[PDF] Crowell and Slesnicks Calculus with Analytic Geometry

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Crowell and Slesnick's

Calculus with Analytic Geometry

The Dartmouth CHANCE Project

1

Version 3.0.3, 5 January 2008

1 Copyright (C) 2008 Peter G. Doyle. This work is freely redistributable under the terms of the GNU Free Documentation License as published by the Free Software Foundation. Derived from `Calculus with Analytic Geometry', Copyright (C) 1963,1965,1968 Richard H. Crowell and William E. Slesnick. LaTeX conversion 2003-2005 by Fuxing Hou (text) and Helen Doyle (exercises), with help from Peter Doyle.

Contents

WARNING: Page references are wrong. 7

1 Functions, Limits, and Derivatives 9

1.1 Real Numbers, Inequalities, Absolute Values. . . . . . . . . . . . . . 9

1.2 Ordered Pairs of Real Numbers, thexy-Plane, Functions. . . . . . . 18

1.3 Operations with Functions. . . . . . . . . . . . . . . . . . . . . . . . 28

1.4 Limits and Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.5 Straight Lines and Their Equations. . . . . . . . . . . . . . . . . . . 49

1.6 The Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

1.7 Derivatives of Polynomials and Rational Functions. . . . . . . . . . . 66

1.8 The Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

1.9 Implicit Dierentiation. . . . . . . . . . . . . . . . . . . . . . . . . . 83

2 Applications of the Derivative 89

2.1 Curve Sketching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.2 Maximum and Minimum Problems. . . . . . . . . . . . . . . . . . . . 100

2.3 Rates of Change with respect to Time. . . . . . . . . . . . . . . . . . 109

2.4 Approximate Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

2.5 Rolle's Theorem and Its Consequences. . . . . . . . . . . . . . . . . . 118

2.6 The Dierential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

2.7 L'H^opital's Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3 Conic Sections 137

3.1 The Circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.2 The Parabola. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.3 The Ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

3.4 The Hyperbola. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4 Integration 167

4.1 The Denite Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.2 Sequences and Summations. . . . . . . . . . . . . . . . . . . . . . . . 179

4.3 Integrability of Monotonic Functions. . . . . . . . . . . . . . . . . . . 188

4.4 Properties of the Denite Integral. . . . . . . . . . . . . . . . . . . . 195

4.5 The Fundamental Theorem of Calculus. . . . . . . . . . . . . . . . . 203

4.6 Indenite Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.7 Area between Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . 223

4.8 Integrals of Velocity and Acceleration. . . . . . . . . . . . . . . . . . 233

3

4CONTENTS

5 Logarithms and Exponential Functions 243

5.1 The Natural Logarithm. . . . . . . . . . . . . . . . . . . . . . . . . . 243

5.2 The Exponential Function. . . . . . . . . . . . . . . . . . . . . . . . 253

5.3 Inverse Function Theorems. . . . . . . . . . . . . . . . . . . . . . . . 262

5.4 Other Exponential and Logarithm Functions. . . . . . . . . . . . . . 267

5.5 Introduction to Dierential Equations. . . . . . . . . . . . . . . . . . 277

6 Trigonometric Functions 285

6.1 Sine and Cosine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

6.2 Calculus of Sine and Cosine. . . . . . . . . . . . . . . . . . . . . . . . 294

6.3 Other Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . 304

6.4 Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . 313

6.5 Algebraic and Transcendental Functions. . . . . . . . . . . . . . . . . 323

6.6 Complex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

6.7 The Complex Exponential Functionez. . . . . . . . . . . . . . . . . 335

6.8 Dierential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7 Techniques of Integration 353

7.1 Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7.2 Integrals of Trigonometric Functions. . . . . . . . . . . . . . . . . . . 361

7.3 Trigonometric Substitutions. . . . . . . . . . . . . . . . . . . . . . . 373

7.4 Partial Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

7.5 Other Substitutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

8 The Denite Integral (Continued) 401

8.1 Average Value of a Function. . . . . . . . . . . . . . . . . . . . . . . 401

8.2 Riemann Sums and the Trapezoid Rule. . . . . . . . . . . . . . . . . 409

8.3 Numerical Approximations (Continued). . . . . . . . . . . . . . . . . 418

8.4 Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

8.5 Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

8.6 Integration of Discontinuous Functions. . . . . . . . . . . . . . . . . 447

8.7 Improper Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

9 Innite Series 467

9.1 Sequences and Their Limits. . . . . . . . . . . . . . . . . . . . . . . . 468

9.2 Innite Series: Denition and Properties. . . . . . . . . . . . . . . . 475

9.3 Nonnegative Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

9.4 Alternating Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

9.5 Absolute and Conditional Convergence. . . . . . . . . . . . . . . . . 496

9.6 Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

9.7 Functions Dened by Power Series. . . . . . . . . . . . . . . . . . . . 511

9.8 Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

10 Geometry in the Plane 531

10.1 Parametrically Dened Curves. . . . . . . . . . . . . . . . . . . . . . 531

10.2 Arc Length of a Parametrized Curve. . . . . . . . . . . . . . . . . . . 540

10.3 Vectors in the Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

10.4 The Derived Vector of a Parametrized Curve. . . . . . . . . . . . . . 559

10.5 Vector Velocity and Acceleration. . . . . . . . . . . . . . . . . . . . . 567

CONTENTS5

10.6 Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

10.7 Area and Arc Length in Polar Coordinates. . . . . . . . . . . . . . . 589

11 Dierential Equations 601

11.1 Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

11.2 First-Order Linear Dierential Equations. . . . . . . . . . . . . . . . 610

11.3 Linear Dierential Operators. . . . . . . . . . . . . . . . . . . . . . . 616

11.4 Homogeneous Dierential Equations. . . . . . . . . . . . . . . . . . . 626

11.5 Nonhomogeneous Equations. . . . . . . . . . . . . . . . . . . . . . . 635

11.6 Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Appendix A. Properties of Limits 651

Appendix B. Properties of the Denite Integral 655 Appendix C. Equivalent Denitions of the Integral 659

6CONTENTS

WARNING: Page references

in the text are wrong. The page references are left over from the original printed version, and are therefore wrong. References to theorems, etc., are also hardwired, though they may still be correct. Someone should go in and put in real live links. 7

8CONTENTS

Chapter 1

Functions, Limits, and

Derivatives

1.1 Real Numbers, Inequalities, Absolute Values.

Calculus deals with numerical-valued quantities and, in the beginning, with quan- tities whose values are real numbers. Some understanding of the basic setRof all real numbers is therefore essential. Areal numberis one that can be written as a decimal: positive or negative or zero, terminating or nonterminating. Examples are

1;5;0;14;

23
= 0:666666:::;38 = 0:375; p2 = 1:4142:::; =3:141592:::;

176355:14233333::::

The most familiar subset ofRis the setZofintegers. These are the numbers :::;3;2;1;0;1;2;3;::::(1.1) Another subset is the setQof all rational numbers. A real numberrisrationalif it can be expressed as the ratio of two integers, more precisely, ifr=mn , wherem andnare integers andn6= 0. Since every integermcan be writtenm1 , it follows that every integer is also a rational number. A scheme, analogous to (??), which lists all the positive rational numbers is the following: 11 ;21 ;31 ;41 ; ::: 12 ;22 ;32 ;42 ; ::: 13 ;23 ;33 ;43 ; ::: ...(1.2) 9

10CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Of course there are innitely many repetitions in this presentation since, for exam- ple, 21
=42 =63 =::::An unsophisticated guess would be that all real numbers are rational. There are, however, many famous proofs that this is not so. For example, a very simple and beautiful argument shows thatp2 is not rational. (See Problem ??at the end of this section.) It is not hard to prove that a real number is rational if and only if its decimal expansion beyond some digit consists of a nite sequence of digits repeated forever. Thus the numbers

1:71349213213213213213:::(forever);

1:500000000:::(forever)

are rational, but

0:101001000100001000001:::(etc.)

is not. The fundamental algebraic operations on real numbers are addition and mul- tiplication: For any two elementsaandbinR, two elementsa+bandabinR are uniquely determined. These elements, called thesumandproductofaandb, respectively, are dened so that the following six facts are true:

Axiom 1(Associative Laws).

a+ (b+c) = (a+b) +c; a(bc) = (ab)c:

Axiom 2(Commutative Laws).

a+b=b+a; ab=ba:

Axiom 3(Distributive Law).

