number of the
form
p
q
, where
p
and
q
are integers and
q
=
0
.
For example,
2
3
,
−
7
3
and
27
125
are all rational
numbers. Notice that every integer
n
is also a rational number, since we can write it as the
quotient of two integers:
n
=
n
1
.
The
irrational numbers
are all those real numbers that cannot be written in the form
p
q
,
where
p
and
q
are integers. Recall that rational numbers have decimal expansions that either
terminate or repeat. For instance,
1
2
=
0
.
5
,
1
3
=
0
.
3333
¯
3
,
1
8
=
0
.
125 and
1
6
=
0
.
16666
¯
6 are
all rational numbers. By contrast, irrational numbers have decimal expansions that do
not repeat or terminate. For instance, three familiar irrational numbers and their decimal
expansions are
√
2
=
1
.
41421 35623
...,
π
=
3
.
14159 26535
...
and
e
=
2
.
71828 18284
... .
We picture the real numbers arranged along the number line displayed in Figure 0.2
(the
real line
). The set of real numbers is denoted by the symbol
R
.
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03
SECTION 0.1
.
.
Polynomials and Rational Functions
3
0
5
4
3
2
1
1
2
3
4
5
2
3
p
e
FIGURE 0.2
The real line
For real numbers
a
and
b
,
where
a
<
b
,
we define the
closed interval
[
a
,
b
] to be the
set of numbers between
a
and
b
,
including
a
and
b
(the
endpoints
), that is,
[
a
,
b
]
= {
x
∈
R

a
≤
x
≤
b
}
,
as illustrated in Figure 0.3, where the solid circles indicate that
a
and
b
are included in
[
a
,
b
].
a
b
FIGURE 0.3
A closed interval
a
b
FIGURE 0.4
An open interval
Similarly, the
open interval
(
a
,
b
) is the set of numbers between
a
and
b
,
but
not
including the endpoints
a
and
b
,
that is,
(
a
,
b
)
= {
x
∈
R

a
<
x
<
b
}
,
as illustrated in Figure 0.4, where the open circles indicate that
a
and
b
are not included in
(
a
,
b
).
You should already be very familiar with the following properties of real numbers.
THEOREM 1.1
If
a
and
b
are real numbers and
a
<
b
, then
(i) For any real number
c
,
a
+
c
<
b
+
c
.
(ii) For real numbers
c
and
d
, if
c
<
d
, then
a
+
c
<
b
+
d
.
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 Spring '09
 Dr.RichardR.Reynolds
 Calculus, candy bar

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