Tom M Apostol CALCULUS VOLUME One-Variable Calculus, with an Introduction to Linear Algebra SECOND EDITION New York l John Wiley & Sons, Inc Santa Barbara l London l Sydney l Toronto C O N S U L T I N G EDITOR George Springer, Indiana University XEROX @ is a trademark of Xerox Corporation Second Edition Copyright 01967 by John WiJey & Sons, Inc First Edition copyright 1961 by Xerox Corporation Al1 rights reserved Permission in writing must be obtained from the publisher before any part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system ISBN 471 00005 Library of Congress Catalog Card Number: 67-14605 Printed in the United States of America TO Jane and Stephen PREFACE Excerpts from the Preface to the First Edition There seems to be no general agreement as to what should constitute a first course in calculus and analytic geometry Some people insist that the only way to really understand calculus is to start off with a thorough treatment of the real-number system and develop the subject step by step in a logical and rigorous fashion Others argue that calculus is primarily a tool for engineers and physicists; they believe the course should stress applications of the calculus by appeal to intuition and by extensive drill on problems which develop manipulative skills There is much that is sound in both these points of view Calculus is a deductive science and a branch of pure mathematics At the same time, it is very important to remember that calculus has strong roots in physical problems and that it derives much of its power and beauty from the variety of its applications It is possible to combine a strong theoretical development with sound training in technique; this book represents an attempt to strike a sensible balance between the two While treating the calculus as a deductive science, the book does not neglect applications to physical problems Proofs of a11 the important theorems are presented as an essential part of the growth of mathematical ideas; the proofs are often preceded by a geometric or intuitive discussion to give the student some insight into why they take a particular form Although these intuitive discussions Will satisfy readers who are not interested in detailed proofs, the complete proofs are also included for those who prefer a more rigorous presentation The approach in this book has been suggested by the historical and philosophical development of calculus and analytic geometry For example, integration is treated before differentiation Although to some this may seem unusual, it is historically correct and pedagogically sound Moreover, it is the best way to make meaningful the true connection between the integral and the derivative The concept of the integral is defined first for step functions Since the integral of a step function is merely a finite sum, integration theory in this case is extremely simple As the student learns the properties of the integral for step functions, he gains experience in the use of the summation notation and at the same time becomes familiar with the notation for integrals This sets the stage SO that the transition from step functions to more general functions seems easy and natural vii WI Preface Prefuce to the Second Edition The second edition differs from the first in many respects Linear algebra has been incorporated, the mean-value theorems and routine applications of calculus are introduced at an earlier stage, and many new and easier exercises have been added A glance at the table of contents reveals that the book has been divided into smaller chapters, each centering on an important concept Several sections have been rewritten and reorganized to provide better motivation and to improve the flow of ideas As in the first edition, a historical introduction precedes each important new concept, tracing its development from an early intuitive physical notion to its precise mathematical formulation The student is told something of the struggles of the past and of the triumphs of the men who contributed most to the subject Thus the student becomes an active participant in the evolution of ideas rather than a passive observer of results The second edition, like the first, is divided into two volumes The first two thirds of Volume deals with the calculus of functions of one variable, including infinite series and an introduction to differential equations The last third of Volume introduces linear algebra with applications to geometry and analysis Much of this material leans heavily on the calculus for examples that illustrate the general theory It provides a natural blending of algebra and analysis and helps pave the way for the transition from