Calculo Parte

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Chapter

1

PRELIMINARIES
OVERVIEW This chapter reviews the basic ideas you need to start calculus. The topics include the real number system, Cartesian coordinates in the plane, straight lines, parabolas,
circles, functions, and trigonometry. We also discuss the use of graphing calculators and
computer graphing software.

1.1Real Numbers and the Real Line
This section reviews real numbers, inequalities, intervals, and absolute values.

Real Numbers
Much of calculus is based on properties of the real number system. Real numbers are
numbers that can be expressed as decimals, such as
3
= - 0.75000 Á
4
1
= 0.33333 Á
3

22 = 1.4142 Á
-

The dots Á in each case indicate that the sequence of decimal digitsgoes on forever.
Every conceivable decimal expansion represents a real number, although some numbers
have two representations. For instance, the infinite decimals .999 Á and 1.000 Á represent the same real number 1. A similar statement holds for any number with an infinite tail
of 9’s.
The real numbers can be represented geometrically as points on a number line called
the real line.
–2

–1 –3
4

0

1
3

1

2

2

3

4

The symbol denotes either the real number system or, equivalently, the real line.
The properties of the real number system fall into three categories: algebraic properties, order properties, and completeness. The algebraic properties say that the real numbers can be added, subtracted, multiplied, and divided (except by 0) to produce more real
numbersunder the usual rules of arithmetic. You can never divide by 0.

1
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley

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Chapter 1: Preliminaries

The order properties of real numbers are given in Appendix 4. The following useful
rules can be derived from them, where the symbol Q means “implies.”Rules for Inequalities
If a, b, and c are real numbers, then:

5.

a6bQa+c6b+c
a6bQa-c6b-c
a 6 b and c 7 0 Q ac 6 bc
a 6 b and c 6 0 Q bc 6 ac
Special case: a 6 b Q - b 6 - a
1
a70Qa70

6.

If a and b are both positive or both negative, then a 6 b Q

1.
2.
3.
4.

1
1
6a
b

Notice the rules for multiplying an inequality by a number. Multiplying by a positive numberpreserves the inequality; multiplying by a negative number reverses the inequality.
Also, reciprocation reverses the inequality for numbers of the same sign. For example,
2 6 5 but - 2 7 - 5 and 1> 2 7 1> 5.
The completeness property of the real number system is deeper and harder to define
precisely. However, the property is essential to the idea of a limit (Chapter 2). Roughly
speaking, it says thatthere are enough real numbers to “complete” the real number line, in
the sense that there are no “holes” or “gaps” in it. Many theorems of calculus would fail if
the real number system were not complete. The topic is best saved for a more advanced
course, but Appendix 4 hints about what is involved and how the real numbers are constructed.
We distinguish three special subsets of real numbers.1.
2.
3.

The natural numbers, namely 1, 2, 3, 4, Á
The integers, namely 0, ; 1, ; 2, ; 3, Á
The rational numbers, namely the numbers that can be expressed in the form of a
fraction m> n, where m and n are integers and n Z 0. Examples are
1
,
3

-

4
-4
4
=
=
,
9
9
-9

200
,
13

and

57 =

57
.
1

The rational numbers are precisely the real numbers withdecimal expansions that are
either
(a) terminating (ending in an infinite string of zeros), for example,
3
= 0.75000 Á = 0.75
4

or

(b) eventually repeating (ending with a block of digits that repeats over and over), for
example
23
= 2.090909 Á = 2.09
11

The bar indicates the
block of repeating
digits.

Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley...