Differential And Integral Calculus

Citation preview

N. PISKUNOV

DIFFEREN TIAL and IN TEGRAL CALCULUS

MIR P UBLISHERS MOSCOW

N. PISKUNOV

DIFFERENTIAL

and INTEGRAL CALCULUS

MI R P U B L I S H E R S Mo s c o w 1969

TRANSLATED FROM t h e RU SSIAN BY G. YAN KO VSKY

H. C. ITHCKyHOB ^ H d x D E P E H U H A J lb H O E H H H T E T P A Jlb H O E HCMHCJ1EHM5I

Ha am/iuucKOM tobiKe

CONTENTS P r e f a c e ............................................................................................................

.

11

Chapter /. NUMBER. VARIABLE. FUNCTION

1. 2. 3. 4. 5.

Real Numbers. Real Numbers as Points on aNumber Scale . . . . 13 The Absolute Value of a Real N u m b er........................................................... 14 Variables and C on stan ts.........................................................................................16 The Range of a V a r ia b le .....................................................................................16 Ordered Variables. Increasing and Decreasing Variables. Bounded Variables ..................................................................................................................18 6. Function ..................................................................................................................19 7. Ways of Representing F u n ctio n s....................................................................... 20 8. Basic Elementary Functions. Elementary F u n c t io n s ..................................22 9. Algebraic F u n ctio n s................................. ’ ............................................................ 26 10. Polar Coordinate System .................................................................................. 28 Exercises on Chapter I ................................................................................................ 30 Chapter //. LIMIT. CONTINUITY OF A FUNCTION

1. 2. 3. 4. 5.

The Limit of a Variable. An Infinitely Large V a ria b le..............................32 The Limit of a F u n ctio n ........................................................................................35 A Function that Approaches Infinity. Bounded Functions 38 Infinitesimals and Their Basic P r o p e r tie s ...................................................... 42 Basic Theorems on L im it s ....................................................................................45 sin x

6. The Limit of the Function----- a s * —*• 0 .......................................................50 x 7. The Number e .........................................................................................................51 8. Natural Logarithm s................................................................................................ 56 9. Continuity of F u n ctio n s........................................................................................57 10. Certain Properties of Continuous F u n c t io n s .................................................. 61 11. Comparing Infinitesim als........................................................................................63 Exercises on Chapter 7 7 ................................................................................................ 66 Chapter // / . DERIVATIVE AND DIFFERENTIAL

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1*

Velocity of M o t i o n ................................................................................................ 69 Definition of D eriv a tiv e........................................................................................ 71 Geometric Meaning of the D er iv a tiv e............................................................... 73 Differentiability of Functions ........................................................................... 74 Finding the Derivatives of Elementary Functions. The Derivative of the Function y —xnt Where n Is Positive and Integral . . . . . . . 76 Derivatives of the Functions y = s m x \ y = c o s x .......................................... 78 Derivatives of: a Constant, the Product of a Constant by a Function, a Sum, a Product, and a Q u o tie n t................................................................... 80 The Derivative of a LogarithmicF u n c t io n .....................................................84 The Derivative of a Composite Function . ...................................................85 Derivatives of the Functions y = ianx, y = cot xt y — In | x | ..................... 83

Contents

4

11. An Implicit Function and Its D iffe re n tia tio n ..............................................89 12. Derivatives of a Power Function for an Arbitrary Real Exponent, of an Exponential Function, and a Composite Exponential Function . . 91 13. An Inverse Function and Its Differentiation..................................................94 14. Inverse Trigonometric Functions and Their D ifferen tia tio n ..................... 98 15. Table of Basic Differentiation F o r m u la s .................................................... 102 16. Parametric Representation of a F u n c t io n .....................................................103 17. The Equations of Certain Curves in Parametric F o r m ............................105 18. The Derivative of a Function Represented Parametrically ................... 108 19. Hyperbolic F u n c tio n s......................................................................................... 110 20. The D ifferential......................................................................................................113 21. The Geometric Significance of the D ifferential............................................ 117 22. Derivatives of Different O r d ers......................................................................... 118 23. Differentials of Various O rd ers......................................................................... 121 24. Different-Order Derivatives of Implicit Functions and of Functions Represented P ara m etrica lly ............................................................................. 122 25. The Mechanical Significance of the Second D e r iv a t iv e ............................124 26. The Equations of a Tangent and of a Normal. The Lengths of the Subtangent and the Subnorm al......................................................................... 126 27. The Geometric Significance of the Derivative of the Radius Vector with Respect to tne Polar A n g le ..................................................................... 129 Exercises on Chapter I I I ..............................................................................................130 Chapter IV . SOME THEOREMS ON DIFFERENTIABLE FUNCTIONS

1. A Theorem on the Roots of a Derivative (Rollers Theorem) . . . . 140 2. A Theorem on Finite Increments (Lagrange's T heorem )............................142 3. A Theorem on the Ratio of the Increments of Two Functions ......................................... '...............................................143 (Cauchy's Theorem) 4. The Limit of a Ratio of Two Infinitesimals ^Evaluation of Indeter­ minate Forms of the Type

J

.....................................................................144

5. The Limit of a Ratio of Two Infinitely Large Quantities ^Evaluation of Indeterminate Forms of

00

the Type —

\

00J

147

6. Taylor's F orm u la..................................................................................................152 7. Expansion of the Functions e*t sinx, and cos x in a Taylor Series . 156 Exercises on Chapter I V ............................................................................................. 159 Chapter V. INVESTIGATING THE BEHAVIOUR OF FUNCTIONS

1. 2. 3. 4.

Statement of the P rob lem ................................................................................. 162 Increase and Decrease of a F u n c t io n .............................................................163 Maxima and Minima of F u n c tio n s .................................................................164 Testing a Differentiable Function for Maximum and Minimum with a First D e r iv a t iv e ...............................................................................................171 5. Testing a Function for Maximum and Minimum with a Second D e r iv a tiv e ...............................................................................................................174 6. Maxima and Minima of a Function on an Interval ................................ 178

Con tenls

5

7. Applying the Theory of Maxima and Minima of Functions to the Solution of P r o b le m s .......................................................................................... 179 8. Testing a Function for Maximum and Minimum by Means of Taylor's F o r m u la ................................................................................................................... 181 9. Convexity and Concavity of a Curve. Points of I n f le c t io n ....................183 10. Asymptotes .......................................~.................................................................. 189 11. General Plan for Investigating Functions and Constructing Graphs 194 12. Investigating Curves Represented P a ra m etrica lly ...................................... 199 Exercises on Chapter V .................................................................................. ... . 203 Chapter VI. THE CURVATURE OF A CURVE

1. 2. 3. 4. 5.

The Length of an Arc and Its D er iv a tiv e......................................... 208 Curvature ............................ 210 Calculation of C urvature..................................................................................... 212 Calculation of the Curvature of a Line Represented Parametrically . . 215 Calculation of the Curvature of a Line Given by an Equation of Polar C oordinates................................................................................................. 215 6. The Radius and Circle of Curvature. Centre of Curvature. Evolute and I n v o lu t e .................................................................................................................. 217 7. The Properties of an E v o l u t e .............................................................. 221 8. Approximating the Real Roots of an E q u a t io n ........................................225 Exercises on Chapter VI ........................................................................-. . . . 229 Chapter VII. COMPLEX NUMBERS. POLYNOMIALS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Complex Numbers. Basic D efin itio n s............................................................ 233 Basic Operations on Complex Numbers : .................................................... 234 .................................................... 237 Powers and Roots of Complex Numbers Exponential Function with Complex Exponent and ItsProperties . . 240 Euler's Formula. The Exponential Form of a ComplexNumber . . . 243 Factoring a P o ly n o m ia l.....................................................................................244 The Multiple Roots of a P o ly n o m ia l....................................... 247. Factorisation of a Polynomial in the Case of Complex Roots . . . . 248 Interpolation. Lagrange'sInterpolation F orm u la.......................................... 250 On the Best Approximation of Functions by Polynomials. Chebyshev's T h e o r y ....................................... ' ............................................................................252 Exercises on Chapter V I I ......................................................................................... 253 Chapter VIII. FUNCTIONS OF SEVERAL VARIABLES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Definition of a Function of Several V a r ia b le s............................................255 Geometric Representation of a Function of Two V a r ia b le s ................... 255 Partial and Total Increment of a F u n c t io n ................................................259 Continuity of a Function of Several V a ria b les............................................260 Partial Derivatives of a Function of Several V aria b les........................... 263 The Geometric Interpretation of the Partial Derivatives of a Func­ tion of Two V ariab les.........................................................................................264 Total Increment and Total Differentials ....................................................265 Approximation by Total D ifferentials............................................................ 268 Error Approximation by D ifferentials............................................................270 The Derivative of a Composite Function. The Total Derivative . . . 273 The Derivative of a Function Defined I m p l ic it ly ................................... 276 Partial Derivatives of Different O rd ers........................................................ 279

Contents

6

13. 14. 15. 16. 17. 18.

Level S u r f a c e s .................................................................................................. . 283 Directional D e r iv a t iv e s ............................. . ....................................................284 G ra d ie n t................................................................................................................... 286 Taylor's Formula for a Function of Two V aria b les....................................290 Maximum and-Minimum of a Function of Several Variables . . . . 292 Maximum and Minimum of a Function of Several Variables Related by Given Equations (Conditional Maxima and M in im a )............................300 19. Singular Points of a C u r v e ..................................................................................305 Exercises on Chapter V I I I .......................................................................................... 310 Chapter IX. APPLICATIONS OF DIFFERENTIAL CALCULUS TO SOLID GEOMETRY