onevariable calculus to multivariable calculus, discussed in Volume II Further development of linear algebra Will occur as needed in the second edition of Volume II Once again acknowledge with pleasure my debt to Professors H F Bohnenblust, A Erdélyi, F B Fuller, K Hoffman, G Springer, and H S Zuckerman Their influence on the first edition continued into the second In preparing the second edition, received additional help from Professor Basil Gordon, who suggested many improvements Thanks are also due George Springer and William P Ziemer, who read the final draft The staff of the Blaisdell Publishing Company has, as always, been helpful; appreciate their sympathetic consideration of my wishes concerning format and typography Finally, it gives me special pleasure to express my gratitude to my wife for the many ways she has contributed during the preparation of both editions In grateful acknowledgment happily dedicate this book to her T M A Pasadena, California September 16, 1966 CONTENTS INTRODUCTION Part Historical Introduction 11.1 1.2 1.3 *1 1.4 1.5 1.6 The two basic concepts of calculus Historical background The method of exhaustion for the area of a parabolic segment Exercises A critical analysis of Archimedes’ method The approach to calculus to be used in this book Part 12.1 2.2 12.3 2.4 2.5 Some Basic Concepts of the Theory of Sets Introduction to set theory Notations for designating sets Subsets Unions, intersections, complements Exercises Part 3 8 10 11 12 12 13 15 A Set of Axioms for the Real-Number System 13.1 Introduction 3.2 The field axioms *1 3.3 Exercises 3.4 The order axioms *1 3.5 Exercises 3.6 Integers and rational numbers 17 17 19 19 21 21 ix X Contents 3.7 Geometric interpretation of real numbers as points on a line 3.8 Upper bound of a set, maximum element, least upper bound (supremum) 3.9 The least-Upper-bound axiom (completeness axiom) 3.10 The Archimedean property of the real-number system 3.11 Fundamental properties of the supremum and infimum *1 3.12 Exercises *1 3.13 Existence of square roots of nonnegative real numbers *1 3.14 Roots of higher order Rational powers *1 3.15 Representation of real numbers by decimals - 22 23 25 25 26 28 29 30 30 Part Mathematical Induction, Summation Notation, and Related Topics 14.1 An example of a proof by mathematical induction 4.2 The principle of mathematical induction *1 4.3 The well-ordering principle 4.4 Exercises *14.5 Proof of the well-ordering principle 4.6 The summation notation 4.7 Exercises 4.8 Absolute values and the triangle inequality 4.9 Exercises *14.10 Miscellaneous exercises involving induction 32 34 34 35 37 37 39 41 43 44 THE CONCEPTS OF INTEGRAL CALCULUS 1.1 The basic ideas of Cartesian geometry 1.2 Functions Informa1 description and examples *1.3 Functions Forma1 definition as a set of ordered pairs 1.4 More examples of real functions 1.5 Exercises 1.6 The concept of area as a set function 1.7 Exercises 1.8 Intervals and ordinate sets 1.9 Partitions and step functions 1.10 Sum and product of step functions 1.11 Exercises 1.12 The definition of the integral for step functions 1.13 Properties of the integral of a step function 1.14 Other notations for integrals 48 50 53 54 56 57 60 60 61 63 63 64 66 69 Contents 1.15 Exercises 1.16 The integral of more general functions 1.17 Upper and lower integrals 1.18 The area of an ordinate set expressed as an integral 1.19 Informa1 remarks on the theory and technique of integration 1.20 Monotonie and piecewise monotonie functions Definitions and examples 1.21 Integrability of bounded monotonie functions 1.22 Calculation of the integral of a bounded monotonie function 1.23 Calculation of the integral Ji xp dx when p is a positive integer 1.24 The basic properties of the integral 1.25 Integration of polynomials 1.26 Exercises 1.27 Proofs of the basic properties of the integral xi 70 72 74 75 75 76 77 79 79 80 81 83 84 SOME APPLICATIONS OF INTEGRATION 2.1 Introduction 2.2 The area of a region between two graphs expressed as an integral 2.3 Worked examples 2.4 Exercises 2.5 The trigonometric functions 2.6 Integration formulas for the sine and cosine 2.7 A geometric description of the sine and cosine functions 2.8 Exercises 2.9 Polar coordinates 2.10 The integral for area in polar coordinates 2.11 Exercises 2.12 Application of integration to the calculation of volume 2.13 Exercises 2.14 Application of integration to the concept of work 2.15 Exercises 2.16 Average value of a function 2.17 Exercises 2.18 The integral as a function of the Upper limit Indefinite integrals 2.19 Exercises 88 88 89 94 94 97 102 104 108 109 110 111 114 115 116 117 119 120 124 CONTINUOUS FUNCTIONS 3.1 3.2 Informa1 