(a+b)c=ac+bc: Axiom 4(Existence of Identities). Rcontains two distinct elements0and1with the properties that0 +a=aand1
a=afor everyainR. Axiom 5(Existence of Subtraction).For everyainR, there is an element inR denoted byasuch thata+ (a) = 0.

Note.abis an abbreviation ofa+ (b).

Axiom 6(Existence of Division).For everya6= 0inR, there is an element inR denoted bya1or1a such thataa1= 1. Note. ab is an abbreviation ofab1. Addition and multiplication are here introduced as binary operations. However, as a result of the associative law of addition,a+b+cis dened to be the common value of (a+b)+canda+(b+c). In a like manner we may dene the triple product abcand, more generally,a1+:::+ananda1:::an. Many theorems of algebra are consequences of the above six facts, and we shall assume them without proof. They are, in fact, frequently taken as part of a set of axioms forR.

1.1. REAL NUMBERS, INEQUALITIES, ABSOLUTE VALUES.11

Another essential property of the real numbers is that of order. We writea < b as an abbreviation of the statement thatais less thanb. Presumably the reader, given two decimals, knows how to tell which one is the smaller. The following four facts simply recall the basic properties governing inequalities. On the other hand, they may also be taken as axioms for an abstractly dened relation between elements ofR, which we choose to denote by<. Axiom 7(Transitive Law).Ifa < bandb < c, thena < c. Axiom 8(Law of Trichotomy).For every real numbera, one and only one of the following alternatives holds:a= 0, ora <0, or0< a.

Axiom 9.Ifa < b, thena+c < b+c.

Axiom 10.Ifa < band 0< c, thenac < bc.

Note that each of the above Axioms except??remains true when restricted to the setZof integers. Moreover, all the axioms are true for the setQof rational numbers. Hence as a set of axioms forR, they fail to distinguish between two very dierent sets:Rand its subsetQ. Later in this section we shall add one more item to the list, which will complete the algebraic description ofR. A real numberais ifpositive0< aandnegativeifa <0. Since the relation \greater than" is just as useful as \less than," we adopt a symbol for it, too, and abbreviate the statement thatais greater thanbby writinga > b. Clearlya > bif and only ifb < a. Axiom??, when translated into English, says that the direction of an inequality is preserved if both sides are multiplied by the same positive number. Just the opposite happens if the number is negative: The inequality is reversed.

That is,

1.1.1.Ifa < bandc <0, thenac > bc.

Proof.Sincec <0, Axioms??,??, and??imply

0 =c+ (c)<0 + (c) =c:

Socis positive. Hence by (x), we getac <bc. By Axiom??again, ac+ (bc+ac)<bc+ (bc+ac):

Hencebc < ac, and this is equivalent toac > bc.Two more abbreviations complete the mathematician's array of symbols for

writing inequalities: abmeansa < bora=b, abmeansa > bora=b. The geometric interpretation of the setRof all real numbers as a straight line is familiar to anyone who has ever used a ruler, and it is essential to an understanding of calculus. To describe the assignment of points to numbers, consider an arbitrary straight lineL, and choose on it two distinct points, one of which we assign to, or identify with, the number 0, and the other to the number 1. (See Figure??.) The rest is automatic. The scale onLis chosen so that the unit of distance is the length of the line segment between the points 0 and 1. Every positive numberais

12CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Figure 1.1: A lineLwith two distinguished point 0 and 1. assigned the point on the side of 0 containing 1 which isaunits of distance from

0. Every negative numberais assigned the point on the side of 0 not containing 1

which isaunits of distance from 0. Note that ifLis oriented so that 1 lies to the right of 0, thenfor any two numbersaandb(positive, negative, or zero),a < bif and only ifalies to the left ofb.A line which has been identied withRunder a correspondence such as the one just described is called areal number line. (See

Figure??.)

Figure 1.2: A real number line.

Anintervalis a subsetIofRwith the property that wheneveraandcbelong toIandabc, thenbalso belongs toI. Geometrically an interval is a connected piece of a real number line. A number d is called alower boundof a setSof real numbers ifdsfor everysinS. It is anupper boundofSifsdfor every sinS. A given subset ofR, and in particular an interval, is calledboundedif it has both an upper and lower bound. There are four dierent kinds of bounded intervals: (a;b), the set of all numbersxsuch thata < x < b; [a;b], the set of all numbersxsuch thataxb; [a;b), the set of all numbersxsuch thatax < b; (a;b], the set of all numbersxsuch thata < xb. In each case the numbersaandbare called theendpointsof the interval. The set [a;b] contains both its endpoints, whereas (a;b) contains neither one. Clearly [a;b) contains its left endpoint but not its right one, and an analogous remark holds for (a;b]. It is important to realize that there is no element1(innity) in the setR. Nevertheless, the symbols1and1are commonly used in denoting unbounded intervals. Thus (a;1) is the set of all numbersxsuch thata < x; [a;1) is the set of all numbersxsuch thatax; (1;a) is the set of all numbersxsuch thatx < a; (1;a] is the set of all numbersxsuch thatxa; (1;1) is the entire setR.

1.1. REAL NUMBERS, INEQUALITIES, ABSOLUTE VALUES.13

The symbols1and1also appear frequently in inequalities although they are really unnecessary, because, for example, 1< x < ais equivalent tox < a; ax <1is equivalent toax; etc. Since1is not an element ofR, we shall never use the notations [a;1];x 1, etc. An unbounded interval has either one endpoint or none; in each of the above cases it is the numbera. We call an intervalopenif it contains none of its endpoints, andclosedif it contains them all. Thus, for example, (a;b) and (1;a) are open, but [a;b] and [a;1) are closed. The intervals [a;b) and (a;b] are neither open nor closed, although they are sometimes called half-open or half-closed. Since (1;1) has no endpoints, it vacuously both does and does not contain them. Hence (1;1) has the dubious distinction of being both open and closed.

Figure 1.3: Types of intervals.