1. The Equations of a Curve in S p a c e ................................................................. 314 2. The Limit and Derivative of the Vector Function of a Scalar Argument. The Equation of a Tangent to a Curve. The Equation of a Normal Plane .......................................................................................................317 3. Rules for Differentiating Vectors(Vector F u n c tio n s).................................. 322 4. The First and Second Derivatives of a Vector with Respect to the Arc Length. The Curvature of a Curve. The Principal Normal . . . . 324 331 5. Osculating Plane. Binormal.T o r s io n ....................................... 6. A Tangent Plane and Normal to a S u r f a c e .................................................336 Exercises on Chapter I X .............................................................................................. 340 Chapter X, INDEFINITE INTEGRALS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Antiderivative and the Indefinite In te g ra l.....................................................342 Table of In teg ra ls.................................................................................................. 344 Some Properties of an Indefinite In te g ra l.....................................................346 Integration by Substitution (Change of Variable) ............................... 348 Integrals of Functions Containing a Quadratic T rin o m ia l........................351 Integration by Parts ..........................................................................................354 Rational Fractions. Partial Rational Fractions andTheir Integration 357 Decomposition of a Rational Fraction into PartialFractions . . . . 361 Integration of Rational F ra c tio n s.................... 365 Ostrogradsky's M e th o d .......................................................................................... 368 371 Integrals of Irrational F u n c tio n s ............................ Integrals of the Form ^ R (xt Y a x 2-p bx + c) d x ........................................ 372 Integration of Binomial D iffe r e n tia ls.............................................................375 Integration of Certain Classes of Trigonometric F u n c t io n s ....................378 Integration of Certain Irrational Functions by Means of Trigonometric S u b stitu tio n s...........................................................................................................383 16. Functions Whose Integrals Cannot Be Expressed in Terms of Elementary F u n ction s.......................................................................................... 385 Exercises on Chapter X .............................................................................................. 386 Chapter X t. THE DEFINITE INTEGRAL

1. 2. 3. 4. 5.

Statement of the Problem. The Lower and. Upper Integral Sums . . . 396 The Definite Integral ............................................................................. 398 Basic Properties oT the Definite In te g ra l......................................................404 Evaluating a Definite Integral.Newton-Leibniz F orm ula......................... 407 Changing the Variable in theDefinite In teg ra l............................................ 412

Contents

7

(5. Integration by P a r t s ......................................................................................... 413 7. Improper In te g ra ls............................................................................................. 416 8. Approximating Definite I n te g r a ls ..................................................................424 9. Chebyshev’s F orm u la......................................................................................... 430 10. Integrals Dependent on a P aram eter............................................................ 435 Exercises on Chapter X I ............................................................................................. 438 Chapter X II. GEOMETRIC AND MECHANICAL APPLICATIONS OF THE DEFINITE INTEGRAL

1. 2. 3. 4.

Computing Areas in Rectangular C oordinates.............................................. 442 The Area of a Curvilinear Sector in Polar C oordinates...................* . 445 The Arc Length of a C u r v e ..............................................................................447 Computing the Volume of a Solid from the Areas of Parallel Sections (Volumes by S lic in g )......................................................................................... 453 5. The Volume of a Solid of Revolution ................................................. . 455 6. The Surface of a Solid of R evolu tion ..............................................................455 7. Computing Work by the Definite In te g r a l..................................................457 8. Coordinates of the Centre of G r a v ity ..............................................................459 Exercises on Chapter X I I .........................................................................................462 Chapter X III. DIFFERENTIAL EQUATIONS

1. Statement of the Problem. The Equation of Motion of a Body with Resistance of the Medium Proportional to the Velocity. The Equation of a C a te n a r y .....................................................................................469 2. Definitions .............................................................................................................472 3. First-Order Differential Equations (General N o t i o n s ) ..............................473 4. Equations with Separated and Separable Variables. The Problem of the Disintegration of R a d iu m ......................................................................... 478 5. Homogeneous First-Order E q u a tio n s............................................................. 482 6. Equations Reducible to Homogeneous E q u a tio n s ......................................484 7. First-Order Linear Equations ......................................................................... 487 8. Bernoulli’s Equation .........................................................................................490 9. Exact Differential E q uation s............................................................................. 492 10. Integrating F a c to r .................................................................................................495 11. The Envelope of a Family of C u rves..............................................................497 12. Singular Solutions of a First-Order Differential E q u a tio n ...................... 504 13. Clairaut’s E q u a t io n .............................................................................................505 14. Lagrange’s E q u a tio n .............................................................................................507 15. Orthogonal and Isogonal Trajectories............................................................. 509 16. Higher-Order Differential Equations (Fundam entals)..................................514 17. An Equation of the Form y'n) = f ( x ) ..................... 516 18. Some Types of Second-Order Differential Equations Reducible to First-Order E q u ation s............................................................................ . . 518 19. Graphical Method of Integrating Second-Order Differential Equations 527 20. Homogeneous Linear Equations. Definitions and General Properties 528 21. Second-Order Homogeneous Linear Equations with Constant Coeffi­ cients ................... ( .................................................................................................. 535 22. Homogeneous Linear Equations of the nth Order with Constant C o efficie n ts............................................................................................................ 539 23. Nonhomogeneous Second-Order Linear E q u ation s......................................541 24. Nonhomogeneous Second-Order Linear Equations with Constant C o efficie n ts............................................................................................................ 545

8

Contents

25. 26. 27. 28. 29. 30. 31. 32.