description of continuity The definition of the limit of a function 126 127 Contents xii 3.3 The definition of continuity of a function 3.4 The basic limit theorems More examples of continuous functions 3.5 Proofs of the basic limit theorems 3.6 Exercises 3.7 Composite functions and continuity 3.8 Exercises 3.9 Bolzano’s theorem for continuous functions 3.10 The intermediate-value theorem for continuous functions 3.11 Exercises 3.12 The process of inversion 3.13 Properties of functions preserved by inversion 3.14 Inverses of piecewise monotonie functions 3.15 Exercises 3.16 The extreme-value theorem for continuous functions 3.17 The small-span theorem for continuous functions (uniform continuity) 3.18 The integrability theorem for continuous functions 3.19 Mean-value theorems for integrals of continuous functions 3.20 Exercises 130 131 135 138 140 142 142 144 145 146 147 148 149 150 152 152 154 155 DIFFERENTIAL CALCULUS 4.1 4.2 4.3 4.4 4.5 4.6 Historical introduction A problem involving velocity The derivative of a function Examples of derivatives The algebra of derivatives Exercises 4.7 4.8 4.9 4.10 Geometric interpretation of the derivative as a slope Other notations for derivatives Exercises The chain rule for differentiating composite functions 4.11 Applications of the chain rule Related rates and implicit differentiation 4.12 Exercises 4.13 Applications of differentiation to extreme values of functions 4.14 The mean-value theorem for derivatives 4.15 Exercises 4.16 4.17 4.18 4.19 Applications of the mean-value theorem to geometric properties of functions Second-derivative test for extrema Curve sketching Exercises 156 157 159 161 164 167 169 171 173 174 176 cc 179 181 183 186 187 188 189 191 Answers to exercises 653 16.12 Exercises (page 596) (a) The identity matrix Z = (gjk), where 6j, = ifj = k, and iii, = ifj # k (b) The zero matrix = (ajk) where each entry ujk = (c) The matrix (cBjk), where (gjk) is the identity matrix of part (a) (a) [A : ‘j W [: y] (c) [ i % a] (a) -5i + 7j, 9i - 12j (b) [: -y]* [; y] (c) [-: -;]3 [: y] -1 (a) -1 -1 -5 3i + 4j + 4k; nullity 0, rank [u (a) (5,0, -1); nullity 0, rank Cc) T(4i - j + k) = (0, -2); nullity 1, rank (c) (a) el = i, e,=i+j, u -:] (d) e, =j, e,=k, Cb) ea=i, [: w1 w,=(l,O,l), w,=(O,O,2), 1 =(I,I), -:] w2=(1, w,=(O,l,O) = (0, 1, o>, w3 = (090, 1) (a) ( - , -3, -1); nullity 0, rank (b) (cl e, = i , e2 = j - i , w1 = u,o, l), w2 10 (a) e, - e,; nullity 0, rank : L 10 11 ( c ) a=5, b=4 -1) Answers to exercises 654 o - 1 0 0 o - ’ 0 17 [; -;], [; [; 0 -1 -;] 18 -2 o-1 0 16 : -;]y [ -1 o-1 Ï! 7; -‘:] ‘0 0 0 0 0 0 0 I -48 20 O - 0 0 0 (f) 0 Choose (x3, x2, x, 1) as a basis for V, and (x2, x) as a basis for W Then the matrix of TD is [ 06 00 00 16.16 Exercises (page 603) AB = -28 28 6-4 [ a b (a) (a) -15 AC = I 15 0’ a = 9, [ya a and b arbitrarq b = 6, yi c = 1, y!] (b) (b) d = [ :! [ -2a a -2b b ’ (b) a=l, -,i -i] a and b arbitrary b = 6, c = 0, d = -2 Answers to exercises 655 A” = An = -sin nO I1 cas n0 n A” = n(n + 1) 01 n 00 , where b and c are arbitrary, and a is any solution of the equation a2 = -bc where b and c are arbitrary and a is any solution of the equation a2 = - bc 14 (b) (A + B)2 = A2 + AB + BA + B2; (A + B)(A - B) = A2 + BA - AB - B2 (c) For those which commute 16.20 Exercises (page 613) (x, y, z) = ( = C-1,-, 4, :-y, 0) + r(4, - 11, 7,22) -1 12 14 -3 - 5% 13 Fi [ -1 1 -$ -Q 15 I -2 l-2 0 10 0 l-2 656 Answers to exercises - * o-1 1- 0 0 0 -1 0 0 16 -3 0 0 + o-3 0! - 16.21 Miscellaneous exercises on matrices (page 614) P= [ 1 [o i;, c’y],and [I “d, b and c are arbitrary and a is any solution of the where quadratic equation a2 - a + bc = (a) [-: [-: :1* [: I:l* [I: -:1* -:1 [ i :]T [: -:12 [-i I:l? [Ii -:]y INDEX 407 Abel’s partial summation formula, 407 Abel’s test for convergence, 408 Abscissa, 48 Absolute convergence of series, Absolute maximum and minimum, 150 Absolute values, 41, 363 Acceleration, 160, 521 in polar coordinates, 541 normal and tangential components of, 527 Addition formulas for the sine and cosine, 96 Additive property: of arc length, 532 of area, 59 of averages, 119 (Exercise 13) of convergent series, of derivatives, 164 of finite sums, 40 of the integral, 66, 67, 80, 514 of the supremum and infimum, 27 of volume, 112 of work, 115 Alternating series, 403 Analytic geometry, 48, 471 Analytic mode1 of Euclidean geometry, 471 Angles : in a Euclidean space, 564 in n-space, 458 radian measurement of, 102 Angular acceleration, 545 (Exercise 19) Angular speed, 522, 545 (Exercise 19) Angular velocity, 545 (Exercise 19) Antiderivative (primitive), 205 APPOLONIUS, 498 Approximations: by polynomials, 272-304, 575 by trigonometric