Example 1.Draw the intervals [0;1], [1;4), (2;1), (1;1], (1;3), and iden- tify them as open, closed, neither, or both (see Figure??). It is frequently necessary to talk about the size of a real number without regard to its sign, not caring whether it is positive or negative. This happens often enough to warrant a denition and special notation: Theabsolute valueof a real number ais denoted byjajand dened by jaj=aifa0; aifa <0: Thusj3j= 3,j0j= 0,j 3j= 3. Obviously,the absolute value of a real number cannot be negative.Geometrically,jajis the distance between the points 0 anda on the real number line. A generalization that is of extreme importance is the fact

14CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

thatjabjis the distance between the pointsaandbon the real number line for any two numbersaandbwhatsoever.Probably the best way to convince oneself that this is true is to look at a few illustrations (see Figure??). Figure 1.4: Computing distances with the absolute value. Example 2.Describe the setIof all real numbersxsuch thatjx5j<3. For any numberx, the numberjx5jis the distance betweenxand 5 on a real number line (see Figure??). That distance will be less than 3 if and only ifxsatises the Figure 1.5: An open ball in a one-dimensional space. inequalities 2< x <8. We conclude thatIis the open interval (2;8). There is an alternative way of writing the denition of the absolute value of a numberawhich requires only one equation: We do not have to give separate denitions for positive and negativea. This denition uses a square root, and before proceeding to it, we call attention to the following mathematical custom: Although every positive real number a has two square roots, in this bookthe expressionpa

always denotes the positive root. Thus the two solutions of the equationx2= 5 arep5 andp5. Note that the two equations

x 2=a and x=pa are not equivalent. The second implies the rst, but not conversely. On the other hand, x 2=a and jxj=pa are equivalent. Having made these remarks, we observe that

1.1. REAL NUMBERS, INEQUALITIES, ABSOLUTE VALUES.15

1.1.2.

jaj=pa 2: The formulation??is a handy one for establishing two of the basic properties of absolute value. They are

1.1.3.

jabj=jajjbj:

1.1.4.

ja+bj  jaj+jbj: Proof.Since (ab)2=a2b2and since the positive square root of a product of two positive numbers is the product of their positive square roots, we get jabj=p(ab)2=pa

2b2=pa

2pb

2=jajjbj:

To prove??, we observe, rst of all, thatab jabj. Hence a

2+ 2ab+b2a2+ 2jabj+b2=jaj2+ 2jajjbj+jbj2:

Thus, ja+bj2= (a+b)2(jaj+jbj)2: By taking the positive square root of each side of the inequality (see Problem??), we get??.As remarked above, our list of Axioms??through??about the setRof real numbers is incomplete. One important property of real numbers that together with the others gives a complete characterization is the following: Axiom 11(Least Upper Bound Property).Every nonempty subset ofRwhich has an upper bound has a least upper bound. SupposeSis a nonempty subset ofRwhich has an upper bound. What Axiom ??says is that there is some numberbwhich (1) is an upper bound, i.e.,sbfor everysinS, and (2) ifcis any other upper bound ofS, thenbc. It is hard to see at rst how such a statement can be so signicant. Intuitively it says nothing more than this: If you cannot go on forever, you have to stop somewhere. Note, however, that the rational numbers donothave this property. The set of all rational numbers less than the irrational numberp2 certainly has an upper bound. In fact, each of the numbers 2, 1:5, 1:42, 1:415, 1:4143, and 1:41422 is an upper bound. However, for every rational upper bound, there will always exist a smaller one. Hence there is no rational least upper bound.

16CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Problems

1. Draw the following intervals and identify them as bounded or unbounded,

closed or open, or neither: (2;4), [3;5], (1;2], [1:5;2:5), (p2;).

2. Draw each of the following subsets ofR. For those that are given in terms of

absolute values write an alternative description that does not use the absolute value. (a) Set of allxsuch that 4< x7:5. (b) Set of allxsuch that 0< x <1. (c) Set of allxsuch that 5x <8. (d) Set of allxsuch thatjxj>2. (e) Set of allysuch that 10. (h) Set of allusuch that 13. Prove the following facts about inequalities. [Hint:Use??,??,??,??, and the meanings ofand. In each problem you will have to consider several cases separately, e.g.a >0 anda= 0.] (a) Ifab, thena+cb+c. (b) Ifab, thena+cb+c. (c) Ifabandc0, thenacbc. (d) Ifabandc0, thenacbc.

4. Prove thatais positive (negative) if and only if1a

is positive (negative).

5. If 0< a < b, prove that1b

<1a .

6. Ifa > candb <0, prove thatab

7. Ifa < b < c, prove that bc 0; bc >ba ifc <0:

8. Does the setZof integers have the Least Upper Bound Property? That is, if

a nonempty subset ofZhas an upper bound, does it have a smallest one?

9. Show that if 0ab, then 0papb.

10. Prove thata=bif and only ifabandba.

11. Show that the Least Upper Bound Property implies the Greatest Lower Bound

Property. That is, using??, prove that if a nonempty subset ofRhas a lower bound, then it has a greatest lower bound.

12. Verify the assertion made in the text that if an interval is bounded it must be

one of four types: (a;b), [a;b], (a;b], or [a;b). (Hint:See Problem??.)

1.1. REAL NUMBERS, INEQUALITIES, ABSOLUTE VALUES.17

13. Prove that

p2 is irrational. (Hint:The proof, which is elegant and famous, starts by assuming thatp2 = pq , wherepandqare integers not both even. A contradiction can then be derived.)

18CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

1.2 Ordered Pairs of Real Numbers, thexy-Plane,

Functions.

The set whose members consist of just the two elementsaandbis denotedfa;bg. The notation is not perfect because it suggests that the membersaandbhave been ordered:ais written rst andbsecond. Actually no ordering is present because fa;bg=fb;ag. Note also that ifa=b, thenfa;bg=fb;ag=fag. It can happen, however, that the ingredient of order is essential. We therefore introduce the notion of anordered pair(a;b) whose rst member isaand whose second member isb.

The characteristic property of ordered pairs is

(a;b) = (c;d) if and only ifa=candb=d. In particular (a;b) = (b;a)if and only ifa=b. In Section??we saw that the setRof all real numbers can be thought of as a straight line. We shall now show that every ordered pair (a;b) of real numbersaandbcan be identied with a point in a plane. This brings up a notational problem: Is (5;7) the ordered pair of real numbers or is it the open interval consisting of allxsuch that 5< x <7? The answer is that it is impossible to tell out of context|just as it is impossible to tell whether the word \well" is the noun or the adverb. Consider two distinct real number lines drawn in a plane so that they intersect at the number 0 on each line. One of the lines is traditionally drawn horizontal and called thex-axis, and the other is made perpendicular to it and called they-axis. The orientation is chosen so that the number 1 on thex-axis lies to the right of 0, and the number 1 on they-axis is above 0. It is also customary to use the same scale of distances on both axes. For every ordered pair (a;b) of real numbers, let L abe the line parallel to they-axis that cuts thex-axis ata, and letMbbe the line parallel to thex-axis that cuts they-axis atb. We assign the point of intersection ofLaandMbto the ordered pair (a;b) (see Figure??). The numbersaandbare called thecoordinatesof the point.ais thex-coordinate(orabscissa) andbis they-coordinate(orordinate).

Figure 1.6:

If the pairs (a;b) and (c;d) are not equal, then the points in the plane assigned to them will be dierent. In addition, every point in the plane has a number pair assigned to it: Starting with a point, draw the two lines through it which are parallel to thex-axis and they-axis. One line cuts thex-axis at a numbera, and the other

1.2. ORDERED PAIRS OF REAL NUMBERS, THEXY-PLANE, FUNCTIONS.19

cuts they-axis atb. The ordered pair (a;b) has the original point assigned to it. It follows that our assignment pair!point is a one-to-one correspondence between the set of all ordered pairs of real numbers, which we denote byR2, and the set of all points of the plane. It is convenient simply to identifyR2with the plane together with the two axes. Example 3.Plot the points (1;2);(2;3);(0;1);(4;0);(2;3), and (2;3) on thexy-plane (see Figure??).

