2314-FM_R2_140512 College Mathematics This page intentionally left blank 2314-FM_R2_140512 College Mathematics Fourth Edition Algebra Discrete Mathematics Precalculus Calculus Frank Ayres, Jr., Ph.D. Formerly Professor and Head Department of Mathematics Dickinson College Philip A. Schmidt, Ph.D. Associate Provost for Academic Programs Western Governors University Salt Lake City, Utah Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto 2314-FM_R2_140512 FRANK AYRES, Jr., Ph.D., was formerly Professor and Head of the Department of Mathematics at Dickinson College, Carlisle, Pennsylvania. He is the author or coauthor of eight Schaum’s Outlines, including Calculus, Trigonometry, Differential Equations, and Modern Abstract Algebra. PHILIP A. SCHMIDT, Ph.D., has a B.S. from Brooklyn College (with a major in mathematics), an M.A. in mathematics, and a Ph.D. in mathematics education from Syracuse University. He is currently the Associate Provost for Academic Programs for Western Governors University in Salt Lake City, Utah. Dr. Schmidt is the co-author of Elementary Algebra. Copyright # 1958 by The McGraw-Hill Companies, Inc. under the title Schaum’s Outline of Theory and Problems of First Year College Mathematics. All rights reserved. Schaum’s Outline of COLLEGE MATHEMATICS Copyright # 2010, 2003, 1992 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the 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. 2 3 4 5 6 7 8 9 0 CUS/CUS 1 0 9 8 7 6 5 4 3 2 ISBN 978-0-07-162647-7 MHID 0-07-162647-6 McGraw-Hill books are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at
[email protected] Library of Congress Cataloging-in-Publication Data applied for. 2314-FM_R2_140512 Contents Preface PART I vii REVIEW OF ALGEBRA 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 1 Elements of Algebra Functions Graphs of Functions Linear Equations Simultaneous Linear Equations Quadratic Functions and Equations Inequalities The Locus of an Equation The Straight Line Families of Straight Lines The Circle 3 9 14 20 25 35 44 50 58 64 69 PART II TOPICS IN DISCRETE MATHEMATICS 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 79 Arithmetic and Geometric Progressions Infinite Geometric Series Mathematical Induction The Binomial Theorem Permutations Combinations Probability Determinants of Orders Two and Three Determinants of Order n Systems of Linear Equations 81 90 94 99 106 112 117 125 131 138 PART III TOPICS IN PRECALCULUS 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 145 Angles and Arc Length Trigonometric Functions of a General Angle Trigonometric Functions of an Acute Angle Reduction to Functions of Positive Acute Angles Graphs of the Trigonometric Functions Fundamental Relations and Identities Trigonometric Functions of Two Angles Sum, Difference, and Product Formulas Oblique Triangles Inverse Trigonometric Functions Trigonometric Equations Complex Numbers 147 153 161 170 175 181 188 200 205 216 227 237 v 2314-FM_R2_140512 Contents vi 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. The Conic Sections Transformation of Coordinates Points in Space Simultaneous Equations Involvin