polynomials, 575 in a Euclidean space, 574 Arbitrary constant, 211, 307 ARBOGAST, LOUIS, 171 Arc cosine, 254 Archimedean property of real numbers, 26 ABEL, NIELS HENRIK, 2-9, 26 Arc length: as an integral, 534 definition of, 530, 531 function, 533 in polar coordinates, 544 (Exercise 4) Arc sine, 253 Arc tangent, 255 Area : and similarity transformations, 92 axiomatic definition of, 57-59 in polar coordinates, 110 of an ordinate set, 75 of a radial set, 110 of a region between two graphs, 88 Argument of a complex number, 363 Arithmetic mean, 46, 117 Associative law : for addition of numbers, 18, 359 for addition of vectors, 447 for composition of functions, 141, 584 for multiplication of numbers, 18, 359 for union and intersection of sets, 14 in a linear space, 551, 552 Asymptote, 190 of a hyperbola, 506 Asymptotically equal, 396 Average, 46, 149 of a function, 117-l 19 rate of change, 160 velocity, 157 weighted, 118 Axes, 48, 197 Axiom(s) : completeness (continuity), field, 17 for a linear space, 551, 552 for area, 58, 59 for the real-number system, 17-25 for volume, 112 least-Upper-bound, 25 order, 20 ARCHIMEDES, 657 658 Index Axiomatic development : of area, 57-59 of inner products, 561 of the real-number system, 17-25 of vector algebra, 551, 552 of volume, 11 l-l 12 BARROW, ISAAC, 157 Base of logarithms, 232 Basis, 327, 466 Bernoulli : differential equation, 312 inequality, 46 (Exercise 14) polynomials, 225 (Exercise 35) BERNOULLI, JOHANN, 235, 292, 305, 331 BERNSTEIN, SERGEI, 437 Bernstein’s theorem, 437 Bessel functions, 443 (Exercise 10) Binary scale, 393 Binomial coefficient, 44, 383, 442 Binomial series, 377, 441 Binomial theorem, 44 (Exercise 4), 378, 442 BOLYAI, JOHANN, 474 BOLZANO, BERNARD, 143 Bolzano’s theorem, 143 BOOLE, GEORGE, 11 Bound: greatest lower, 25 least Upper, 24 Upper and lower, 23-25 Bounded function, 73 Bounded sequence, Bounded set of real numbers, 23 Boundedness of continuous functions, 150 BROUNCKER, WILLIAM, 377, 390 Calculation: of e, 281 of logarithms, 240-242 of rr, 285 (Exercise 10) of square roots, 444 (Exercises 20, 21) Calculus, fundamental theorems of, 202, 205, 515 CANTOR, GEORG, 11, 17 CARDANO, HIERONIMO, Cartesian equation, 49, 475, 494 Cartesian geometry, 48 CAUCHY, AUGUSTIN-LOUIS, 3, 42, 127, 172, 186, 284, 368, 378, 397, 399,411, 452 Cauchy-Schwarz inequality, 42, 452, 563 Cauchy’s mean-value formula, 186 Cauchy’s remainder in Taylor’s formula, 284 CAVALIER& BONAVENTURA, 3, 111 Cavalieri solid, 111 Cavalieri’s principle, 111, 112 ARTHUR, 446 Center of mass, 118 Centrifugal, 522 Centripetal, 522 Chain rule, 174, 514 Characteristic equation, 327 Circle, 49, 521 of convergence, 428 Circular helix, 523 Circular motion, 521 Class of sets, 14 Closed interval, 60 Closure axioms, 551 Coefficient matrix, 605 Column matrix (column vector), 592, 598 Commutative law: for addition of numbers, 18, 359 for addition of vectors, 447 for dot products, 451 for inner products, 561 for multiplication of numbers, 18, 359 for union and intersection of sets, 14 in a linear space, 551 Comparison tests for convergence: of improper integrals, 418 of series, 394-396 Comparison theorem for integrals, 67, 81 Complement of a set, 14 Complex Euclidean space, 562 Complex function, 368 Complex linear space, 552 Complex numbers, 358-373 Complex vector space, 468 Composite function, 140, 584 continuity of, 141 differentiation of, 174, 514 Composition of transformations, 584 Concave function, 122, 189 Conditional convergence, 406 Congruence of sets, 58 Conic sections, 497-507 Conjugate complex number, 364 Constant function, 54 Continuous functions: definition of, 130, 369, 513 integrability of, 153 theorems on, 132, 141-154 Contour lines, 197 Convergence : of improper integrals, 416, 418 of sequences, 379 of series, 384-425 pointwise, 422 tests for, 394-408 uniform, 424 Convex function, 122, 189 CAYLEY, Index Convex set, 112 Coordinates : cylindrical, 543 polar, 108, 540 rectangular, 48, 197 Copernican theory, 545 Cosine function : continuity of, 134, 139 differential equation for, 323 differentiation of, 162 integration of, 100, 207 power series for, 436 properties of, 96 Cotangent function, 103 Crame?s rule, 491 Critical point, 188 Cross product (vector product), 483 Curvature, 537 Curve : definition of, 517 length of, 529-535 nonrectifiable, 530, 536 (Exercise 22) rectifiable, 530 Cycloid, 536 (Exercise 20) Cylindrical coordinates, 543 Damped vibrations, 335 498 Decimal expansion of real numbers, 30, 393 Decreasing function, 76 Decreasing sequence, 381 DEDEKIND, RICHARD, 17 Deductive systems, Definite integral: definition of, 73 properties of, 80, 81 De Moivre’s theorem, 371 (Exercise 