Figure 1.7:

The usefulness of the idea of an ordered pair is by no means limited to pairs of real numbers. In plane geometry, for example, we may consider the set of all ordered pairs (T;p) in whichTis a triangle andpis the point of intersection of its medians. In the three-dimensional extension of thexy-plane, the setR3of all ordered triples (a;b;c) of real numbers is identied with the set of all points in three-dimensional space. The denition of an ordered triple can be reduced to that of an ordered pair by dening (a;b;c) to be ((a;b);c). LetP= (a;b) andQ= (c;d) be arbitrary elements in the setR2of all ordered pairs of real numbers. We dene thedistancebetweenPandQby the formula distance(P;Q) =p(ac)2+ (bd)2:(1.3)

Three simple corollaries of this denition are:

1.2.1.distance(P;Q)0; i.e., distance is never negative.

1.2.2.distance(P;Q) = 0if and only ifP=Q.

1.2.3.distance(P;Q) = distance(Q;P).

Another consequence of (1) is that it is no longer simply a matter of tradition and convenience that we draw they-axis perpendicular to thex-axis. It follows from consideration of the Pythagorean Theorem and its converse (see Figure??) that the above denition of distance between elements ofR2corresponds with our geometric notion of the distance between points in the Euclidean plane if and only if the two coordinate axes are perpendicular and the scales are the same on both.

20CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Figure 1.8:

Figure 1.9:

Example 4.LetCbe the subset of thexy-plane consisting of all points whose distance from (1,1) is equal to 2. ThusCis the circle shown in Figure??. If

(x;y) is an arbitrary point in thexy-plane, its distance from (1;1) is equal top(x1)2+ (y1)2. Hence, (x;y) belongs toCif and only if

p(x1)2+ (y1)2= 2:(1.4) Numbersxandysatisfy (2) if and only if they satisfy (x1)2+ (y1)2= 4:(1.5) ThusCis the set of all ordered pairs (x;y) that satisfy (3)|or that satisfy (2). Either (2) or (3) is therefore calledan equation of the circleC. The set of all points (x;y) in the plane that satisfy a given equation is called thegraphof the equation. Hence, in the above example, the circleCis the graph of the equation (x1)2+ (y1)2= 4. Example 5.LetLbe the set of all ordered pairs (x;y) such thaty= 2x3. For each real numberx, there is one and only one numberysuch that (x;y) belongs to

1.2. ORDERED PAIRS OF REAL NUMBERS, THEXY-PLANE, FUNCTIONS.21xy= 2x3-1-5

0-3 1-1 21
33

Table 1.1:

yx 00 11 24

Table 1.2:

L:y= 2x3. To see whatLlooks like, we plot ve of its points (see Table??). As shown in Figure??, all these points lie on a straight line. In Section??we shall justify the natural conjecture that this straight line is the setL.

Figure 1.10:

Example 6.The set of all pairs (x;y) such thaty2=xis the curve shown in Figure??. This curve is a parabola, one of the conic sections, which are studied in greater detail in Chapter??. At present we shall be satised with plotting a few points and connecting them with a smooth curve (see Table??). Afunctionfis any setfof ordered pairs such that whenever (a;b) and (a;c) belong tof, thenb=c. Note that every subset of thexy-plane is a set of ordered pairs, but not every subset is a function. In particular, the parabola in Example?? is not, because it contains both (4, 2) and (4;2). On the other hand, the straight

22CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Figure 1.11:

line in Example??is a function. This condition that a function must never contain two pairs (a;b) and (a;c) withb6=cmeans geometrically that a subset of thexy- plane is a function if and only if it never intersects a line parallel to they-axis in more than one point. Hence it is an easy matter to decide which of the following sets are functions and which are not: (i) The setfof all pairs (x;y) such thaty=x+ 1. (ii) The setgof all pairs (x;y) such thatx2+y2= 1. (iii) The setFof all pairs (x;y) such thaty=x2+ 2x+ 2. (iv) The sethof all pairs (x;y) such that 2x+ 3y= 1. (v) The setGof all pairs (x;y) such thaty=px+ 2. (vi) The setHof all pairs (x;y) such thaty4=x. The setsf,F,h, andGare functions, butgandHare not. Thedomainof a functionfis the set of all elementsafor which there is a correspondingbsuch that (a;b) belongs tof. Analogously, therangeoffis the set of all elementsbfor which there is anasuch that (a;b) belongs tof. In (i), the domain offis the setRof all real numbers and so is the range. On the other hand, in (iii), although the domain ofFis equal toR, the range is the interval consisting of all real numbersy1, because we can writex2+ 2x+ 2 = (x+ 1)2+ 11. If a pair (a;b) belongs to a functionf, we callbthevalue offataand write b=f(a). Note that the meaning off(a) is unambiguous only because the denition of a function forbids having (a;b) and (a;c) both belong tofifb6=c. Therefore the second member of any ordered pair that belongs tofis determined by the rst member.

Example 7.In (i),

f(x) =x+ 1; f(a) =a+ 1; f(0) = 1; f(3 + 4) = (3 + 4) + 1 = 8; f(1) =1 + 1 = 0; f(a+b) =a+b+ 1:

1.2. ORDERED PAIRS OF REAL NUMBERS, THEXY-PLANE, FUNCTIONS.23

In (v),

G(x) =px+ 2; G(2x+y) =p2x+y+ 2;

G(0) =p2; G(2) = 0;

G(2) = 2; G(3)is not dened:

To each elementain the domain of a functionfthere corresponds a valuef(a) in the range. This correspondence between domain and range, which is pictured in Figure??, is the central idea in the denition of a function. Thus the functionfthat

Figure 1.12:

consists of all ordered pairs (x;y) such thaty=x2and1x2 is interpreted as the rule of correspondence which assigns to each number in the interval [1;2] its square. We can describefcompletely and simply by writing f(x) =x2;1x2:

Examples of other functions are

g(x) =px1;1x <1;

F(x) =x2;1< x <1;

h(x) =xx+ 2; x6=2: Note that the functionsfandFimmediately above arenotequal, althoughfis a subset ofF. Two functions are equal if they are one and the same set of ordered pairs. It follows that

1.2.4.Functionsfandgare equal if and only if they have the same domainDand

f(x) =g(x)for every elementxinD. Thus any complete description of a function must include a description of its domain. Sometimes this information is in fact omitted. We shall adopt the con- vention that if no explicit description of the domain of a function is given, then its domain is assumed to be the largest set of real numbers that makes sense. For example, the domain of the functionHdened by

H(x) =1x

2x2=1(x+ 1)(x2)

is assumed to be the entire set of real numbers with the exception of1 and 2.

24CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Figure 1.13: A computing machine.

It is sometimes helpful to think of a function as a computing machine. Imagine a computing machine, namedf, which is provided with an input tape, an output tape, and a button (see Figure??). One writes a numberxon the input tape and pushes the button. Ifxis one of the inputs which the machine will accept, i.e., ifx is in the domain off, the machine whirs contentedly and prints an output, which we denotef(x), on the output tape. Ifxis not in the prescribed domain, either nothing happens or a red light ashes. We have already seen that one of the best ways of describing a subset ofR2is to draw a picture of it. If this subset happens to be a function, we call the picture the graph of the function. More specically, if a functionfis a subset ofR2, its graphis the set of all points in the plane that correspond to ordered pairs of the form (x;f(x)). Note that the graph offdepends on the correspondence between ordered pairs and points; i.e., it depends on the choice of axes. To illustrate this, in Figure??we have drawn the graph of the functionfdened byf(x) =x3for two

Figure 1.14: Two graphs of the functionf(x) =x3.

sets of axes. For a single choice of axes, we simply identify ordered pairs and points, and under this identication a function and its graph become the same thing. Most of the functions encountered in an introduction to calculus are dened by means of a single equation; e.g.,h(x) =x3+ 3. It is a bad mistake, however, to assume that this is always true. The functionFgiven by

F(x) =x2+ 1 ifx0;

x2 ifx <0; requires two equations for its denition. The graph ofFis shown in Figure??.

