Materi Kalkulus

PRELIMINARIES
Chapter
1
Real 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
The dots in each case indicate that the sequence of decimal digits goes on forever.
Every conceivable decimal expansion represents a real number, although some numbers
have two representations. For instance, the infinite decimals and 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.
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
numbers under the usual rules of arithmetic. You can never divide by 0.


–2 –1 0 1 2 3
4 34
13
–
2
.999 Á 1.000 Á
Á
22 = 1.4142 Á
1
3 = 0.33333 Á
- 3
4 = -0.75000 Á
1.1
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.”
2 Chapter 1: Preliminaries
Rules for Inequalities
If a, b, and c are real numbers, then:
1.
2.
3.
4.
Special case:
5.
6. If a and b are both positive or both negative, then a 6 b Q
1
b 6 1a
a 7 0 Q
1a
7 0
a 6 b Q -b 6 -a
a 6 b and c 6 0 Q bc 6 ac
a 6 b and c 7 0 Q ac 6 bc
a 6 b Q a - c 6 b - c
a 6 b Q a + c 6 b + c
Notice the rules for multiplying an inequality by a number. Multiplying by a positive number
preserves the inequality; multiplying by a negative number reverses the inequality.
Also, reciprocation reverses the inequality for numbers of the same sign. For example,
but and
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 that there 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. The natural numbers, namely 1, 2, 3, 4
2. The integers, namely
3. The rational numbers, namely the numbers that can be expressed in the form of a
fraction , where m and n are integers and Examples are
The rational numbers are precisely the real numbers with decimal expansions that are
either
(a) terminating (ending in an infinite string of zeros), for example,
(b) eventually repeating (ending with a block of digits that repeats over and over), for
example
The bar indicates the
block of repeating
digits.
23
11 = 2.090909Á = 2.09
3
4 = 0.75000Á = 0.75 or
1
3
, - 4
9 = -4
9 = 4
-9
, 200
13
, and 57 = 57
1
.
m>n n Z 0.
0, ;1, ;2, ;3, Á
, Á
2 6 5 -2 7 -5 1>2 7 1>5.
A terminating decimal expansion is a special type of repeating decimal since the ending
zeros repeat.
The set of rational numbers has all the algebraic and order properties of the real numbers
but lacks the completeness property. For example, there is no rational number whose
square is 2; there is a “hole” in the rational line where should be.
Real numbers that are not rational are called irrational numbers. They are characterized
by having nonterminating and nonrepeating decimal expansions. Examples are
and Since every decimal expansion represents a real number, it
should be clear that there are infinitely many irrational numbers. Both rational and irrational
numbers are found arbitrarily close to any point on the real line.
Set notation is very useful for specifying a particular subset of real numbers. A set is a
collection of objects, and these objects are the elements of the set. If S is a set, the notation
means that a is an element of S, and means that a is not an element of S. If S
and T are sets, then is their union and consists of all elements belonging either to S
or T (or to both S and T). The intersection consists of all elements belonging to both
S and T. The empty set is the set that contains no elements. For example, the intersection
of the rational numbers and the irrational numbers is the empty set.
Some sets can be described by listing their elements in braces. For instance, the set A
consisting of the natural numbers (or positive integers) less than 6 can be expressed as
The entire set of integers is written as
Another way to describe a set is to enclose in braces a rule that generates all the elements
of the set. For instance, the set
is the set of positive integers less than 6.
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. For example, the set of all
real numbers x such that is an interval, as is the set of all x such that
The set of all nonzero real numbers is not an interval; since 0 is absent, the set fails to contain
every real number between and 1 (for example).
Geometrically, intervals correspond to rays and line segments on the real line, along
with the real line itself. Intervals of numbers corresponding to line segments are finite intervals;
intervals corresponding to rays and the real line are infinite intervals.
A finite interval is said to be closed if it contains both of its endpoints, half-open if it
contains one endpoint but not the other, and open if it contains neither endpoint. The endpoints
are also called boundary points; they make up the interval’s boundary. The remaining
points of the interval are interior points and together comprise the interval’s interior.
Infinite intervals are closed if they contain a finite endpoint, and open otherwise.
The entire real line is an infinite interval that is both open and closed.
Solving Inequalities
The process of finding the interval or intervals of numbers that satisfy an inequality in x is
called solving the inequality.


