calculo, Apuntes de Ciencias Ambientales

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Tom M. Apostol

CALCULUS

VOLUME 1

One-Variable Calculus, with an

Introduction to Linear Algebra

SECOND EDITION

John Wiley & Sons, Inc. New York l 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 0 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 0 471 00005 1 Library of Congress Catalog Card Number: 67- Printed in the United States of America. 1 0 9 8 7 6 5 4 3 2

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 applica- tions 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 impor- tant 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 dis- cussions 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 develop- ment 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

CONTENTS

1. INTRODUCTION

Part 1. Historical Introduction

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 2. Some Basic Concepts of the Theory of Sets

Introduction to set theory Notations for designating sets Subsets Unions, intersections, complements Exercises

Part 3. A Set of Axioms for the Real-Number System

Introduction The field axioms Exercises The order axioms Exercises Integers and rational numbers

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X Contents

1 3.7 Geometric interpretation of real numbers as points on a line 1 3.8 Upper bound of a set, maximum element, least upper bound (supremum) 1 3.9 The least-Upper-bound axiom (completeness axiom) 1 3.10 The Archimedean property of the real-number system 1 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

Part 4. Mathematical Induction, Summation Notation,

and Related Topics

14.1 An example of a proof by mathematical induction 1 4.2 The principle of mathematical induction *1 4.3 The well-ordering principle 1 4.4 Exercises *14.5 Proof of the well-ordering principle 1 4.6 The summation notation 1 4.7 Exercises 1 4.8 Absolute values and the triangle inequality 1 4.9 Exercises *14.10 Miscellaneous exercises involving induction

1. 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

• 22 23 25 25 26 28 29 30 30

xii Contents

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

4. DIFFERENTIAL CALCULUS

4.1 Historical introduction 156 4.2 A problem involving velocity 1 5 7 4.3 The derivative of a function 1 5 9 4.4 Examples of derivatives 161 4.5 The algebra of derivatives 1 6 4 4.6 Exercises 1 6 7 4.7 Geometric interpretation of the derivative as a slope 1 6 9 4.8 Other notations for derivatives 1 7 1 4.9 Exercises 1 7 3 4.10 The chain rule for differentiating composite functions 1 7 4 - 4.11 Applications of the chain rule. Related rates and implicit differentiation^176 cc 4.12 Exercises 1 7 9 4.13 Applications of differentiation to extreme values of functions 181 4.14 The mean-value theorem for derivatives 1 8 3 4.15 Exercises 1 8 6 4.16 Applications of the mean-value theorem to geometric properties of functions 1 8 7 4.17 Second-derivative test for extrema 1 8 8 4.18 Curve sketching 1 8 9 4.19 Exercises 191

Contents x...

4.20 Worked examples of extremum problems 191 4.21 Exercises 194 “4.22 Partial derivatives 1 9 6 “4.23 Exercises 201

5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION

5.1 The derivative of an indefinite integral. The first fundamental theorem of calculus 202 5.2 The zero-derivative theorem 204 5.3 Primitive functions and the second fundamental theorem of calculus 205 5.4 Properties of a function deduced from properties of its derivative 207 5.5 Exercises 208 5.6 The Leibniz notation for primitives 210 “-. 5.7 Integration by substitution 212 5.8 Exercises 216 5.9 Integration by parts 217 - 5.10 Exercises 220

*5.11 Miscellaneous review exercises 222

6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS

6.1 Introduction 6.2 Motivation for the definition of the natural logarithm as an integral 6.3 The definition of the logarithm. Basic properties 6.4 The graph of the natural logarithm 6.5 Consequences of the functional equation L(U~) = L(a) + L(b) 6.6 Logarithms referred to any positive base b # 1 6.7 Differentiation and integration formulas involving logarithms 6.8 Logarithmic differentiation 6.9 Exercises 6.10 Polynomial approximations to the logarithm 6.11 Exercises 6.12 The exponential function 6.13 Exponentials expressed as powers of e 6.14 The definition of e” for arbitrary real x 6.15 The definition of a” for a > 0 and x real

Contents xv

8.5 Exercises 3 1 1 8.6 Some physical problems leading to first-order linear differential equations 313 8.7 Exercises 319 8.8 Linear equations of second order with constant coefficients 322 8.9 Existence of solutions of the equation y” + ~JJ = 0 323 8.10 Reduction of the general equation to the special case y” + ~JJ = 0 324 8.11 Uniqueness theorem for the equation y” + bu = 0 324 8.12 Complete solution of the equation y” + bu = 0 326 8.13 Complete solution of the equation y” + ay’ + br = 0 326 8.14 Exercises 328 8.15 Nonhomogeneous linear equations of second order with constant coeffi- cients 329 8.16 Special methods for determining a particular solution of the nonhomogeneous equation y” + ay’ + bu = R 8.17 Exercises