1.2. ORDERED PAIRS OF REAL NUMBERS, THEXY-PLANE, FUNCTIONS.25

Another function, which is so wild that it is impossible to draw its graph, is the Figure 1.15: A function not dened by a simple formula. following: g(x) =0 ifxis rational,

1 ifxis irrational.

The ordered pairs that comprise a function are not necessarily pairs of numbers. An example is the function, mentioned earlier in this section, which assigns to each triangle the point of intersection of its medians. It is possible for the domain of a function to be a set of ordered pairs. Consider the functionfconsisting of all ordered pairs ((x;y);z), wherex;y, andzare numbers that satisfyxyand z= 2x2+y2. We describe this function simply as follows: f(x;y) = 2x2+y2; xy:(1.6) As a nal example of a function, consider the rule of correspondence that assigns to each person his or her male parent. As we have indicated, the denition of a function is appallingly general. One of our tasks is to delineate properly the kinds of functions studied in calculus. To begin with, a functionfis said to bereal-valuedif its range is a subset ofR, the set of real numbers. If the domain offis a subset ofR, we callfafunction of a real variable. The functionf(x;y) dened in (??) has as its domain a subset ofR2. It is a real-valued function of two real variables. For the most part, a rst course in calculus is a study of real-valued functions of one real variable.

26CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Problems

1. Plot the following point in thexy-plane: (0;2), (1;3), (3;1), (4;4), and

(5;0).

2. In thexy-plane plot the points (1;2) and (2;1), (3;2) and (2;3), (2;3)

and (3;2). Describe the relative positions of the points (a;b) and (b;a) for arbitraryaandb.

3. Thex-axis and they-axis divideR2into four quadrants, as shown in Figure

??. Let (a;b) be a point for which neitheranorbis zero. How can you recognize instantly which quadrant (a;b) belongs to?

4. Find the distance between (1;2) and (3;4); (2;3) and (3;2); (3;4) and

(1;2); (2;1) and (2;1). In each case plot the points inR2.

5. Verify Proposition??.

6. Plot the subsets of thexy-plane dened in (i) through (vi).

7. In each of the following, plot the subset ofR2that consists of all pairs (x;y)

such that the given equation (or conditions) is satised. (a) 3x+ 2y= 3 (b)x+y= 1 (c)y=jxj (d)y=px (e)x2+y2= 4 (f)x2+ 4y2= 4 (g)x2+y2= 1 andy0 (h) 4x2y2= 4 (i)y= 2x2+x2 (j)y=jx3j (k)y= largest integer less than or equal tox (l)y=2x+ 3; x0 x 22
; x <0:

8. In Problem??, which subsets are functions?

9. Letfandgbe two functions dened, respectively, by

f(x) =x2+x+ 1;1< x <1; g(x) =x+1x1;for every real numberxexceptx= 1: Find: (a)f(2),f(0),f(a),f(a+b),f(ab). (b)g(0),g(1),g(10),g(5 +t),g(x3).

1.2. ORDERED PAIRS OF REAL NUMBERS, THEXY-PLANE, FUNCTIONS.27

10. Give an example of a functionfand a functiongthat satisfy each of the

following conditions. (a) domainf= domaing, but rangef6= range g. (b) domainf6= domaing, but rangef= range g. (c) domainf= domaingand rangef= rangeg, butf6=g. (d)f(a) =g(a) for everyathat belongs to both domains, butf6=g.

11. What is the assumed domain of each of the following functions?

(a)f(x) =5x3 (b)f(x) =x2+2x 22 (c)g(x) =x+3x

2+x12

(d)f(x) = 5 (e)f(t) =q1 5t (f)F(x) =px

28x20

(g) The set of all ordered pairs (x;y) such that xyx2x9= 7:

28CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

1.3 Operations with Functions.

Iffandgare two functions, a new functionf(g), called thecompositionofgwith f, is dened by (f(g))(x) =f(g(x)):

For example, iff(x) =x31 andg(x) =x+1x1, then

(f(g))(x) =f(g(x)) = (g(x))31 (1.7) =x+ 1x1 3 1 =2(3x2+ 1)(x1)3:(1.8) The composition of two functions is the function obtained by applying one after the other. Iffandgare regarded as computing machines, thenf(g) is the composite machine constructed by feeding the output ofginto the input offas indicated in

Figure??.

Figure 1.16:

In general it is not true thatf(g) =g(f). In the above example we have (g(f))(x) =g(f(x)) =f(x) + 1f(x)1(1.9) = (x31) + 1(x31)1=x3x

32;(1.10)

and the two functions are certainly not the same. In terms of ordered pairs the compositionf(g) ofgwithfis formally dened to be the set of all ordered pairs (a;c) for which there is an elementbsuch thatb=g(a) andc=f(b). Iffandgare two real-valued functions, we can perform the usual arithmetic operations of addition, subtraction, multiplication, and division. Thus for the func- tionsf(x) =x31 andg(x) =x+1x1, we have f(x) +g(x) =x31 +x+ 1x1; f(x)g(x) =x31x+ 1x1; f(x)g(x) = (x31)x+ 1x1; = (x2+x+ 1)(x+ 1) ifx6= 1; f(x)=g(x) =x31x+1x1 = (x31)(x1)x+ 1:

1.3. OPERATIONS WITH FUNCTIONS.29xf(x)2f(x)024

112 200
312

Table 1.3:

Just as with the composition of two functions, each arithmetic operation provides a method of constructing a new function from the two given functionsfandg. The natural notations for these new functions aref+g,fg,fg, andfg . They are dened by the formulas (f+g)(x) =f(x) +g(x); (fg)(x) =f(x)g(x); (fg)(x) =f(x)g(x); fg (x) =f(x)g(x)ifg(x)6= 0: The product functionfgshould not be confused with the composite functionf(g). For example, iff(x) =x5andg(x) =x3, then we have (fg)(x) =f(x)g(x) = x

5
x3=x8, whereas

(f(g))(x) =f(g(x)) = (x3)5=x15: We may also form the productafof an arbitrary real numberaand real-valued functionf. The product function is dened by (af)(x) =af(x): Example 8.Let functionsfandgbe dened byf(x) =x2 andg(x) =x2

5x+ 6. Draw the graphs off,g, 2f, andf+g. We compute the function values

corresponding to several dierent numbersxin Tables??and??. The resulting graphs offandgare, respectively, the straight line and parabola shown in Figure ??(a). It turns out that the graphs of 2fandf+gare also a straight line and a parabola. They are drawn in Figure??(b). To see why the graph off+gis a parabola, observe that (f+g)(x) =f(x) +g(x) = (x2) + (x25x+ 6) =x24x+ 4 = (x2)2: It follows thatf+gis very much like the function dened byy=x2. Instead of simply squaring a number,f+grst subtracts 2 and then squares. Its graph will be just like that ofy=x2except that it will be shifted two units to the right. Up to this point we have used the lettersf,g,h,F,G, andHto denote functions, and the lettersx,y,a,b, andcto denote elements of sets|usually real numbers. However, the letters in the second set are sometimes also used as functions. This

30CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Figure 1.17:

xg(x)06 56
52
14 12 42

Table 1.4:

occurs, for example, when we speak ofxas a real variable. As such, it not only is the name of a real number but also can take on many dierent values: 5, or7, or, or .... Thus the variablexis a function. Specically, it is the very simple function that assigns the value 5 to the number 5, the value7 to the number7, the valueto, .... For every real numbera, we have x(a) =a:

This function is called theidentity function.