-1
x 7 6 -2 … x … 5.
A = 5x ƒ x is an integer and 0 6 x 6 66
50, ;1, ;2, ;3, Á 6.
A = 51, 2, 3, 4, 56.
¤
S ¨ T
S ´ T
a H S a x S
p, 22, 23 5, log10 3.
22
1.1 Real Numbers and the Real Line 3
EXAMPLE 1 Solve the following inequalities and show their solution sets on the real
line.
(a) (b) (c)
Solution
(a)
Add 1 to both sides.
Subtract x from both sides.
The solution set is the open interval (Figure 1.1a).
(b)
Multiply both sides by 3.
Add x to both sides.
Subtract 3 from both sides.
- Divide by 7. 3
7 6 x
-3 6 7x
0 6 7x + 3
-x 6 6x + 3
- x
3 6 2x + 1
s -q, 4d
x 6 4
2x 6 x + 4
2x - 1 6 x + 3
6
- x - 1 Ú 5 x
3 6 2x + 1 2x - 1 6 x + 3
4 Chapter 1: Preliminaries
TABLE 1.1 Types of intervals
Notation Set description Type Picture
Finite: (a, b) Open
[a, b] Closed
[a, b) Half-open
(a, b] Half-open
Infinite: Open
Closed
Open
Closed
(set of all real
numbers) Both open
and closed
s -q, qd

s -q, b] 5x ƒ x … b6
s -q, bd 5x ƒ x 6 b6
[a, qd 5x ƒ x Ú a6
sa, qd 5x ƒ x 7 a6
5x ƒ a 6 x … b6
5x ƒ a … x 6 b6
5x ƒ a … x … b6
5x ƒ a 6 x 6 b6
a b
a b
a b
a
a
b
b
b
a
0
0
0 1
1
1 4
(a)
–37
(b)
11
5
(c)
x
x
x
FIGURE 1.1 Solution sets for the
inequalities in Example 1.
The solution set is the open interval (Figure 1.1b).
(c) The inequality can hold only if because otherwise
is undefined or negative. Therefore, is positive and the inequality will be preserved
if we multiply both sides by and we have
Multiply both sides by
Add 5 to both sides.
The solution set is the half-open interval (1, ] (Figure 1.1c).
Absolute Value
The absolute value of a number x, denoted by is defined by the formula
EXAMPLE 2 Finding Absolute Values
Geometrically, the absolute value of x is the distance from x to 0 on the real number
line. Since distances are always positive or 0, we see that for every real number x,
and if and only if Also,
on the real line (Figure 1.2).
Since the symbol always denotes the nonnegative square root of a, an alternate
definition of is
It is important to remember that Do not write unless you already
know that
The absolute value has the following properties. (You are asked to prove these properties
in the exercises.)
a Ú 0.
2a2 = ƒ a ƒ. 2a2 = a
ƒ x ƒ = 2x2 .
ƒ x ƒ
2a
ƒ x - y ƒ = the distance between x and y
ƒ x ƒ = 0 x = 0.
ƒ x ƒ Ú 0
ƒ 3 ƒ = 3, ƒ 0 ƒ = 0, ƒ -5 ƒ = -s -5d = 5, ƒ - ƒ a ƒ ƒ = ƒ a ƒ
ƒ x ƒ = e
x, x Ú 0
-x, x 6 0.
ƒ x ƒ,
11>5
Or x … 11
5
.
11
5 Ú x.
11 Ú 5x
6 Ú 5x - 5 sx - 1d .
6
x - 1 Ú 5
sx - 1d,
sx - 1d
6>sx - 1d Ú 5 x 7 1, 6>sx - 1d
s -3>7, qd
1.1 Real Numbers and the Real Line 5