Higher-Order Nonhomogeneous Linear E q u a t io n s ....................................... 551 The Differential -Equation of Mechanical V ib ra tio n s............................... 555 Free O scillation s................ .................................................................................... 557 Forced Oscillations .............................................................................................559 Systems of Ordinary Differential E q u a tio n s................................................... 563 Systems of Linear Differential Equations with Constant Coefficients 569 On Lyapunov’s Theory of S ta b ilit y ............................................................... 576 Euler’s Method of Approximate Solution of First-Order Differential E q u a tio n s.......................................................................................... ...................... 581 33. A Difference Method for Approximate Solution of Differential Equa­ tions Based on Taylor’s Formula. Adams M e th o d ................................... 584 34. An Approximate Method for Integrating Systems of First-Order Differential E q u a tio n s............................................................................................591 Exercises on Chapter X I I I ............................................................................................595 Chapter X IV . MULTIPLE INTEGRALS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Double In te g ra ls........................................ 608 Calculating Double In te g ra ls............................................................................... 610 Calculating Double Integrals (C ontinued).......................................................617 Calculating Areas and Volumes by Means of Double Integrals . . . . 623 The Double Integral in Polar C o o r d in a te s ................................................... 626 Changing Variables in a Double Integral (General C a s e ) .......................633 Computing the Area of aS u r fa c e ...................................................................... 638 The Density of Distribution of Matter and the DoubleIntegral . . . 642 The Moment of Inertia of the Area of a Plane F ig u r e ...........................643 The Coordinates of the Centre of Gravity of the Area ofa Plane Figure 648 ' Triple In teg ra ls...................................................................................................650 Evaluating a Triple In te g ra l............................................................................... 651 Change of Variables in a Triple In te g r a l....................................................... 656 The Moment of Inertia and the Coordinates of the Centre of Gravity of a S o l i d ................................................................................................................... 660 15. Computing Integrals Dependent on a P a ram eter....................................... 662 Exercises on Chapter XIV . . . . - ........................................................................... 663 Chapter XV. LINE INTEGRALS AND SURFACE INTEGRALS

1. 2. 3. 4.

Line In te g ra ls........................................................................................................... 670 Evaluating a Line In te g r a l............................................................................... 673 Green’s F orm u la....................................................................................................... 679 Conditions for a Line Integral Being Independent of the Path of I n te g r a tio n ............................................................................................................... 681 5. Surface Integrals ................................................................................................... 687 6. Evaluating Surface In te g r a ls............................................................................... 689 7. Stokes’ F orm u la.......................................................................................................692 8. Ostrogradsky’s F o r m u la .......................................................................................697 9. The Hamiltonian Operator and Certain Applications of I t ...................700 Exercises on Chapter X V ...........................................................................................703 Chapter X V I. SERIES

710 1. Series. Sum of a S e r i e s ................................................................................ 2. Necessary Condition for Convergence of a S e r i e s .......................................713 3. Comparing Series with Positive T e r m s................................., . ......................716

Contents

9

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

D’Alembert's T e s t ....................................................................................................718 Cauchy’s T e s t ............................................................................................................721 The Integral Test for Convergence of a S e r ie s ........................................... 723 Alternating Series. Leibniz’ T h eo rem ............................................................... 727 PIus-and-Minus Series. Absolute and Contitional Convergence . . . . 729 Functional S e r i e s ................................................................................................... 733 Majorised S e r ie s ....................................................................................................... 734 The Continuity of the Sum of a S e r ie s ........................................................... 736 Integration and Differentiation of S e r i e s ....................................................... 739 Power Series. Interval of C onvergence...........................................................742 Differentiation of Power S e r ie s ....................................................................... 747 Series in Powers of x — a ................................................................................... 748 Taylor’s Series and Maclaurin’s S e r i e s ........................................................ 750 Examples of Expansion of Functions in S e r i e s ...........................................751 Euler’s F o rm u la ....................................................................................................... 753 The Binomial S e r ie s ........................................................................................... 754 Expansion of the Function In (1 + x ) in a Power Series. Computing L o g a r ith m s............................................................................................................... 756 21. Integration by Use of Series (Calculating Definite Integrals) . . . . 758 22. Integrating Differential Equations by Means of Series . . . . . . . 760 23. Bessel's E q u a tio n ........................ .. ................................................................ 763 Exercises on Chapter X V I ............................................................................................768 Chapter X V II. FOURIER SERIES