8.18 Examples of physical problems leading to linear second-order equations with constant coefficients 8.19 Exercises 8.20 Remarks concerning nonlinear differential equations 8.21 Integral curves and direction fields 8.22 Exercises 8.23 First-order separable equations 8.24 Exercises 8.25 Homogeneous first-order equations 8.26 Exercises 8.27 Some geometrical and physical problems leading to first-order equations 8.28 Miscellaneous review exercises

9. COMPLEX NUMBERS

9.1 Historical introduction 9.2 Definitions and field properties 9.3 The complex numbers as an extension of the real numbers 9.4 The imaginary unit i 9.5 Geometric interpretation. Modulus and argument 9.6 Exercises 9.7 Complex exponentials 9.8 Complex-valued functions 9.9 Examples of differentiation and integration formulas 9.10 Exercises

xvi Contents

10. SEQUENCES, INFINITE SERIES,

IMPROPER INTEGRALS

10.1 Zeno’s paradox 10.2 Sequences 10.3 Monotonie sequences of real numbers 10.4 Exercises 10.5 Infinite series 10.6 The linearity property of convergent series 10.7 Telescoping series 10.8 The geometric series 10.9 Exercises “10.10 Exercises on decimal expansions 10.11 Tests for convergence 10.12 Comparison tests for series of nonnegative terms 10.13 The integral test 10.14 Exercises 10.15 The root test and the ratio test for series of nonnegative terms 10.16 Exercises 10.17 Alternating series 10.18 Conditional and absolute convergence 10.19 The convergence tests of Dirichlet and Abel 10.20 Exercises *10.21 Rearrangements of series 10.22 Miscellaneous review exercises 10.23 Improper integrals 10.24 Exercises

11. SEQUENCES AND SERIES OF FUNCTIONS

11.1 Pointwise convergence of sequences of functions 422 11.2 Uniform convergence of sequences of functions 423 11.3 Uniform convergence and continuity 424 11.4 Uniform convergence and integration 425 11.5 A sufficient condition for uniform convergence 427 11.6 Power series. Circle of convergence 428 11.7 Exercises 430 11.8 Properties of functions represented by real power series 4 3 1 11.9 The Taylor’s series generated by a function 434 11.10 A sufficient condition for convergence of a Taylor’s series 435

... xv111 Contents

13.10 The cross product expressed as a determinant 13.11 Exercises 13.12 The scalar triple product 13.13 Cramer’s rule for solving a system of three linear equations 13.14 Exercises 13.15 Normal vectors to planes 13.16 Linear Cartesian equations for planes 13.17 Exercises 13.18 The conic sections 13.19 Eccentricity of conic sections 13.20 Polar equations for conic sections 13.21 Exercises 13.22 Conic sections symmetric about the origin 13.23 Cartesian equations for the conic sections 13.24 Exercises 13.25 Miscellaneous exercises on conic sections

14. CALCULUS OF VECTOR-VALUED FUNCTIONS

14.1 Vector-valued functions of a real variable 14.2 Algebraic operations. Components 14.3 Limits, derivatives, and integrals 14.4 Exercises 14.5 Applications to curves. Tangency 14.6 Applications to curvilinear motion. Velocity, speed, and acceleration 14.7 Exercises 14.8 The unit tangent, the principal normal, and the osculating plane of a curve 14.9 Exercises 14.10 The definition of arc length 14.11 Additivity of arc length 14.12 The arc-length function 14.13 Exercises 14.14 Curvature of a curve 14.15 Exercises 14.16 Velocity and acceleration in polar coordinates 14.17 Plane motion with radial acceleration 14.18 Cylindrical coordinates 14.19 Exercises 14.20 Applications to planetary motion 14.2 1 Miscellaneous review exercises

Contents xix

15. LINEAR SPACES

15.1 Introduction 5 5 1 15.2 The definition of a linear space 5 5 1 15.3 Examples of linear (^) spaces 552 15.4 Elementary (^) consequencesof the axioms^554 15.5 Exercises 555 15.6 Subspaces of a linear space 556 15.7 Dependent and independent sets in a linear space 557 15.8 Bases and dimension 559 15.9 Exercises 560 15.10 Inner products, Euclideanspaces, norms 5 6 1 1 5. 1 1 Orthogonality in a Euclidean space 564 15.12 Exercises 566 15.13 Construction of orthogonal sets. The Gram-Schmidt process 568 15.14 Orthogonal complements. Projections 572 15.15 Best approximation of elements in a Euclidean space by elements in a finite- dimensional subspace 574 15.16 Exercises 576

16. LINEAR TRANSFORMATIONS AND MATRICES

16.1 Linear transformations 16.2 Nul1 space and range 16.3 Nullity and rank 16.4 Exercises 16.5 Algebraic operations on linear transformations 16.6 Inverses 16.7 One-to-one linear transformations 16.8 Exercises 16.9 Linear transformations with prescribed values 16.10 Matrix representations of linear transformations 16.11 Construction of a matrix representation in diagonal form 16.12 Exercises 16.13 Linear spaces of matrices 16.14 Isomorphism between linear transformations and matrices 16.15 Multiplication of matrices 16.16 Exercises 16.17 Systems of linear equations

Calculus