Undergraduate Texts in Mathematics Editors F. W. Gehring P. R. Halmos Advisory Board C. DePrima I. Herstein J. Kiefer E. J. LeCuyer Introduction to College Mathematics with A Programming Language Springer-Verlag New York Heidelberg Berlin Edward J. LeCuyer Western New England College Department of Mathematics Springfield, MA 01119 USA Editorial Board F. W. Gehring P. R. Halmos University of Michigan University of California Department of Mathematics Department of Mathematics Ann Arbor, Michigan 48104 Santa Barbara, CA 93106 USA USA AMS Subject Classifications: 00-01, OOA05, 26-01, 26-04, 26A06, 26A09 Library of Congress Cataloging in Publication Data LeCuyer, E Introduction to college mathematics with A Programming Language (Undergraduate texts in mathematics) Includes indexes. I. Mathematics-1961- 2. Mathematics-Data processing. 3. APL (computer program language) I. Title. QA39.2.L4 510 78-2054 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. IC> 1978 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1978 9 8 7 6 5 432 I ISBN-13:978-1-4613-9424-2 e-ISBN-13:978-1-4613-9422-8 DOl: 10.1007/978-1-4613-9422-8 To my wife, Carol, and children, Karen, Michael, and Todd, whose love, patience, and encouragement made this work possible. To Howard Peelle for all of his help. Preface The topics covered in this text are those usually covered in a full year's course in finite mathematics or mathematics for liberal arts students. They correspond very closely to the topics I have taught at Western New England College to freshmen business and liberal arts students. They include set theory, logic, matrices and determinants, functions and graph ing, basic differential and integral calculus, probability and statistics, and trigonometry. Because this is an introductory text, none of these topics is dealt with in great depth. The idea is to introduce the student to some of the basic concepts in mathematics along with some of their applications. I believe that this text is self-contained and can be used successfully by any college student who has completed at least two years of high school mathematics including one year of algebra. In addition, no previous knowledge of any programming language is necessary. The distinguishing feature of this text is that the student is given the opportunity to learn the mathematical concepts via A Programming Lan guage (APL). APL was developed by Kenneth E. Iverson while he was at Harvard University and was presented in a book by Dr. Iverson entitled A Programming Languagei in 1962. He invented APL for educational purpo ses. That is, APL was designed to be a consistent, unambiguous, and powerful notation for communicating mathematical ideas. In 1966, APL became available on a time-sharing system at IBM. Today, APL is gaining wide acceptance in such fields as business, insurance, scientific research, and education. The reason for this is that APL is one of the most concise, versatile, and powerful computer programming languages yet developed. Programs requiring several steps in other computer languages become very 'A Programming Language by Kenneth E. Iverson, New York: John Wiley and Sons, (1962). vii Preface concise in APL, if a program is needed at all. This is both because many primitive functions are available directly on the APL keyboard and be cause such APL operations as + and X can be applied to arrays of any size (as well as to scalars). Yet, in spite of power and sophistication of APL, it is not a difficult language to learn. One can use APL to solve mathematical problems immediately after only a few minutes of instruc tion. Conventional mathematical notation and APL notation are presented in parallel throughout the text. Thus, if one desires, it is possible to ignore the APL and still use this text as a standard survey-of-mathematics text. Alternatively, one may use the text in conjunction with an APL terminal. APL notation corresponds closely to standard mathematical notation, and many mathematical processes are executed very easily in APL. By using the computer, the student can save a great deal of time doing tedious calculations and can concentrate more on the principles and concepts of the mathematics. In addition, the APL programs tend to reinforce these principles and concepts. It is my experience