1. Definition. Statement of the P ro b le m ............................................................... 776 2. Expansions of Functions in Fourier S e r ie s ................................................... 780 3. A Remark on the Expansion of a Periodic Function in a Fourier Series . . . . -........................................................................................................... 785 4. Fourier Series for Even and Odd F u n c tio n s............................................... 787 5. The Fourier Series for a Function with Period 2 / ................................... 789 6. On the Expansion of a Nonperiodic Function in a Fourier Series . . 791 7. Approximation by a Trigonometric Polynomial of a Function Represented in the Mean ..................................................................................792 8. The Dirichlet In te g r a l........................................................................................... 798 9. The Convergence of a Fourier Series at a Given P o i n t ...........................801 10. Certain Sufficient Conditions for the Convergence of a Fourier Series 802 11. Practical Harmonic A n a ly s is ............................................................................... 805 12. Fourier I n te g r a l....................................................................................................... 810 13. The Fourier Integral in Complex F o r m ....................................................... 810 Exercises on Chapter X V I I ........................................................................................... 812 Chapter X V I I I . EQUATIONS OF MATHEMATICAL PHYSICS

1. Basic Types of Equations of Mathematical Physics . ................................815 2. Derivation of the Equation of Oscillations of a String. Formulation of the Boundary-Value Problem. Derivation of Equations of Electric Oscillations in W i r e s ........................................................................................... 816 3. Solution of the Equation of Oscillations of a String by the Method of Separation of Variables(The Fourier M e th o d )..........................................820 4. The Equation for Propagation of Heat in a Rod. Formulation of the Boundary-Value P ro b le m ................................................ 823 5. Heat Propagation in S p a c e ...............................................................................825 6. Solution of the First Boundary-Value Problem for the HeatConductivity Equation bythe Method of Finite D ifferences..................... 829

10

Contents

7. Propagation ol Heat in an Unbounded R o d ........................................... 831 8. Problems That Reduce to Investigating Solutions of the Laplace Equation. Stating Boundary-Value P r o b le m s ........................................... 836 9. The Laplace Equation in Cylindrical Coordinates. Solution of the Dirichlet Problem for a Ring with Constant Values of the Desired Function on the Inner and Outer Circumferences................................... 841 10. The Solution of Dirichlet’s Problem for a C ir c le ............................... 843 11. Solution of the Dirichlet Problem by the Method of Finite Differences 847 Exercises on Chapter X V I I I .................................................................................. 850 Chapter X IX . OPERATIONAL CALCULUS AND CERTAIN OF ITS APPLICATIONS

1. The Initial Function and Its T r a n sfo r m ................................................... 854 2. Transforms of the Functions.cr0(/), sin/, c o s / ....................................... 855 3. The Transform of a Function with Changed Scale of the Independent Variable. Transforms of the Functions sin at, cos a t ............................... 856 4. The Linearity Property of a Transform .......................................................857 5. The Shift Theorem ...................................................................................... ... 858 6. Transforms of the Functions e~at, sinh at, cosh at, e~ai sin at, e~ a/ cos at 858 7. Differentiation of Transforms.......................................................................860 8. The Transforms of D er iv a tiv es.......................................................................861 9. Table of Transform s............................................................. ~ . .......................862 10. An Auxiliary Equation for a Given Differential E q u a tio n ................864 11. Decomposition Theorem . ............................................................................. 867 12. Examples of Solutions of Differential Equations and Systems of Differential Equations by the Operational /M ethod............................... 869 13. The Convolution T h e o r e m .............................................................................. 871 14. The Differential Equations of Mechanical Oscillations. The Differen­ tial Equations of Electric-Circuit T h eo r y ............................................... 873 15. Solution of the Differential Oscillation E q u a tio n ................................... 874 16. Investigating Free O scilla tio n s.................................................................. 875 17. Investigating Mechanical and Electrical Oscillations in the Case of a Periodic External F o r c e .................................................................................. 876 18. Solving the Oscillation Equation in the Case of Resonance . . . . 878 19. The Delay T heorem .......................................................................................... 879 Exercises on Chapter X I X ...................................................................................... 880 Sub[ect Index

PREFACE This text is designed as a course of mathematics for higher technical schools. It contains many worked examples that illustrate the theoretical material and serve as models for solving problems. The first two chapters “Number. Variable. Function’* and “Limit. Conti* nuity of a Function” have been made as short as possible. Some of the ques­ tions that are usually discussed in these chapters have been put in the third and subsequent chapters without loss of continuity. This has made it possible to take up very early the basic concept of differential calculus— the deriva­ tive— which is required in the study of technical subjects. Experience has shown this arrangement of the material to be the best and most convenient for the student. A large number of problems have been included, many of which illust­ rate the interrelationships of mathematics and other disciplines. The problems are specially selected (and in sufficient number) for each section of the course thus helping the student to master the theoretical material. To a large extent, this makes the use of a separate book of problems unnecessary and extends the usefulness of this text as a course of mathematics for self-instruction. N. S. Piskunov