–5
 5
3


4  1

1  4
 3
–5 0 3
1 4
FIGURE 1.2 Absolute values give
distances between points on the number
line.
Absolute Value Properties
1. A number and its additive inverse or negative have
the same absolute value.
2. The absolute value of a product is the product of
the absolute values.
3.
4. The triangle inequality. The absolute value of the
sum of two numbers is less than or equal to the
sum of their absolute values.
ƒ a + b ƒ … ƒ a ƒ + ƒ b ƒ
The absolute value of a quotient is the quotient
of the absolute values.
` a
b
` =
ƒ a ƒ
ƒ b ƒ
ƒ ab ƒ = ƒ a ƒ ƒ b ƒ
ƒ -a ƒ = ƒ a ƒ
Note that For example, whereas If a and b
differ in sign, then is less than In all other cases, equals
Absolute value bars in expressions like work like parentheses: We do
the arithmetic inside before taking the absolute value.
EXAMPLE 3 Illustrating the Triangle Inequality
The inequality says that the distance from x to 0 is less than the positive number
a. This means that x must lie between and a, as we can see from Figure 1.3.
The following statements are all consequences of the definition of absolute value and
are often helpful when solving equations or inequalities involving absolute values.
-a
ƒ x ƒ 6 a
ƒ -3 - 5 ƒ = ƒ -8 ƒ = 8 = ƒ -3 ƒ + ƒ -5 ƒ
ƒ 3 + 5 ƒ = ƒ 8 ƒ = ƒ 3 ƒ + ƒ 5 ƒ
ƒ -3 + 5 ƒ = ƒ 2 ƒ = 2 6 ƒ -3 ƒ + ƒ 5 ƒ = 8
ƒ a ƒ + ƒ b ƒ. ƒ -3 + 5 ƒ
ƒ a + b ƒ ƒ a ƒ + ƒ b ƒ. ƒ a + b ƒ
ƒ -a ƒ Z -ƒ a ƒ. ƒ -3 ƒ = 3, - ƒ 3 ƒ = -3.
6 Chapter 1: Preliminaries
–a x 0 a
a a

x

FIGURE 1.3 means x lies
between -a and a.
ƒ x ƒ 6 a
Absolute Values and Intervals
If a is any positive number, then
5.
6.
7.
8.
9. ƒ x ƒ Ú a if and only if x Ú a or x … -a
ƒ x ƒ … a if and only if -a … x … a
ƒ x ƒ 7 a if and only if x 7 a or x 6 -a
ƒ x ƒ 6 a if and only if -a 6 x 6 a
ƒ x ƒ = a if and only if x = ;a
The symbol is often used by mathematicians to denote the “if and only if ” logical
relationship. It also means “implies and is implied by.”
EXAMPLE 4 Solving an Equation with Absolute Values
Solve the equation
Solution By Property 5, so there are two possibilities:
Solve as usual.
The solutions of are and
EXAMPLE 5 Solving an Inequality Involving Absolute Values
Solve the inequality ` 5 - 2x
` 6 1.
ƒ 2x - 3 ƒ = 7 x = 5 x = -2.
x = 5 x = -2
2x = 10 2x = -4
Equivalent equations
2x - 3 = 7 without absolute values 2x - 3 = -7
2x - 3 = ;7,
ƒ 2x - 3 ƒ = 7.
3
Solution We have
Property 6
Subtract 5.
Take reciprocals.
Notice how the various rules for inequalities were used here. Multiplying by a negative
number reverses the inequality. So does taking reciprocals in an inequality in which both
sides are positive. The original inequality holds if and only if
The solution set is the open interval ( , ).
EXAMPLE 6 Solve the inequality and show the solution set on the real line:
(a) (b)
Solution
(a)
Property 8
Add 3.
Divide by 2.
The solution set is the closed interval [1, 2] (Figure 1.4a).
(b)
Property 9
Divide by 2.
Add
The solution set is s -q, 1] ´ [2, qd (Figure 1.4b).
3
2
x Ú 2 . or x … 1
x - 3
2 Ú 1
2 or x - 3
2 … - 1
2
2x - 3 Ú 1 or 2x - 3 … -1
ƒ 2x - 3 ƒ Ú 1
1 … x … 2
2 … 2x … 4
-1 … 2x - 3 … 1
ƒ 2x - 3 ƒ … 1
ƒ 2x - 3 ƒ … 1 ƒ 2x - 3 ƒ Ú 1
1>3 1>2
s1>3d 6 x 6 s1>2d.
3 1
3 6 x 6 1
2
.
Multiply by - 1
2
33 7 . 1x
7 2
3 -6 6 - 2x
6 -4
` 5 - 2x
` 6 13 -1 6 5 - 2x
6 1
1.1 Real Numbers and the Real Line 7
1 2
1 2
(a)
(b)
x
x
FIGURE 1.4 The solution sets (a) [1, 2]
and (b) s -q, 1] ´ [2, qd in Example 6.
1.1 Real Numbers and the Real Line 7
EXERCISES 1.1
Decimal Representations
1. Express as a repeating decimal, using a bar to indicate the repeating
digits. What are the decimal representations of ? ?
? ?
2. Express as a repeating decimal, using a bar to indicate the
repeating digits. What are the decimal representations of ?
3>11? 9>11? 11>11?
2>11
1>11
8>9 9>9
2>9 3>9
1>9
Inequalities
3. If which of the following statements about x are necessarily
true, and which are not necessarily true?
a. b.
c. d.
e. f.
g. -6 6 -x 6 2 h. -6 6 -x 6 -2
1 6 ƒ x - 4 ƒ 6 2 6x
6 3
1
6 6 1x
6 1
2
1 6 x
2 6 3
0 6 x 6 4 0 6 x - 2 6 4
2 6 x 6 6,
4. If which of the following statements about y
are necessarily true, and which are not necessarily true?
a. b.
c. d.
e. f.
g. h.
In Exercises 5–12, solve the inequalities and show the solution sets on
the real line.
5. 6.
7. 8.
9. 10.
11. 12.
Absolute Value
Solve the equations in Exercises 13–18.
13. 14. 15.
16. 17. 18.
Solve the inequalities in Exercises 19–34, expressing the solution sets
as intervals or unions of intervals. Also, show each solution set on the
real line.
19. 20. 21.
22. 23. 24.
25. 26. 27.
28. 29. 30.
31. 32. 33.
34. ` 3r
5 - 1 ` 7 2
5
` r + 1
2
ƒ 1 - x ƒ 7 1 ƒ 2 - 3x ƒ 7 5 ` Ú 1
ƒ s + 3 ƒ Ú 1
ƒ 2 ` 2s ƒ Ú 4 2x
- 4 ` 6 3
` 3 - 1x
` 6 1
2
` 3
2 ` z - 1 ` … 2 z
5 - 1 ` … 1
ƒ t + 2 ƒ 6 1 ƒ 3y - 7 ƒ 6 4 ƒ 2y + 5 ƒ 6 1
ƒ x ƒ 6 2 ƒ x ƒ … 2 ƒ t - 1 ƒ … 3
` s
ƒ 2 - 1 ` = 1 8 - 3s ƒ = 9
ƒ 2 1 - t ƒ = 1
ƒ y ƒ = 3 ƒ y - 3 ƒ = 7 ƒ 2t + 5 ƒ = 4
- x + 5
2 … 12 + 3x
4
4
5 sx - 2d 6 1
3 sx - 6d
6 - x
4 6 3x - 4
2
2x - 1
2 Ú 7x + 7
6
5x - 3 … 7 - 3x 3s2 - xd 7 2s3 + xd
-2x 7 4 8 - 3x Ú 5
ƒ y - 5 ƒ 6 1
1
6 6 1y
6 1
4
2 6
y
2 6 3 0 6 y - 4 6 2
y 7 4 y 6 6
4 6 y 6 6 -6 6 y 6 -4
-1 6 y - 5 6 1, Quadratic Inequalities
Solve the inequalities in Exercises 35–42. Express the solution sets as
intervals or unions of intervals and show them on the real line. Use the
result as appropriate.
35. 36. 37.
38. 39. 40.
41. 42.
Theory and Examples
43. Do not fall into the trap For what real numbers a is
this equation true? For what real numbers is it false?
44. Solve the equation
45. A proof of the triangle inequality Give the reason justifying
each of the numbered steps in the following proof of the triangle
inequality.
(1)
(2)
(3)
(4)
46. Prove that for any numbers a and b.
47. If and what can you say about x?
48. Graph the inequality
49. Let and let be any positive number. Prove
that implies Here the notation
means the value of the expression when
This function notation is explained in Section 1.3.
50. Let and let be any positive number. Prove
that Here the notation
means the value of the expression when
(See Section 1.3.)
51. For any number a, prove that
52. Let a be any positive number. Prove that if and only if
or
53. a. If b is any nonzero real number, prove that
b. Prove that
54. Using mathematical induction (see Appendix 1), prove that
ƒ an for any number a and positive integer n.