1-1realnumbers matematika 1

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Prerequisites for CalculusThis chapter contains topics necessary for the study ofcalculus. After a brief review of real numbers, coordinatesystems, and graphs in two dimensions, we turn our

I. I Real Numbersattention to one of the most important concepts mathematics-the notion of junction.

Real numbers are used considerably in precalculus mathematics, and we wassume familiarity with the fundamental properties of addition, subtractiomultiplication, division, exponents and radicals. Throughout this chaptunless otherwise specified, lower-case letters a, b, c, . . . denote real numbeThe positive integers 1, 2, 3, 4, . . . may be obtained by adding the renumber 1 successively to itself. The integers consist of all positive and negatiintegers together with the real number 0. A rational number is a real numbthat can be expressed as a quotient a/b, where a and bare integers and b =I=Real numbers that are not rational are called irrational. The ratio of tcircumference of a circle to its diameter is irrational. This real numberdenoted by n and the notation n ~ 3.1416 is used to indicate that n is aproximately equal to 3.1416. Another example of an irrational number is JReal numbers may be represented by nonterminating decimals. Fexample, the decimal representation for the rational number 7434/2310found by long division to be 3.2181818 . . . , where the digits 1 and 8 repeindefinitely. Rational numbers may always be represented by repeatidecimals. Decimal representations for irrational numbers may also obtained; however, they are nonterminating and nonrepeating.

It is possible to associate real numbers with points on a line l in such a wthat to each real number a there corresponds one and only one point, a


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2 1 Prerequisites for Calculus

(I. I)

conversely, to each point Pthere corresponds precisely one real number. Suan association between two sets is referred to as a one-to-one correspondenWe first choose an arbitrary point 0, called the origin, and associate withthe real number 0. Points associated with the integers are then determined considering successive line segments of equal length on either side of 0illustrated in Figure 1.1. The points corresponding to rational numbers suas 253 and - ta re obtained by subdividing the equal line segments. Poiassociated with certain irrational numbers, such as J2, can be foundgeometric construction. For other irrational numbers such as n, no costruction is possible. However, the point corresponding to n can be appromated to any degree of accuracy by locating successively the points corrponding to 3, 3.I, 3.I4, 3.I4I, 3.1415, 3.I4I59, . . . . It can be shown thatevery irrational number there corresponds a unique point on l and, coversely, every point that is not associated with a rational number corresponto an irrational number.

0 B A ~ l r 17\; -3 -2 0 4 \ 5 b a.J i 7rI 3 23-2 2 5FIGURE I .I

The number a that is associated with a point A on I is called the coordinof A. An assignment of coordinates to points on I is called a coordinsystem for /, and I is called a coordinate line, or a real line. A direction can assigned to l by taking the positive direction to the right and the negatidirection to the left. The positive direction is noted by placing an arrowheon las shown in Figure I. I.

The real numbers which correspond to points to the right of 0 in FiguI. I are called positive real numbers, whereas those which correspond points to the left of 0 are negative real numbers. The real number 0 is neithpositive nor negative. The collection of positive real numbers is closrelative to addition and multiplication; that is, if a and b are positive, thso is the sum a + b and the product ab.

I f a and bare real numbers, and a - b is positive, we say that a is greathan h and write a > b. An equivalent statement is his less than a, writtb < a. The symbols > or < are called inequality signs and expressions suas a > b orb < a are called inequalities. From the manner in which we costructed the coordinate line l in Figure I. I, we see that if A and Bare poinwith coordinates a and b, respectively, then a > b (orb < a) ifand only ilies to the right ofB. Since a - 0 = a, it follows that a > 0 if and only if apositive. Similarly, a < 0 means that a is negative. The following propertof inequalities can be proved.

If a > b and b > c, then a > c.If a > b, then a + c > b + c.If a >band c > 0, then ac > be.If a >band c < 0, then ac < be.

Analogous properties for" less than" can also be established.


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1-41=4 141=4, - - - - - - " - . . . ~+ I I I + I I I + I -5 -4 -3 -2 -1 0 1 2 3 4 S IFIGURE 1.2

Definition (1.2)

(1.3)

(1.4)

The Triangle Inequality (1 .5)

Real Numbers I. IThe symbol a ~ b, which is read a is greater than or equal to h, means theither a > b or a = b. The symbol a < b < c means that a < b and b b if and only if a > b or a < - bIa I = b if and only if a = b or a = - b.