CHAPTER I NUMBER. VARIABLE. FUNCTION SEC. 1. REAL NUMBERS. REAL NUMBERS AS POINTS ON A NUMBER SCALE

Number is one of the basic concepts of mathematics. It originated in ancient times and has undergone expansion and generalisation over the centuries. Whole numbers and fractions, both positive and negative, together with the number zero are called rational numbers. Every rational number may be represented in the form of a ratio, —, of two integers p and q; for example,

In particular, the integer p may be regarded as a ratio of the integers -- ; for example,

Rational numbers may be represented in the form of periodic terminating or nonterminating fractions. Numbers represented by nonterminating, but nonperiodic, decimal fractions are called irrational numbers', such are the nurfibers Y 2, V 3, 5 —]/2, etc. The collection of all rational and irrational numbers makes up the set of real numbers. The real numbers are ordered in magnitude', that is to say, for each pair of real numbers x and y there is one, and only one, of the following relations: x< y,

x = y,

x> y.

Real numbers may be depicted as points on a number scale. A number scale is an infinite straight line on which are chosen: 1) a certain point 0 called the origin, 2) a positive direction indicated by an arrow, and 3) a suitable unit of length. We shall' usually make the number scale horizontal and take the positive direction to be from left to right. If the number xt is positive, it is depicted as a point M, at a distance OMt = x, to the right of the origin O', if the number xx is negative, it is represented by a point M, to the left of 0 at a

14

Number. Variable . Function

distance 0M 2= — x2 (Fig. 1). The point 0 represents the number zero. It is obvious that every real number is represented by a definite point on the number scale. Two different real numbers are represented by different points on the number scale. The following assertion is also true: each point on the number scale represents only one real number (rational or irrational). To summarise, all real numbers and all points on the number scale are in one-to-one correspondence: to each number there cor­ responds only one point, and conversely, to each point there cor­ responds only one number. This frequently enables us to regard “the number x” and “the point x” as, in a certain sense, equivalent expressions. We shall make wide use m2 , , | | t Mi of this circumstance in our course. ' -2-1 12 3 ° We state without proof the followFia j ing important property of the set ** ' of real numbers: both rational and irrational numbers may be found between any two arbitrary real numbers. In geometrical terms, this proposition reads thus: both rational and irrational points may be found between any two arbitrary points on the number scale. In conclusion we give the following theorem, which, in a certain sense, represents a bridge between theory and practice. Theorem. Every irrational number a may be expressed, to any degree of precision, with the aid of rational numbers. Indeed, let the irrational number a > 0 and let it be required to evaluate a with an accuracy of ~ ^for example, and so forth No matter what a is, it lies between two integral numbers N and N + l . We divide the segment between N and N 1 into n parts; then a will lie somewhere between the rational numbers N + — and Since their difference is equal to , each of them expresses a to the given degree of accuracy, the former being smaller and the latter greater. Example. The irrational number Y 2 is expressed by rational numbers: 1.4 and 1.5 to one decimal place, 1.41 and 1.42 to two decimal places, 1.414 and 1.415 to three decimal places, etc. SEC. 2. THE ABSOLUTE VALUE OF A REAL NUMBER

Let us introduce a concept which we shall need later on: the absolute value of a real number.

The Absolute Value of a Real Number

15

Definition. The absolute value (or modulus) of a real number x (written \x\) is a nonnegative real number that satisfies the con­ ditions \x\ = x i f * ^ 0; \x\ = — x i f a : < 0. Examples. |2 | = 2;

| — 5 | = 5;

|0 | = 0 .

From the definition it follows that the relationship x x (holds for any x. Let us examine some of the properties of absolute values. 1. The absolute value of an algebraic sum of several real numbers is no greater than the sum of the absolute values of the terms Proof. Let Jt + t/SsO, then l* + «/l = * + l / < l * l +\ y \ (since and t/< [r/|). Let x -j- y whence 1* 1—I t / K I * —yl thus completing the proof. 3. The absolute value of a product is equal to the product of the absolute values of the factors: \xyz I= l*| \y\ M4. The absolute value of a quotient is equal to the quotient of the absolute values of the dividend and the divisor: |£ I

y

\y\'

The latter two properties follow directly from the definition of absolute value.