Suppose, for example, thatsis used to denote the distance that a stone falling freely in space has fallen. The value ofsincreases as the stone falls and depends on the length of timetthat it has fallen according to the equations=12 gt2, whereg is the constant gravitational acceleration. (This formula assumes no air resistance, that the stone was at rest at timet= 0, and that distance is measured from the starting point.) Thusshas the value92 gifthas the value 3, and, more generally,

1.3. OPERATIONS WITH FUNCTIONS.31

the value 12 ga2whenthas the valuea. If we considertto be another name for the identity function, thensmay be regarded as the function whose value is s(a) =12 ga2=12 g(t(a))2 for every real numbera. The original equations=12 gt2then states the relation between the two functionssandt. The fact thatsandttake on dierent values is also expressed by referring to them as variables. Avariableis simply a name of a function. In our examplesis called a dependent variable, andtan independent variable, because the values ofsdepend on those oftaccording tos=12 gt2. Thus anindependent variableis a name for the identity function, and adependent variableis one that is not independent. A real variable is therefore a name of a real-valued function. Since the arithmetic operations of addition, subtraction, multiplication, and division have been dened for real-valued functions, they are automatically dened for real variables. We shall generally use the letterxto denote an independent variable. This raises the question: How does one tell whether an occurrence ofxdenotes a real number or the identity function? The answer is that the notation alone does not tell, but the context and the reader's understanding should. However, a more practical reply is that it doesn't really make much dierence. We may regardf(x) as either the value of the functionfat the numberxor as the composition offwith the variable x. Ifxis an independent variable, the functionf(x) is then the same thing asf. Example 9.The conventions that we have adopted concerning the use of vari- ables give our notations a exibility that is both consistent and extremely useful.

Consider, for example, the equation

y= 2x23x: On the one hand, we may consider the subset ofR2, pictured in Fifure??, that consists of all ordered pairs (x;y) such thaty= 2x23x. This subset is a function fwhose value at an arbitrary real numberxis the real numberf(x) = 2x23x. Alternatively, we may regardxas an independent variable, i.e., the identity function. The composition offwithxis then the functionf(x) = 2x23x, whose value at

2, for instance, is

(f(x))(2) =f(x(2)) =f(2) = 86 = 2: A third interpretation is thatyis a dependent variable that depends onxaccording to the equationy= 2x23x. That is,yis the name of the function 2x23x. Example 10.LetFbe the function dened byF(x) =x3+x+1. Ifu=px2, then

F(u) =u3+u+ 1

= (x2)3=2+ (x2)1=2+ 1:

If we denote the functionF(x) byw, then

u+w=px2 +x3+x+ 1;

32CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Figure 1.18:

uw= (x2)1=2(x3+x+ 1): On the other hand, we may letGbe the function dened byG(x) =px2 for every real numberx2. ThenG+FandGFare the functions dened, respectively, by (G+F)(x) =G(x) +F(x) =px2 +x3+x+ 1; (GF)(x) =G(x)F(x) = (x2)1=2(x3+x+ 1): To say thatais a realconstantmeans rst that it is a real number. Second, it may or may not matter which real numberais, but it is xed for the duration of the discussion in which it occurs. Similarly, aconstant functionis one which takes on just one value; i.e., its range consists of a single element. For example, consider the constant functionfdened by f(x) = 5;1< x <1: The graph offis the straight line parallel to thex-axis that intersects they-axis in the point (0, 5); see Figure??. We shall commonly use lower-case letters at the beginning of the alphabet, e.g.,a,b,c,..., to denote both constants and constant functions. Example 11.Consider the functionax+b, whereaandbare constants,a6= 0, andxis an independent variable. The graph of this function is a straight line that cuts they-axis atband thex-axis atba . It is drawn in Figure??. This function is the sum of the constant functionband the function which is the product of the constant functionaand the identity functionx.

1.3. OPERATIONS WITH FUNCTIONS.33

Figure 1.19:

Figure 1.20:

34CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

Problems

1. Let functionsfandgbe dened by

f(x) =x34x2+ 5x2 = (x2)(x22x+ 1); g(x) =1x :

Findh(x) if

(a)h=f(g) (b)h=f+g (c)h=g(f) (d)h=fg (e)h= 5fg2.

2. What is the domain and range of the functionsfandgin Problem??? What

is the domain of each of the functionsh?

3. Iff(x) =x+ 1 andg(x) =x1, plot the graph of the functionfg

.

4. Plot the graph of the composite functionF(g), whereFandgare the functions

dened byg(x) =x2 andF(x) =1x .

5. Iff,g, andhare functions, show thatf(g(h)) = (f(g))(h). This is the

Associative Law for the Composition of Functions.

6. Iffis a real-valued function, how would you dene the functions 3f? How

would you denepf?

7. The velocityvof a freely falling body depends on the distancesthat it has

fallen according to the equationv=p2gs, wheregis the constant gravita- tional acceleration. (a) Using ans-axis and av-axis, plot the dependent variablevas a function of the independent variables. (b) Ifsdepends on the timetaccording to the equations=12 gt2, how does vdepend ont? Note that the variablevin??, which depends ons, is not the same function as the variablevin??, which depends ont. Without knowing which is referred to, the meaning of the value ofvat 2 is ambiguous.

8. Ifw=u2+u+ 1,u=x2+ 2, andv=x1, what is the value of each of the

following functions at an arbitrary real numberx? (a)u+v (b)w+v (c)wu.

9. IfF(x) =x3+x+ 2 andu=x2+ 1 andw=x+1x

, then (a) (F(u))(x) =

1.3. OPERATIONS WITH FUNCTIONS.35

(b)F(w(x)) = (c) (u+v)(x) =

10. The equationy= 2x+ 1 denesyas a function ofx. It also denesxas a

function ofy. Describe the latter function in two ways.

11. Draw the graph of the functionf(x) =ax1 for four dierent values of the

constanta.

12. Iffandgare two real-valued functions, give the denitions of the sumf+g

and the productfgin terms of ordered pairs.

13. Letfandgbe two real-valued functions. In terms of domainfand domain

g, what are: (a) domainf(g)? (b) domain (f+g)? (c) domainfg?

36CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

1.4 Limits and Continuity.

Consider the functionfdened by

f(x) =x23x+ 2x2; x6= 2: The domain offis the set of all real numbers with the exception of the number 2, which has been excluded because substitution ofx= 2 in the expression forf(x) yields the undened term 00 . On the other hand,x23x+ 2 = (x1)(x2) and (x1)(x2)x2=x1;providedx6= 2:(1.11) The proviso is essential. Without it, (1) is false because, ifx= 2, the left side is undened and the right side is equal to 1. We therefore obtain f(x) =x1; x6= 2: The graph of the functionx1 is a straight lineL; so the graph offis the punctured line obtained fromLby omitting the one point (2;1) (see Figure??).

Figure 1.21:

Although the functionfis not dened atx= 2, we know its behavior for values ofxnear 2. The graph makes it clear that ifxis close to 2, thenf(x) is close to 1. In fact, the valuesf(x) can be brought arbitrarily close to 1 by takingxsuciently close to 2. We express this fact by writing lim x!2x

23x+ 2x2= 1;

which is translated:The limit ofx23x+2x2is1asxapproaches2.

Example 12.Evaluate limx!3pxp3

x3. The functionpxp3 x3is not dened at x= 3. The following algebraic manipulation puts the function in a form in which its behavior close to 3 can be read o easily: pxp3 x3=pxp3 x3px+p3px+p3

1.4. LIMITS AND CONTINUITY.37

= x3x31px+p3 =

1px+p3

;ifx6= 3: Again note the provisox6= 3: Whenx= 3, the last quantity in the preceding equations is equal to 1p3+ p3 , but the rst quantity is not dened. However, by taking values ofxclose to 3, it is clear that the corresponding values of1px+p3 can be brought as close as we please to 12 p3 . We conclude that lim x!3pxp3 x3=12 p3 :

In words: The limit of

pxp3 x3, asxapproaches 3, is12 p3 .

Example 13.Iff(x) =1x

, evaluate limx!0f(x). The functionfis not dened at

0 (i.e., the number 0 is not in the domain off). From the graph offand the list

of ordered pairs (x;f(x)) shown in Figure??, it is clear that there are values ofx

Figure 1.22:

arbitrarily close to 0 for which the corresponding values off(x) are arbitrarily large in absolute value (see Table??). We conclude that limx!01x does not exist. Thus far our examples have been conned to the problem of nding the limit of a function at a number which happens to lie outside the domain of the function. If it happens that the numberais in the domain off, then it is frequently possible to determine lim x!af(x) at a glance. Consider, for example, the functionf(x) =

2x2x2. Asxtakes on values closer and closer to 3, the corresponding value of

2x2approaches 18, the value ofxapproaches3, and the constant2 does not

change. We conclude that lim x!3(2x2x2) = 13;

38CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVESxf(x) =1x

11 0.110

0.01100

0.0011000

0.000110000

-1-1 -0.1-10 -0.01-100 -0.001-1000 -0.0001-10000

Table 1.5:

or that, for this particular function, lim x!3f(x) =f(3). Example 14.It would be incorrect to suppose that ifais in the domain off, then it always happens that lim x!af(x) =f(a). Consider the two functionsfandg dened by f(x) =x2+ 1 ifjxj>0

2 ifx= 0

g(x) =x2+ 1 ifx0; x21 ifx <0: Both these functions are dened on the whole real line; i.e., domainf= domain g=R(see Figure??). Furthermore, f(0) = 2 andg(0) = 1: Asxapproaches 0, however, it is clear thatx2+ 1 approaches 1 and not 2. Hence lim x!0f(x) = 16=f(0): [Note that in computing lim x!af(x), we consider values off(x) for allxarbitrarily close toabut not equal toa. This point will be made explicit when we give the formal denition.] Turning tog, we see that the value ofg(x) near 0 depends on whetherx is positive or negative. For any small positive numberx, the corresponding number g(x) is close to 1, but ifxis small in absolute value and negative, theng(x) is close to1. Since there is no reason to prefer numbers of one sign to those of the other, we conclude that there is no limit. Thus lim x!0g(x) does not exist: The reader may feel that Example??loses force because the functions used to make the point were in some sense articial. There is some truth in the objection. Recall, however, that one of our major objectives is to reduce the class of all func- tions to those we wish to study in this course. After dening lim x!af(x) precisely,

1.4. LIMITS AND CONTINUITY.39x-axisy-axis

f(0) = 2 f(x) = x + 1 if x = 02 y-axis x-axis g(x) = x +1, if x > 0 g(x) = -x -1, if x < 022Figure 1.23: we shall turn our point of view around and use this denition as the major tool in the problem of deciding what does constitute a well-behaved function. The conceptual problems in trying to give an exact meaning to the expression lim x!af(x) =brevolve around phrases such as \arbitrarily close," \suciently near," and \arbitrarily small." After all, there is no such thing in any absolute sense as asmallpositive real number. The number 0:000001 is small in most contexts, but in comparison with 0:000000000001 it is huge. However, we can assert that one number issmaller thananother. Moreover, the actual closeness of one numberxto another numberais just the distance between them: It isjxaj. One way to say that a functionftakes on values arbitrarily close to a numberbis to state that, for any positive real number, there are numbersxsuch thatjf(x)bj< . We are stating that no matter what positive numberis selected, 1017, or 1017, or 10127, there are numbersxso that the distance betweenf(x) andbis smaller than. Thus the diculty inherent in the phrase \arbitrarily close" has been circumvented by the prex \for any." To nish the denition, we want to be able to say thatf(x) is arbitrarily close tobwheneverxis suciently close, but not equal, toa. What does \suciently close" mean? The answer is this: If an arbitrary >0 is chosen with which to measure the distance betweenf(x) andb, then it must be the case that there is a number >0 such that wheneverxis in the domain offand within a distanceofa, but not equal toa, then the distance betweenf(x) andbis less than. The situation is pictured in Figure??. First >0 is chosen arbitrarily. There must then exist a number >0 such that wheneverxlies in the interval (a;a+), andx6=athen the point (x;f(x)) lies in the shaded rectangle. We summarize by giving the denition: Letfbe a real-valued function of a real

40CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVESab

(a,b)

δ δ

ε

εFigure 1.24:

variable. Thenthe limit asxapproachesaoff(x)isb, written lim x!af(x) =b; if, for any >0, there exists >0 such that wheneverxis in the domain off and 00 such that, for any >0, there is a numberxin the domain ofgsuch that 0Hencejg(x)1j>212 .

The basic limit theorem is the following:

1.4.1.Iflimx!af(x) =b1andlimx!ag(x) =b2, then

(i) lim x!a[f(x) +g(x)] =b1+b2: (ii) lim x!acf(x) =cb1: (iii) lim x!af(x)g(x) =b1b2: (iv) lim x!af(x)g(x)=b1b

2providedb26= 0.

1.4. LIMITS AND CONTINUITY.41y-axis

x-axis

ε = 1/2

ε

δ δ

(-δ/2, -δ /4 -1) 2

This distance is

greater than 1/2Figure 1.25: The proofs are given in Appendix A. They are not dicult, and (i) and (ii) espe- cially follow directly from the denition of limit and the properties of the absolute value. Some ingenuity in algebraic manipulation is required for (iii) and (iv). Note that we have already assumed that this theorem is true. For example, the assertion that lim x!3(2x2x2) = 13 is a corollary of (i), (ii), and (iii). If a functionfis dened for everyxinRand if its graph contains no breaks, then it is apparent from looking at the graph that lim x!af(x) =f(a). Logically, however, this intuitive point of view is backward. So far, we have constructed the graph of a functionfby plotting a few isolated points and then joining them with a smooth curve. In so doing we are assuming that ifxis close toa, thenf(x) is close tof(a). That is, we are assuming that limx!af(x) =f(a). Now that we have given a formal denition of limit, we shall reverse ourselves and use it to say precisely what is meant by a function whose graph has no breaks. Sueh a function is ealled continuous. The denitions are as follows: A real-valued functionfof a real variable iscontinuous ataifais in the domain offand limx!af(x) =f(a). The functionfis simply said to becontinuousif it is continuous at every number in its domain. A continuous function whose domain is an interval is one whose graph has no breaks, but the graph need not be a smooth curve. For example, the function with

42CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

the sawtooth graph shown in Figure??is continuous.y-axis x-axisFigure 1.26: Many functions that are not continuous fail to be so at only a few isolated places. Thus the functionfin Example??, whose graph is drawn in Figure??(a), has its only discontinuity at 0. It is continuous everywhere else. Finally, we emphasize the fact that there are two conditions in the denition of continuity. Even though lim x!1x31x1= 3, the functionx31x1is not continuous atx= 1 simply because it is not dened there. If two functionsfandgare continuous ata, then it is not dicult to prove that the sumf+gis also continuous ata. To begin with,ais in the domain off+gsince we have (f+g)(a) =f(a) +g(a). Furthermore, we know that limx!af(x) =f(a) and that lim x!ag(x) =g(a). It follows by Theorem??(i) that lim x!a[f(x) +g(x)] =f(a) +g(a):

Sincef(x) +g(x) = (f+g)(x), we get

lim x!a(f+g)(x) = (f+g)(a); which proves the continuity off+gata. The other parts of the basic limit theorem ??imply similar results about the products and quotients of continuous functions.

We summarize these in

1.4.2.If two functionsfandgare continuous ata, then so are

(i)f+g. (ii)cf;for any constantc. (iii)fg. (iv)fg ;providedg(a)6= 0. A real-valued functionfof one real variable is called apolynomialif there exist a nonnegative integernand real numbersa0;a1;:::;ansuch that, for every real numberx, f(x) =a0+a1x+:::+anxn: The following functions are all examples of polynomials: f(x) = 24x+ 3x2; f(y) = 117y239+32 y+; f(x) =x; f(x) = 5; g(s) = (s2+ 2)(s51) =s7+ 2s5s22:

1.4. LIMITS AND CONTINUITY.43

It is equally important to be able to recognize that a given function is not a poly- nomial. Examples of functions which are not polynomials are f(x) =jxj; f(x) =1x ; f(x) =x2+x+ 3x2; f(x) =px;

F(y) = (y21)3=2:

Algebraically the set of all polynomials is much like the set of integers: The sum, dierence, and product of any two polynomials is again a polynomial, but, in gen- eral, the quotient of two polynomials is not a polynomial. Moreover, the algebraic axioms??through??listed in Section??also hold. Just as a rational number is one which can be expressed as the ratio of two integers, arational functionis one which can be expressed as the ratio of two polynomials. Examples are the functions f(x) =x3+ 2x+ 2x 4+ 1; g(x) =x3=1x 3; f(x) =x2+ 2x+ 1 =x2+ 2x+ 11 ; g(x) =: The domain of every polynomial is the entire setRof real numbers. Similarly, the domain of a given rational function p(x)q(x), wherep(x) andq(x) are polynomials, is the whole setRwith the exception of those numbersxfor whichq(x) = 0.

Furthermore, we have

1.4.3.Every polynomial is a continuous function, and every rational functionp(x)q(x)

is continuous except at those values ofxfor whichq(x) = 0. Proof.The identity functionxis clearly continuous, and so is every constant func- tion. Since every polynomial can be constructed from the identity functionxand from constants using only the sums and products of these and the resulting func- tions, it follows from Theorem??that every polynomial is continuous. The assertion about the continuity of rational functions then follows from part (iv) of Theorem ??.It is occasionally useful to modify the denition of lim x!af(x) to allowxto approachafrom only one side:- eitheror a x x a

44CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

When this is done, we speak of either the limit from the right or the limit from the left and write either lim x!a+f(x) or limx!af(x); according as the additional condition isx > aorx < a. Thus for the function f(x) =x1; x2; x

21; x <2;

whose graph is shown in Figure??, the limit off(x) asxapproaches 2 does not

Figure 1.27:

exist. Nevertheless, we obtain lim x!2+f(x) = 1;limx!2f(x) = 3: Similarly, for the functiongin Figure??(b), we have limx!0+g(x) = 1, limx!0g(x) = 1.

The graph of the rational function

f(x) =x+ 1x = 1 +1x ; x6= 0; together with a list of some of the ordered pairs (x;f(x)) that comprisefis shown in Figure??. From both Figure??and Table??it is clear that asxincreases without bound,f(x) becomes arbitrarily close to 1. We express this fact by writing lim x!+1x+ 1x = 1: Sincef(x) also becomes arbitrarily close to 1 asxdecreases without bound, i.e.,

1.4. LIMITS AND CONTINUITY.45

Figure 1.28:

xf(x)12 101.1

1001.01

1,0001.001

1,000,0001.000001

...Table 1.6: asxincreases without bound, we write lim x!1x+ 1x = 1: The denition is as follows: Letfbe a real-valued function of a real variable. Then the limit off(x)isbasxincreases without bound, written lim x!+1f(x) =b; if, for any >0, there exists >0 such that wheneverxis in the domain off and < x, thenjf(x)bj< . The analogous denition for limx!1f(x) =bis obvious. The symbols +1and1can also be used to refer to the behavior of the values of the function as well as the independent variable. If, asxapproachesa, the corresponding valuef(x) of the function increases without bound, we may express the fact by writing limx!1f(x) = +1: The reader should be able to attach the correct meanings to the various other

46CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

possibilities: lim x!af(x) =1; lim x!1f(x) = +1; lim x!a+f(x) =1;etc. It is essential to keep in mind that +1and1are not numbers. They are not elements ofR. They are used simply as convenient abbreviations for describing the unbounded characteristics of certain functions. The symbol +1(or simply1) in an expression for a bound will always mean that the quantity referred to increases without limit in the positive direction. Similarly,1always indicates the negative direction. Thus we shall not say lim x!01x =1. But we do say lim x!0+1x =1; lim x!01x =1; lim x!0 1x  =1:

1.4. LIMITS AND CONTINUITY.47

Problems

1. Compute the following limits.

(a) lim x!1x31x1 (b) lim x!2x25x+6x2 (c) lim x!3x25x+6x3 (d) lim x!1 x2x11x1 (e) lim x!0x32 (f) lim x!0px+11x (g) lim x!0jxj (h) lim h!0 1h2h

2+6h21h

2 (i) lim x!0(a+x)3+2(a+x)a32ax (j) lim h!02(x+h)2(x+h)2x2+xh .

2. For each of the following functions, nd those numbers (if any) at which the

function is not continuous. (a)x3+ 3x1 (b)f(x) =jxj (c) x3+x+1x 2x2 (d)g(x) =x33x2x2 (e)px+ 3 (f)h(x) =jxjx (g)f(x) =jxj;jxj 1

2x2;jxj>1

(h)F(x) =1;ifxis rational

0;ifxis irrational

(i)f(x) =x2+ 2x+ 1;ifx6= 1

1;ifx= 1

3. A functionfis said to have aremovable discontinuityif it is not continuous

ata, but can be assigned a valuef(a) [or possibly reassigned a new valuef(a)] such that it becomes continuous there. (a) Locate the removable discontinuities in Problem??. (b) Show that the only discontinuities a rational function can have are ei- ther removable or innite. That is, ifr(x) is a rational function that is not continuous ata, show that eitherais a removable discontinuity or lim x!ajr(x)j= +1.

48CHAPTER 1. FUNCTIONS, LIMITS, AND DERIVATIVES

4. Using Theorem??, prove that if limx!af(x) =b1and limx!ag(x) =b2, then

lim x!a[f(x)g(x)] =b1b2:

5. Show that

(a) lim x!+1f(x) =bif and only if limt!0+f1t
=b. (b) lim x!1f(x) =bif and only if limt!0f1t
=b.

6. Using Problem??, compute

(a) lim x!111+x (b) lim x!13x+1x (c) lim t!14t23t+1t 2 (d) lim t!13t3+7t22t 3+1.

7. True or false?

(a) If lim x!af(x) =b, then limx!a+f(x) =band limx!af(x) =b. (b) If lim x!a+f(x) =band limx!af(x) =b, then limx!af(x) =b.

8. Dene a functionfand draw its graph such that limx!2+f(x) = 2 and

lim x!2f(x

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