It follows from the first and third properties stated in (1.4) thatlal:::;; b if and only if -b:::;; a:::;; b.

la+ bl :::;; lal + lbl


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5=17-21=12-71~I I I I + I I I I + I -2 -1 0 1 2 3 4 5 6 7 8 I

FIGURE 1.3

Definition (1.6)

A B OC D1 1 + 1 1+ + 1 1 1 1+ 1 -5 -3 0 1 6

FIGURE 1.4

Proof From (1.3), - lal s as lal and - lbl s b s lbl. Addicorresponding sides we obtain

-(lal + lbl) s a+ b s lal + lbl.Using the remark preceding this theorem gives us th e desired conclusion.

We shall use the concept of absolute value to define the distance betweany two points on a coordinate line. Let us begin by noting that the distanbetween the points with coordinates 2 and 7 shown in Figure 1.3 equ5 units on l. This distance is the difference, 7 - 2, obtained by subtractithe smaller coordinate from the larger. If we employ absolute values, thsince 17 - 21 = 12 - 71, it is unnecessary to be concerned about the ordersubtraction. We shall use this as ou r motivation for the next definition.

Let a and b be the coordinates of two points A an d B, respectively, on acoordinate line 1. The distance between A and B, denoted by d(A,B), isdefined by

d(A, B) = lb - al.

The number d(A, B) is also called the length of the line segment AB.Observe that, since d(B,A) =la - bl and lb - al= la - bl, we mwrite

d(A, B) = d(B, A).Also note that the distance between the origin 0 and the point A is

d(O, A)= la - 01 = lal,which agrees with the geometric interpretation of absolute value illustrain Figure 1.2.

Example 2 IfA, B, C, and D have coordinates -5 , -3 , 1, an d 6, respectivefind d(A, B), d(C, B), d(O, A), an d d(C, D).Solution The points are indicated in Figure 1.4.

By Definition (1.6),d(A, B) = 1-3-(-5)1=1-3+51=121=2.d(C, B) = 1-3 - 11=1-41=4.d(O,A) = l-5 - OI = 1-51=5.d(C, D) = 16 - 1I=151 = 5.


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(J.7)

( )a b

( I )-1 0 3

( )0 2 4

FIGURE 1.5 Open intervals (a, b), ( - I, 3),and (2, 4)

(1.8)

Real Numbers 1.1

The concept of absolute value has uses other than that offinding distancbetween points. Generally, it is employed whenever one is interested in tmagnitude or numerical value of a real number without regard to its sign.In order to shorten explanations it is sometimes convenient to use tnotation and terminology of sets. A set may be thought of as a collectionobjects of some type. The objects are called elements of the set. Throughoour work ~ will denote the set of real numbers. I fSis a set, then a E S meathat a is an element of S, whereas a S signifies that a is not an element ofI fevery element of a set Sis also an element of a set T, then Sis called a subof T. Two sets Sand Tare said to be equal, written S = T, if Sand T contaprecisely the same elements. The notation S =I= T means that Sand Tare nequal. I f Sand Tare sets, their union S u T consists of the elements which aeither in S, in T, or in both Sand T. The intersection Sn T consists of telements which the sets have in common.

I f he elements ofa set Shave a certain property, then we write S = {x: .where the property describing the arbitrary element xis stated in the spaafter the colon. For example, {x: x > 3} may be used to represent the setall real numbers greater than 3.Of major importance in calculus are certain subsets of called intervaI f a < b, the symbol (a, b) is sometimes used for all re;il numbers betweenand b. This set is called an open interval. Thus we have:

(a, b) = {x: a < x < b}.The numbers a and b are called the endpoints of the interval.The graph of a set Sof real numbers is defined as the points on a coordinaline that correspond to the numbers in S. In particular, the graph of the opinterval (a, b) consists of all points between the points corresponding to a ab. In Figure 1.5 we have sketched the graphs ofa general open interval (a,and the special open intervals ( - I, 3) and (2, 4). The parentheses in the figuindicate that the endpoints of the intervals are not to be included. Fconvenience, we shall use the terms interval and graph of an interval intchangeably.

I f we wish to include an endpoint of an interval, a bracket is used insteofa parenthesis. I f a< b, then closed intervals, denoted by [a, b], and haopen intervals, denoted by [a, b) or (a, b], are defined as follows.

[a,b] = {x:a ~ x ~ b}[a, b) = {x: a ~ x < b}(a, b] = {x: a< x ~ b}

Typical graphs are sketched in Figure 1.6, where a bracket indicates ththe corresponding endpoint is part of the graph.

[ 3 [ ) ( 3a b a b a bFIGURE 1.6


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(1.9)

In future discussions of intervals, whenever the magnitudes of a andare no t stated explicitly it will always be assumed that a < b. I f an intervaa subset of another interval I it is called a subinterval of/. For example, closed interval [2 , 3] is a subinterval of [O, 5]. We shall sometimes emplthe following infinite intervals.

