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MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
Page 1 of 8
CHAPTER 1: PRELIMINARIES
1.1
Real Numbers and the Real Line
Real Numbers
Much of calculus is based on properties of the real number system.
Real numbers
are
numbers that can be expressed as decimals.
Rules for Inequalities
If a, b and c are real numbers, then:
1.
c
b
c
a
b
a
+
<
+
⇒
<
2.
c
b
c
a
b
a

<

⇒
<
3.
b
a
<
and
bc
ac
0
c
<
⇒
4.
b
a
<
and
ac
bc
0
c
<
⇒
<
Special case:
a
b
b
a

<

⇒
<
5.
0
a
1
0
a
⇒
6.
If a and b are both positive or both negative, then
a
1
b
1
b
a
<
⇒
<
Intervals
A subset of the real line is called an interval if it contains at least two numbers and
contains all the real numbers lying between any two of its elements.
Types of intervals
Notation
Set description
Type
Finite:
)
b
,
a
(
{
}
b
x
a
x
<
<
Open
[
]
b
,
a
{
}
b
x
a
x
≤
≤
Closed
[
)
b
,
a
{
}
b
x
a
x
<
≤
Halfopen
]
b
,
a
(
{
}
b
x
a
x
≤
<
Halfopen
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Page 2 of 8
Infinite:
)
,
a
(
∞
{
}
a
x
x
Open
[
)
,
a
∞
{
}
a
x
x
≥
Closed
)
b
,
(
∞
{
}
b
x
x
<
Open
]
b
,
(
∞
{
}
b
x
x
≤
Closed
)
,
(
∞
∞
ℜ
(set of all real numbers)
Example: Attend lecture.
Absolute Value
The
absolute value
of a number x, denoted by
x
, is defined by the formula
<

≥
=
0
x
,
x
0
x
,
x
x
Absolute Values and Intervals
If a is any positive number, then
1.
a
x
=
if and only if
a
x
±
=
2.
a
x
<
if and only if
a
x
a
<
<

3.
a
x
if and only if
a
x
or
a
x

<
4.
a
x
≤
if and only if
a
x
a
≤
≤

5.
a
x
≥
if and only if
a
x
≥
or
a
x

≤
Example: Attend lecture.
MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
Page 3 of 8
PROBLEM SET 1.1
Inequalities
Solve the following inequalities.
1.
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This note was uploaded on 09/20/2009 for the course MATH 3782378 taught by Professor Qizhang during the Fall '09 term at Pennsylvania State University, University Park.
 Fall '09
 QiZhang
 Calculus, Geometry, Real Numbers, Decimals

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