5) Dependence, linear, 463, 557 Derivatives : and continuity, 163 functions of one variable, 160 functions of several variables, 199-201 notations for, 160, 171, 172, 199, 200 of complex-valued functions, 369 of higher order, 160, 200 of vector-valued functions, partial, 199-201 theorems on, 164 DESCARTES, RENÉ, 48, 446 Determinant, 486 Difference: of functions, 132 of real numbers, 18 of sets, 14 of vectors, 447 DANDELIN, GERMINAL P., 659 Difference quotient, 157, 159, 517 Differential equations, 305-357 first-order linear, 308 homogeneous first-order, 347-350 power-series solutions of, 439-443 second-order linear, 322 separable, 345 Dimension of a linear space, 559 Direction field, 343 Directrix of conic sections, 500 DIRICHLET, PETER GUSTAV LEJEUNE, 407 Dirichlet’s test for convergence, 407 Discontinuity: infinite, 13 jump, 131 removable, 13 Disjoint sets, 14 Distance: between two planes, 495 between two points, 364, 462 from a point to a line, 476, 477 from a point to a plane, 494 Distributive law : for cross products, 483 for inner products, 451, 561 for numbers, 18, 359 for set operations, 16 (Exercise 10) in a linear space, 552 Divergent improper integral, 416, 418 Divergent sequence, 379 Divergent series, 384 Division : of functions, 55 of numbers, 18, 360 Domain of a function, 50, 53, 196, 512, 578 Dot product (inner product), 451, 469, 562 Duodecimal scale, 393 e (base of natural logarithm): computation of, 281 definition of, 231 irrationality of, 282 Earth, 545 Eccentricity of conic sections, 500 Electric circuits, 317, 336 Element : of a determinant, 486 of a matrix, 592, 598 of a set, 11 Elementary function, 282 Ellipse, 498, 500, 506 Elliptic integral, 535 (Exercise 17) Elliptic reflector, 519 Empty set, 13 Endpoints of an interval, 60 660 Index Envelope, 342 Equality : of complex numbers, 358 of functions, 54 of sets, 12 of vectors, 447, 468 Equipotential lines, 351 Error in Taylor’s formula, 278, 280 EUCLID, 9, 471 Euclidean geometry, 9, 471 Euclidean space, 472, 561 EULER, LEONARD, 231, 377, 396, 405,420 Euler’s constant, 405 Even function, 84 (Exercise 25) Even integer, 28 (Exercise 10) Exhaustion, method of, 2-8 property of area, 59 Existence theorems, 308, 323 Exponential function: complex-valued, 367 definition of, 242 derivative of, 243 integral of, 246 power series for, 436 Extremum: definition of, 182 tests for, 182, 188, 189 Extreme-value theorem for continuous functions, 151 Factorials, 44, 52 Family of curves, 341, 351 FERMAT, PIERRE DE, 3, 156 FERRARI, LODOVICO, FIBONACCI (Leonardo of Pisa), 379 Fibonacci numbers, 46 (Exercise 16), 379 Field axioms, 17 Fixed point, 145 (Exercise 5) Focus of a conic section, 498 FOURIER, JOSEPH, 127, 575 Fourier coefficients, 575 Frequency of simple harmonie motion, 339 Function(s): bounded, 73 characteristic, 64 (Exercise 8) complex-valued, 368 concave, 122 constant, 54 continuous, 130 convex, 122 decreasing, 76 defined by an integral, 120 domain of, 53 elementary, 282 even, 84 (Exercise 25) exponential, 242, 367 factorial, 52 forma1 definition of, 53 gamma, 419, 421 (Exercise 19) greatest-integer, 63 hyperbolic, 25 identity, 51 increasing, 76 informa1 description of, 50-52 integrable, 73 inverse, 146, 252 inverse trigonometric, 253-256 linear, 54 logarithmic, 229-235 monotonie, 76 notation for, 50, 196, 512 odd, 84 (Exercise 25) of several variables, 196 periodic, 95 piecewise linear, 123 piecewise monotonie, 7 polynomial, 55 power, 54 range of, 53 rational, 166, 258-266 real-valued, Riemann zeta, 396 step, 52 trigonometric, 95-107 unbounded, 73 vector-valued, 12 Function space, 553 Functional equation, 227 for the exponential function, 243 for the logarithm, 227 Fundamental theorem of algebra, 362 Fundamental theorems of calculus, 202,205,515 498 Gamma function, 419, 421 (Exercise 19) GAUSS, KARL FRIEDRICH, 358, 362, 378, 473 Gauss-Jordan elimination process, 607 Gauss’ test for convergence, 402 (Exercise 17) Geometric interpretation: of derivative as a slope, 169 of integral as area, 65, 75, 89 Geometric mean, 47 (Exercise 20) Geometric series, 388-390 GIBBS, JOSIAH WILLARD, 445 GRAM, J0RGEN PEDERSON, 568 Gram-Schmidt process, 568 Graph of a function, 51 GRASSMANN, HERMANN, 446 Gravitational attraction, Newton’s law of, 546 Greatest-integer function, 63 GALILE:O, Index 390, 403 Gregory’s series, 403 Growth laws, 320, 321 GREGORY, JAMES, JACQUES, 615 Hadamard matrices, 615 (Exercise 10) Half-life, 313 HAMILTON, WILLIAM ROWAN, 358, 445 Harmonie mean, 46 Harmonie motion, 334 Harmonie series, 384 HEAVISIDE, OLIVER, 445 Helix, 523 Heron’s formula, 493 Higher-order derivatives, 160, 200 HILBERT, DAVID, 471 HOLMES, SHERLOCK, Homogeneous