(a, oo) = {x: x >a}(-oo,a) = {x:x


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Signofx-2: --- + + + + + + + + + +Sign of x - 5: - - - - - - - - - + + + +I I I I ) I I ( I -2 -1 0 1 2 3 4 5 6 x

FIGURE 1.7

Real Numbers I .IThe reader should supply reasons for the solutions of the followiinequalities.

Example 3 Solve the inequality 4x + 3 > 2x - 5.Solution The following inequalities are equivalent:

4x + 3 > 2x - 54x > 2x - 82x > -8

x > -4Hence the solutions consist of all real numbers greater than -4 , that is,numbers in the infinite interval ( -4 , oo ).

E l 4 S 1 h . 1. 4 - 3xxamp e o vet e mequa 1ty - 5 < - -2- < 1.Solution We may proceed as follows:

4 - 3x-5 -3 32 14-< x 0.Solution Since the inequality may be written

(x - 5)(x - 2) > 0,it follows that x is a solution if and only if both factors x - 5 and x - 2 positive, or both are negative. The diagram in Figure 1.7 indicates the signsthese factors for various real numbers. Evidently, both factors are positif xis in the interval (5, oo) and both are negative if x is in ( - oo, 2). Henthe solutions consist of all real numbers in the union ( - oo, 2) u (5, oo ).

Among the most important inequalities occurring in calculus are thocontaining absolute values of the type illustrated in the next example.


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1.1 Exercises

Example 6 Solve the inequality Ix - 31 < 0.1.Solution Using (1.4) and (1.1), the given inequality is equivalent to eaof the following:

-0.1 < x - 3 < 0.1-0.1 + 3 < (x - 3) + 3 < 0.1 + 32.9 < x < 3.1.

Thus the solutions are the real numbers in the open interval (2.9, 3.1).

Example 7 Solve I2x - 71 > 3. .Solution By (1.4), x is a solution of I2x - 71 > 3 if and only if either

2x - 7 > 3 or 2x - 7 < - 3.The first of these two inequalities is equivalent to 2x > 10, or x > 5. Tsecond is equivalent to 2x < 4, or x < 2. Hence the solutions of I2x - 71 >are the numbers in the union ( - oo, 2) u (5, oo ).

In Exercises 1 and 2 replace the comma between each pair ofreal numbers with the appropriate symbol , or =.

5 I f A, B, and C are points on a coordinate line with ordinates - 5, - l, and 7, respectively, find the followdistances.1 (a) -2, -5

(c) 6 - l, 2 + 3(e) 2, J4

2 (a) -3, 0(c) 8, -3(e) Ji, 1.4

(b) -2 , 5(d) i. 0.66(f ) n, (b) -8 , -3(d) i - i, i's(f) tm. 3.6513

Rewrite the expressions in Exercises 3 and 4 without usingsymbols for absolute values.3 (a) l2 -51 (b) 1-51+1-21

(c) 151+1-21 (d) l -51-1-21.(e) In - 22/71 (f ) (-2)/1-21(g) I t - o.51 (h) 1(-3)21(i) 15 - x I if x > 5 (j) Ia - b I if a < b

4 (a) 14 - 81 (b) 13 - nl(c) l -41-1-81 (d) 1-4 + 81(e) 1-312 (f ) 12 - J41(g) I -0.671 (h)-1-31(i) lx2 + ll (j) 1-4 - x 2 I

(a) d(A, B)(c) d(C, B)

(b) d(B, C)(d) d(A, C)

6 Rework Exercise 5 if A, B, and C have coordinates 2, -and - 3, respectively.Solve the inequalities in Exercises 7-34 and express the solutioin terms of intervals.7 5x - 6 > 11 8 3x - 5 < 109 2 - 7x $ 16 10 7 - 2x 2':: -3

11 12x + 11 > 5 12 lx+21


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Coordinate Systems in Two Dimensions 1.2

23 17 ~ 3x I:::;; 125 I25x - 81 > 727 3x2 + 5x - 2 < 029 2x2 + 9x + 4 ;:::: 0

I31 2 < 100x3x + 233 - - (l/b .44 I f O < a < b, prove that a2 < b2 Why is the restricti

0 < a necessary?45 I f a < b and c < d, prove that a + c < b + d.46 I f a < band c < d, is it always true tha t ac < bd? Expla47 Prove (1.3).48 Prove (1.4).

J.2 Coordinate Systems in Two DimensionsIn Section 1.1 we discussed how coordinates may be assigned to points online. Coordinate systems can also be introduced in planes by meansordered pairs. The term ordered pair refers to two real numbers, where o

1-1realnumbers matematika 1

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