16

Number. Variable. Function

SEC. 3. VARIABLES AND CONSTANTS

The numerical values of such physical quantities as time, length, area, volume, mass, velocity, pressure, temperature, etc., are deter­ mined by measurement. Mathematics deals with quantities divested of any specific content. From now on, when speaking of quantities, we shall have in view their numerical values. In various phenomena, the numerical values of certain quantities vary, while the numerical values of others remain fixed. For instance, in uniform motion of a point, time and distance change, while the velocity remains constant. A variable is a quantity that takes on various numerical values. A constant is a quantity whose numerical values remain fixed. We shall use the letters *, y, z, u.........etc., to designate variables, and the letters a, b, c, . . . ,etc., to designate constants. Note. In mathematics, a constant is frequently regarded as a special case of variable whose numerical values are the same. It should be noted that when considering specific physical pheno­ mena it may happen that one and the same quantity in one pheno­ menon is a constant while in another it is a variable. For example, the velocity of uniform motion is a constant, while the velocity of uniformly accelerated motion is a variable. Quantities that have the same value under all circumstances are called absolute constants.. For example, the ratio of the circumference of a circle to its dia­ meter is an absolute constant: « = 3.14159. As we shall see throughout this course, the concept of a variable quantity is the basic concept of differential and integral calculus. In “Dialectics of Nature”, Friedrich Engels wrote: “The turning point in mathematics was Descartes’ variable magnitude. With that came motion and hence dialectics in mathematics, and at once, too, of necessity the differential and integral calculus.” SEC. 4. THE RANGE OF A VARIABLE

A variable takes on a series of numerical values. The collection of these values may differ depending on the character of the prob­ lem. For example, the temperature of water heated under ordinary conditions will vary from room temperature (15-18°C) to the boiling point, 100°C. The variable quantity x = cos a can take on all values from— 1 t o + 1. The values of a variable are geometrically depicted as points on a number scale. For instance, the values of the variable x = cos a for all possible values of a are depicted as the set of points of an interval on the number scale, from — 1 to 1, including the points —1 and 1 (Fig. 2).

The Range of a Variable

17

Definition. The set of all numerical values of a variable quantity is called the range of the variable. We shall now define the following ranges of a variable that will be frequently used later on. An open interval is the collection of all numbers x lying between and excluding the given numbers a and b (a 3) y = V 1—x 2; 4) y = sin*; 5) Q== nR 2, and so forth. Here, the functions are represented analytically by means of a single formula (a formula is understood to be the equality of two analytical expressions). In such cases one may speak of the natural domain of definition of the function. The set of values of * for which the analytical expression on the right-hand side has a fully definite value is the natural domain

22

Number . Variable. Function

of definition of a function represented analytically. Thus, the natu­ ral domain of definition of the function y — x4,—2 is the infinite interval—o o < jc < o o , because the function is defined for all x 4-1 values of x. The function **/ = x-— is defined for all values of xf —1 with thfe exception of x = l t because for this value of x the deno­ minator vanishes. For the function y = V I —*2, the natural domain 2 of definition is the closed interval— I ^ j k ^ I , y*x and so on. I Note. It is sometimes necessary to consider / only a part of the natural domain of a function, / and not the whole domain. For instance, the I 1 dependence of the area Q of a circle upon the /; radius R is defined by the function Q = nR2. / The domain of this function, when considering / a given geometrical problem, is the infinite interval 0 0 . Its graph is shown in Fig. 15. Trigonometric functions. In the formulas # = s in * , etc., the independent variable x is expressed in radians. All the enumerated trigonometric functions are periodic. Let us give a general definition of a periodic function. Definition 1. The function y = f(x) is called periodic if there exists a con­ stant C, which, when added to (or sub­ tracted from) the argument x, does not change the value of the function: f(x-\-C) = f(x). The least such number is called the period of the function; it will henceforward be designated as 21. From the definition it follows directly that //= s in x is a periodic function with a period 2n: sinx = = sin (x + 2jt). The period of cos* is likewise 2n. The functions t/ = tanx and y = cot* have a period equal to n. The functions i/=sinx:, y — cosx are defined for all values of x\ the functions y = ta n x and y = sec* are defined everywhere except the points x = {2k-\-\) — 1, 2, ...) ; the functions j/ = cotx and y = cscx are defined for all values of x except the points x = kn(k = 0, 1, 2, ...) . Graphs of trigonometric functions are shown in Figs. 16, 17, 18, and 19. The inverse trigonometric functions will be discussed in more detail later on.

Basic Elementary Functions. Elementary Functions

25

Fig. 16.

Let us now introduce the concept of a function of a function. If y is a function of u, and u (in turn) is dependent on the var­ iable x, then y is also dependent on x. Let y = F (a) u = 9 (x). We get y as a function of x y = F[ , it is necessary to take into account the quad­ rant in which the point is located and then take the correspond­

ing value of cp. The equation q = F ((p) in polar coordinates defines a certain line. Example 1. Equation Q= a, where a = const, defines in polar coordinates a circle with centre in the pole and with radius a. The equation of this circle in a rectangular coordinate system situated as shown in Fig. 24 is

ji

JT

T

2"

Q 0 ^ 0 .7 8 a

=^r 1.57a

a

2 c > 0 be given; for the inequality | (3 * + H — 7 | < e to be fulfilled it is neces­ sary to have the following inequalities fulfilled: |3 * —6 1 < e, \x —2 \ < y ,

— j < x —2 < y -

0 Thus, given any e, for all values of x satisfying the inequality | x — 2 | < - j = = 5, the value of the function 3 x + l will differ from 7 by less than e. And this means that 7 is the limit of the function as x — >-2.

Note 3. For a function to have alimit as x —-a, it is not ne? cessary that the function be defined at the point x = a.When find­ ing the limit we consider the values of the function in the neighbourhood of the point a that are different from a; this is clearly illustrated in the following case. _4 x^__ 4 Example 2. We shall prove that lim ------ ^ = 4 . Here, the function-------^

X-*-2 X *

x *

is not defined for x = 2. It is necessary to prove that for an arbitrary 8, there will be a 6 such that the following inequality will be fulfilled:

x2—4 x—2

4

2.

Let us now consider certain cases of variation of a function as x-+ o o . Definition 2. The function J(x) approaches the limit b a9k-+ o o if for each arbitrarily small positive number e it is possible to indicate a positive number N such that for all values of x that satisfy the inequality the inequality \ f ( x)— 6 | < e will be fulfilled.

38

Limit. Continuity of a Function

Example 3. To prove that

lim X - > CD

(1+ t)! It is necessary to prove that, for an arbitrary e, the following inequality will be fulfilled

N, where N is determined by the choice of e. Inequality (3) is equivalent to the following inequality: |- ^ - |< e , which will be fulfilled if

1*1 > 7 = " ' And this means that

lim X -+ 0 0

(1 -f —) = \

X }

lim X -+ CD

= l (Fig. 33). X

Knowing the meanings of the symbols x -+ oo and x -+■— oo the meanings ol the following expressions are obvious: uf (x) approaches b as x —►-J-oo” and “/ (x) approaches b as x - + — oo”, or, in symbols, lim f (*) = &,

X-++ 00 lim

* -* -0 0

f(x) = b.

SEC. 3. A FUNCTION THAT APPROACHES INFINITY. BOUNDED FUNCTIONS

We have considered cases when the function f(x) approaches a certain limit b as x-+ a or as x-+ oo. Let us now take the case when the function y = f(x) approaches infinity when the argument varies in some way.

39

A Function that Approaches Infinity. Bounded Functions

Definition 1. The function f (x) approaches infinity a s* -* a, i.e., it is an infinitely large quantity as x-+a, if for each positive number M t no matter how large, it is possible to find a 6 > 0 such that for all values of x different from a and satisfying the condition \x —a |< 6 , we have the inequality \ f ( x ) \ >M. If f(x) approaches infinity as x-+a> we write lim f(x) = oo x~+ a

or f(x)-+ oo as x-~+ a. If f (x) approaches infinity as x-+ a and, in the process, assumes only positive or only negative values, the appropriate notation is lim / (x) = + oo or lim f(x) = — oo. x -* a

x— ►o

Example 1. We shall prove that lim

1 -a = (1-*)

+ oo.

Indeed,

for

M > 0 we will have

(i—*)j> provided ( 1 — X )*
I 1 — * |

M


0 we will have

--- x- > M, provided

1*1=1* —0| 0 for x < 0 and ^

< 0 for x > 0 (Fig. 35).

If the function I (x) approaches infinity as x —►oo, we write

lim / (x) = oo, X —► oo

and we may have the particular cases:

lim / (*) = oo, X —► + 00

lim f(x) = oo, X

- OD

lim

/ ( * ) = — oo.

X —► + QD

For example,

lim x2= +oo,

lim xl = — oo

any

any

Limit. Continuity of a Function

40

Note 1. The function y = f(x) as x -►a or as x-+ oo may not approach a finite limit or infinity.

Example 3. The — oo < x < + oo, as infinity (Fig. 36).

function -f oo,

y = sin* defined on the infinite interval does not approach either a finite limit or

i ^_^2n

~7t\

y=sfnx

0 Fig. 36.

Example 4. The function y = sm j

defined for all values of x, except

x = 0, does not approach either a finite limit or infinity as x -+ 0. The graph of this function is shown in Fig. 37.

Definition 2. The function y = f(x) is called bounded in a given range of the argument x if there exists a positive number M such that for all values of x in the range under consideration the

A Function that Approaches Infinity. Bounded Functions

41

inequality |/(*)|==S M will be fulfilled. If there is no such num­ ber M, the function / ( x) is called unbounded in the given range. Example 5. The function y = sin*, defined in the — oo < * < - ) - oo, is bounded, since for all values of x

infinite

interval

| sin x | < 1 = M .

Definition 3. The function f (x) is called bounded as x —>-a if there exists a neighbourhood with centre at the point a, in which the given function is bounded. Definition 4. The function y = f(x) is called bounded as x —>-oo if there exists a number N > 0 such that for all values of x satisfying the inequality |x |> W , the function f(x) is bounded. The boundedness of a function approaching a limit is decided by the following theorem. Theorem 1. i f lim f(x) = b, where b is a finite number, the x -+a

function f(x) is bounded as x —*a. Proof. From the equality lim f{x) = b it follows that for any x-+ a

there will be a 6 such a— 6 < * < a + 6 the inequality e > 0

that

in the neighbourhood

\f{x)-b\