differential equation, 347-350 Homogeneous property : of finite sums, 40 of infinite series, 385 of integrals, 66 Homogeneous system of equations, 605 HOOKE, ROBERT, 50 Hooke’s law, 50, 116 Hyperbola, 498, 500, 506 Hyperbolic function, 251 Hyperbolic paraboloid, 198 HADAMARD, Identity element : for addition, 18 for multiplication, 18 Identity, function, 51 matrix, 600 transformation, 579 Implicit differentiation, 179 Implicit function, 179 Improper integral : of the first kind, 416 of the second kind, 418 Improper rational function, 259 Increasing function, 76 Increasing sequence, Indefinite integral, 120, 134 Indeterminate forms, 2899302 Induction : definition by, 39 proof by, 32-37 Inductive set, 22 Inequality, 20 Bernoulli, 46 Cauchy-Schwarz, 42, 452, 563 for the sine and cosine, 95 triangle, 42, 364, 454, 563 Infimum, 25 Infinite limits, 299, 300 Infinity, 297 Inflection point, 191 Initial condition, 307 Initial-value problem, 307 Inner product, 451, 469, 561 Integer, 22 Integrability: of a continuous function, 153 of a monotonie function, 77 Integral : curve, 341 definite, 73, 211 improper, 416-420 indefinite, 120 lower and Upper, 74 of a bounded function, 73 of a complex-valued function, 369 of a step function, 65 of a vector-valued function, 513 test, 397 Integrand, 74 Integration: by partial fractions, 258-264 by parts, 217-220 by substitution, 212-216 of monotonie functions, 79 of polynomials, 79, 81 of rational functions, 258-264 of trigonometric functions, 100, 207, 264 Intercepts, 190, 495 Intermediate-value theorem: for continuous functions, 144 for derivatives, 187 (Exercise 10) Intersection of sets, 14 Intervals, 60, 310 Inverse : function, 146, 252 matrix, 612 transformation, 585, 586 trigonometric functions, 253-256 Inversion, 146, 253 Invertible transformation, 585-588 Irrational numbers, 17, 22, 28, 31, 282 Isoclines, 344, 348 Isomorphism, 361, 600 Isothermals, 198, 351 Jump discontinuity, 131 Jupiter, 545 498, 545 Kepler’s laws, 545, 546 KEPLER, JOHANNES, 661 662 Index 171, 331, 445 Lagrange’s identity, 483 Lagrange’s remainder in Taylor’s formula, 284 Laplace’s equation, 305 Lattice points, 60 (Exercise 4) Least squares, method of, 196 (Exercise 25) Least-Upper-bound axiom, 25 Left-hand continuity, 126 Left-hand coordinate system, 485 Left-hand limit, 130 Left inverse, 585, 611 LEGENDRE, ADRIE.N-MARIE, 571 Legendre polynomials, 571 LEIBNIZ, GOTTFRIED WILHELM, 3, 113, 157, 172, 210, 222, 305, 403 Leibniz% formula for the nth derivative of a product, 222 (Exercise 4) Leibniz’s notation : for derivatives, 172 for primitives, 210 Leibniz’s rule for alternating series, 404 Length: of a curve, 530, 531 of a vector, 453 other definitions of, 461 (Exercises 17, 18) LEONARD~ OF PISA (Fibonacci), 379 Level curve, 197 L’HƠPITAL, GUILLAUME FRANÇOIS ANTOINE, 292 L’Hôpital’s rule, 292-298 Limit(s) : infinite, 298 left- and right-hand, 129, 130 of a function, 128 of a sequence, 379 of integration, 10, 74 theorems on, 132 Line(s) : Cartesian equation of, 475 definition of, 472 normal vector to, 476 parallelism of, 473 slope of, 169, 475 tangent, 170, 518 vector equation of, 475 Linear combination, 459, 556 Linear dependence and independence, 463, 557 Linear differential equation, 308, 3:!2 Linear function, 54 Linear space (vector space), 551, 552 Linear span, 462, 557 Linear system of equations, 605 Linear transformation, 578 Linearity property: of convergent series, of derivatives, 164 of integrals, 67, 80 LAGRANGE, JOSEPH LOUIS, of Taylor operators, 276 IVANOVICH, 474 Logarithms : base b, 232 base e (Napierian or natural logarithms), 229-232 Logarithmic differentiation, 235 Logarithmic function: calculation of, 240-242 definition of, 227 integration of, 235 power series for, 390, 433 Lorentz transformation, 614 (Exercise 6) Lower bound, 25 Lower integral, 74 LOBATCHEVSKI, NIKOLAI 285 Major axis of an ellipse, 505 Mass density, 118 Mathematical induction, 32-37 Mathematical model, 13 Matrix : algebraic operations on, 598, 601 definition of, 592, 598 diagonal, 595 representation, 592 Maximum element, 23 Maximum of a function : absolute, 150 relative, 182 Mean, 46, 149 arithmetic, 46 geometric, 47 (Exercise 20) harmonie, 46 pth-power, 46, 149 Mean distance from the sun, 546 Mean-value theorem: Cauchy’s extension of, 186 for derivatives, 185 for integrals 154, 219 Measurable set, 58, 111 MERCATOR, NICHOLAS, 377, 390 Mercury, 545 Minimum element, 25 Minimum of a function: absolute, 150 relative, 182 Minor axis of an ellipse, 505 Modulus of a complex number, 363 Moment, 118 of inertia, 119 Monotone property: of area, 59 of averages, 119 (Exercise 13) of volume, 112 of work, 115 MACHIN, JOHN, Index