Frank Ayres - Schaums Outline of Calculus

The late FRANK AYRES, Jr., PhD, was formerly professor and head of the Department at Dickinson College, Carlisle, Pennsylvania. He is the author of eight Schaums Outlines, including Calculus, Differential Equations, 1st Year College Math, and Matrices.Elliott Mendelson, PhD, is professor of mathematics at Queens College. He is the author of Schaums Outline of Beginning Calculus. Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-179554-8
MHID: 0-07-179554-5 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-179553-1, MHID: 0-07-179553-7.

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The purpose of this book is to help students understand and use the calculus. Everything has been aimed toward making this easier, especially for students with limited background in mathematics or for readers who have forgotten their earlier training in mathematics. The topics covered include all the material of standard courses in elementary and intermediate calculus.

The direct and concise exposition typical of the Schaum Outline series has been amplified by a large number of examples, followed by many carefully solved problems. In choosing these problems, we have attempted to anticipate the difficulties that normally beset the beginner. In addition, each chapter concludes with a collection of supplementary exercises with answers. This fifth edition has enlarged the number of solved problems and supplementary exercises. Moreover, we have made a great effort to go over ticklish points of algebra or geometry that are likely to confuse the student. The author believes that most of the mistakes that students make in a calculus course are not due to a deficient comprehension of the principles of calculus, but rather to their weakness in high-school algebra or geometry.

Students are urged to continue the study of each chapter until they are confident about their mastery of the material. A good test of that accomplishment would be their ability to answer the supplementary problems. The author would like to thank many people who have written to me with corrections and suggestions, in particular Danielle Cinq-Mars, Lawrence Collins, L.D. De Jonge, Konrad Duch, Stephanie Happ, Lindsey Oh, and Stephen B. Soffer. He is also grateful to his editor, Charles Wall, for all his patient help and guidance.

ELLIOTT MENDELSON

A linear coordinate system is a graphical representation of the real numbers as the points of a straight line. To each number corresponds one and only one point, and to each point corresponds one and only one number. To set up a linear coordinate system on a given line: (1) select any point of the line as the origin and let that point correspond to the number 0; (2) choose a positive direction on the line and indicate that direction by an arrow; (3) choose a fixed distance as a unit of measure. If x is a positive number, find the point corresponding to x by moving a distance of x units from the origin in the positive direction.

If x is negative, find the point corresponding to x by moving a distance of x units from the origin in the negative direction. (For example, if x = 2, then x = 2 and the corresponding point lies 2 units from the origin in the negative direction.) See . Fig. 11 The number assigned to a point by a coordinate system is called the coordinate of that point. We often will talk as if there is no distinction between a point and its coordinate. Thus, we might refer to the point 3 rather than to the point with coordinate 3.

The absolute value |x| of a number x is defined as follows: