Calculo

Tom M. Apostol

CALCULUS
VOLUME 1

One-Variable Calculus, with an Introduction to Linear Algebra
SECOND EDITION

New York

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John Wiley & Sons, Inc. Santa Barbara l London l Sydney

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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:67-14605 Printed in the United States of America. 1 0 9 8 7 6 5 4 3 2

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 thereal-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 adeductive 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 thetwo. 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 intuitivediscussions 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 ishistorically 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

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Preface

Prefuce to the Second Edition
The second edition differs from the first in many respects. Linear algebra has beenincorporated, 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...