Monotonie function, 76 Monotonie sequence, 381 Motion : along a curve, 521 of a rocket, 337 simple harmonie, 334 Multiplication: of functions, 55, 63, 132 of matrices, 601 of numbers, 17, 44, 358 of transformations, 584 of vectors (cross product), 483 of vectors (inner product), 451, 469, 552 of vectors by scalars, 447, 552 232 Napierian (natural) logarithms, 229-232 Necessary and sufficient conditions, 394 Neighborhood, 127 NEWTON, ISAAC, 3, 157, 171, 305, 377, 498, 522 Newton’s law : of cooling, 315 of motion, 314, 546 of universal gravitation, 546 Non-Archimedean geometries, 26 Non-Euclidean geometries, 474 Nonsingular matrix, 611 Norm : of a vector, 453 of an element in a linear space, 563 Normal : to a line, 476 to a plane, 493 to a plane curve, 529 (Exercise 14) to a space curve, 526 Notations: for derivatives, 160, 171, 172, 200 for integrals, 10, 65, 69, 210, 211, 513 for products, 44 for sets, 12 for sums, 37 for vectors, 446, 512 n-space, 446 nth derivative, 160 nth root, 30, 145 Nul1 space, 580 Nullity, 581 Number : complex, 358 irrational, 17, 22 rational, 17, 22, 393 real, 17 NAPIER, JOHN, Odd function, 84 (Exercise 25) Odd integer, 28 (Exercise 10) One-sided limits, 129-l 30 One-to-one correspondence, 360, 412 One-to-one transformation, 587 o-notation, 286 Open interval, 60 Operator : difference, 172 differentiation, 172, 329, 579 integration, 579 linear, 578 Taylor, 274 Orbits of planets, 545-548 Order axioms, 20 Ordered pairs, 48, 53, 358 Ordinate, 48 Ordinate set, 58, 60, 61, 75 Origin of coordinates, 48 Orthogonal basis, 466, 568 Orthogonal complement, 573 Orthogonal matrix, 615 (Exercise 8) Orthogonal trajectory, 351 Orthogonality: in a Euclidean space, 564 of curves, 351 of lines, 170 of planes, 496 of the sine and cosine, 106 (Exercise 31) of vectors, 455 Orthonormal set, 466, 564 Osculating plane, 526 Parabola, 2, 54, 498, 500, 507 Parabolic mirrors, 519 Parabolic segment, area of, Paradox, Zeno’s, 374-377 Parallelepiped, 112 Parallelism: of lines, 473 of planes, 479 of vectors, 450 Parallelogram law, 362, 449 Parameter, 17 Parametric equations, 475, 517 of a circle, 521 of a helix, 523 of a hyperbola, 524 (Exercise 12) of a line, 475 of an ellipse, 522 PARSEVAL, MARK-ANTOINE, 566 Parseval’s formula, 566 Partial derivatives, 196-201 Partial fractions, integration by, 258-264 Partial sums, 375, 383 Partition, 61 PASCAL, BLAISE, 663 664 Pascal’s triangle, 44 (Exercise 3) PEANO, GUISEPPE, 17 Peano postulates for the integers, 17 Periodic function, 95 Periodic motion, 335, 339, 546 Permutation, 412 Perpendicularity : of lines, 170 of planes, 496 of vectors, 455 77 (pi) : computation of, 285 (Exercise 10) definition of, 91 Piecewise monotonie functions, 77 Planes, 478-482 Polar coordinates, 108, 540 Polar form of complex numbers, 367 Polynomial approximations, 272-304, 575 Polynomial functions, 55 continuity of, 133 differentiation of, 166 integration of, 79, 81 of two variables, 264 Population growth, 320, 321 Position function, 521, 540 Power, functions, 54, 80 series, 428-436 circle of convergence, 428 differentiation and integration of, 432 interval of convergence, 431 Prime numbers, 36 (Exercise ll), 50 Primitive (antiderivative), 205, 210-219 Product(s) : cross, 483 dot (inner), 451, 469, 561 notation for, 44 of functions, 55 of numbers, 17, 44, 358 scalar triple, 488 Projections, 457, 458, 574 Proper rational function, 259 pth-power mean, 46, 149 Pursuit problems, 352 Pythagorean identity, 96, 455, 469, 573 Pythagorean theorem, 49, 196 Quadrant, 48 Quadratic equation, 362 Quadratic polynomial, 54 Quotient, of functions, 55 of numbers, 18, 360 JOSEF LUDWIG, 402 Raabe’s convergence test, 402 (Exercise 16) RAABE, Index Radial acceleration, 542, 546 Radial set, 109 Radian measure, 102 Radioactive decay, 13 Radius : of convergence, 428 of curvature, 537 of gyration, 119 Range of a function, 53, 578 Rank, 581 Rate of change, 160 Rational function, 166 of two variables, 264 Rational number, 17, 22, 393 Rational powers, 30, 135, 166, 206 Ratio test, 400 Real function (real-valued function), 51 Real line (real axis), 22 Real linear space, 552 Real numbers, axioms for, 18-25 Rearrangements of series, 411-413 Reciprocal, 18, 360 Rectifiable curves, 530 Recursion formula, 220 (Exercise 8), 264, 379 Recursive definition, 39 Refinement of a partition, 62 Related rates, 177 Relative maximum and minimum, 182, 183 Remainder in Taylor’s formula, 278-287 Removable discontinuity, 13 Ricatti differential equation, 312 (Exercise 19) RIEMANN, GEORG FRIEDRICH BERNHARD, 3, 396, 413 Riemann’s rearrangement theorem, 413 Riemann zeta function, 396 Right-hand continuity, 126, 13 Right-hand coordinate system, 485 Right-hand limit, 129 Right inverse, 586 ROBERVAL, GILES PERSONE DE, ROBINSON, ABRAHAM, 172 Rocket with variable mass, 337 ROLLE, MICHEL, 184 Rolle’s theorem, 184 Root mean square, 46 Root test, 400 Roots of complex numbers, 372 (Exercise 8) Row, matrix (row vector), 598 operations, 608 Scalar, 447, 468 Scalar product (dot product), Scalar triple product, 488 SCHMIDT, ERHARD, 568 Sections of a cane, 497 451 Index 423 Separable differential equation, 345 Sequence, 378-381,422-426 Series : absolutely convergent, 406 alternating, 403 conditionally convergent, 406 convergent and divergent, 384 differentiation of, 427, 432 exponential, 436 geometric, 388 harmonie, 384 integration of, 432 logarithmic, 433 pointwise convergence of, 425 power, 389, 428 sine and cosine, 436 Taylor’s, 434 telescoping, 386 uniformly convergent, 425 Set, function, 57 theory, l-16 Similarity transformation, 91, 349 Simple harmonie motion, 334 Simultaneous linear equations, 490, 605 Sine function: complex-valued, 372 (Exercise 9) continuity of, 134, 139 differential equation for, 323 differentiation of, 162 integration of, 100, 207 power series for, 436 properties of, 96 Singular matrix, 613 Skew-symmetry, 483 Slope of a curve, 170 Slope of a line, 169 Small-span theorem (uniform continuity), 152 Solution of a differential equation, 306 Space spanned by a set of vectors, 556 Speed, 521 Sphere, volume of, 114 Square roots, 29 computation of, 444 (Exercises 20, 21) Squeezing principle for limits, 133 Step function, 62 integral of, 65 Step region, 58 STOKES, GEORGE GABRIEL, 423 Straight lines in n-space, 472 Subsets, 12 Subspace of a linear space, 556 Substitution, integration by, 212-216 Sum: of a convergent series, of functions, 55, 63, 132 SEIDEL, PHILLIPP LUDVIG VON, of numbers, 18, 358 of vectors, 447, 551 Summation notation, 37 Surface, 197 Systems of linear equations, 605 Tangent function, 103 Tangent line, 170, 518 Tangent vector, 518 TARTAGLIA, TAYLOR, BROOK, 274 Taylor polynomial, 274-277 Taylor’s formula with remainder, 278 Tavlor’s series of a function 434 Teiescoping p r o p e r t y : ’ of finite sums 40 of infinite series, 386 of products, 45 TORRICELLI, EVANGELISTA, Tractrix, 353 Transpose of a matrix, 615 (Exercise 7) Transverse axis of a hyperbola, 505 Triangle inequality : in a Euclidean space, 563 for complex numbers, 364 for real numbers, 42 for vectors, 454 Trigonometric functions: complex-valued, 372 (Exercise 9) continuity of, 134, 139 differentiation of, 162 fundamental properties of, 95 geometric description of, 102-104 graphs of, 107 integration of, 100 power series for, 436 Unbounded function, 73, 416 Unbounded sequence, Undetermined coefficients, 332, 333, 441 Uniform continuity theorem, 152 Uniform convergence, 424 Union of sets, 13 Uniqueness theorems, 309, 324 Unit coordinate vectors, 459 Unit tangent vector, 525 Unitary space, 562 Upper bound, 23 Upper integral, 74 Variation of parameters, 331 Vector(s) : addition and subtraction, 447 angle between, 458, 470 (Exercise 7) components of, 446 665 666 Vector(s) (Contd.) cross product of, 483 direction of, 450 dot product (inner product) of, 451 equality of, 447 geometric, 448 length (norm) of, 453 multiplication by scalars, 447 orthogonality of, 455 parallelism of, 450 Vector space (linear space), 552 Velocity, 159, 521 in polar coordinates, 541 Venn diagram, 13 Venus, 545 Vertex,of ellipse or hyperbola, 505 of a parabola, 507 Vibrations, 335 Volume : axiomatic definition of, 112 solids of known cross section, 113 solids of revolution, 113, 114 Index WALLIS, JOHN, WEIERSTRASS, KARL, 17, 423, 427 Weierstrass M-test for uniform convergence, 427 Weighted average of a function, 118 Weighted mean-value theorem, 154 Well-ordering principle, 34 Work, 115, 116 WRONSKI, J M HOENE, 328 Wronskian, 328 (Exercise 21), 330 374 Zeno’s paradox, 374-377 Zero, 18 complex number, 359 element in a linear space, 552 matrix, 599 transformation, 579 vector, 447 Zero-derivative theorem, 205, 369 Zeta function of Riemann, 396 ZENO, ... the mean-value theorem to geometric properties of functions Second- derivative test for extrema Curve sketching Exercises 15 6 15 7 15 9 16 1 16 4 16 7 16 9 17 1 17 3 17 4 17 6 cc 17 9 18 1 18 3 18 6 18 7 18 8 18 9... 2 .19 Exercises 88 88 89 94 94 97 10 2 10 4 10 8 10 9 11 0 11 1 11 4 11 5 11 6 11 7 11 9 12 0 12 4 CONTINUOUS FUNCTIONS 3 .1 3.2 Informa1 description of continuity The definition of the limit of a function 12 6... 3 .18 The integrability theorem for continuous functions 3 .19 Mean-value theorems for integrals of continuous functions 3.20 Exercises 13 0 13 1 13 5 13 8 14 0 14 2 14 2 14 4 14 5 14 6 14 7 14 8 14 9 15 0 15 2