that by using APL, the student may learn the mathematical concepts better while finding the learning of mathematics meaningful and enjoyable. As an important bonus, he will be learning a powerful programming language which he will then be able to use in many other courses as well as in the "real world." The mathematical concepts and the APL notation are presented in parallel throughout the text because I believe that the APL can best be learned as needed in the development of the mathematics rather than as a separate topic. However, it might also be quite useful to have an APL reference for those who have not previously been exposed to the APL language. Therefore, I have included as an appendix an introduction to APL, including the writing and revising of APL programs. This appendix can be quickly perused at the start of the course and then referred to as needed throughout the course. Finally, I would like to express my appreciation to Dr. Howard A. Peelle of the University of Massachusetts for his encouragement and his numerous valuable suggestions on ways to improve upon this text. Also, I would like to thank the many students at the University of Massachusetts and at Western New England College who used the preliminary versions of this test for their preserverance, encouragement, and suggestions. July, 1977 Edward J. LeCuyer, Jr. viii Contents Chapter 1 Set theory 1 l.l Sets 1 1.2 Operations with Sets 5 1.3 A set theory drill and practice program (optional) 13 1.4 Boolean algebra 15 1.5 The number of elements in a set 20 Chapter 2 Logic 25 2.1 Statements and logical operations 25 2.2 Conditional statements 32 2.3 Logical equivalence 37 2.4 Arguments 41 Chapter 3 Vectors and matrices 45 3.1 Vectors 45 3.2 Operations with vectors 49 3.3 Matrices 53 3.4 Operations with matrices 59 3.5 Properties of matrices 66 Chapter 4 Systems of linear equations 71 4.1 Linear equations 71 4.2 Two-by-two systems of linear equations 73 ix Contents 4.3 Elementary row operations 78 4.4 Larger systems of linear equations 80 4.5 Row reduced form 85 4.6 The inverse of a matrix 91 4.7 Inverses in APL 98 4.8 Applications 100 Chapter 5 Determinants 103 5.1 Definition of a determinant 103 5.2 A Program for evaluation of determinants 107 5.3 Cofactors 1I0 5.4 Adjoints and inverses 1I5 5.5 Cramer's rule 120 Chapter 6 Functions and graphing 125 6.1 Definition of a function 125 6.2 Graphing 133 6.3 Linear functions 142 6.4 Quadratic functions 152 6.5 Polynomials 161 6.6 Rational functions 170 Chapter 7 Exponential and logarithmic functions 177 7.1 Exponential functions 177 7.2 Applications of exponential functions 183 7.3 Logarithmic functions 188 7.4 Properties and applications of logarithms 193 Chapter 8 Differential calculus 199 8.1 The limit of a function 199 8.2 Slope of a curve and the definition of derivative at a point 206 8.3 Differentiating polynomials 21I 8.4 Applications of derivatives 217 8.5 More rules of differentiation (optional) 223 8.6 Theory of maxima, minima 227 8.7 Applied maxima, minima 233 8.8 Curve sketching using derivatives 237 x Contents Chapter 9 Integral calculus 242 9.1 Antidifferentiation 242 9.2 Some formulas for antidifferentiation 246 9.3 Area under a curve 252 9.4 The definite integral 257 9.5 The fundamental theorem of calculus 263 9.6 More applications of integration 267 Chapter 10 Probability 272 10.1 Axioms of probability 272 10.2 More rules of probability 271 10.3 Permutations and combinations 283 10.4 The hypergeometric distribution 289 10.5 The binomial distribution 294 10.6 The Poisson distribution 298 Chapter 11 Statistics 303 11.1 Random samples and frequency distributions 303 11.2 Measures of central tendency 311 11.3 Measures of dispersion 314 11.4 The normal distribution 318 11.5 The sampling distribution of the mean 326 Chapter 12 The trigonometric functions 330 12.1 Angles 330 12.2 The trigonometric functions 335 12.3 The trigonometric functions in APL 344 12.4 Graphs of the trigonometric functions 346 12.5 The inverse trigonometric functions 351 12.6 Solving right triangles 354 12.7 Solving oblique triangles 358 Appendix 363 A.O Using APL on a computer terminal 363 A.I Intr6duction to APL 365 A.2 Program definition 371 A.3 Branching 374 A.4. _ Program revision and editing procedures 378 A.5 The trace command 382 xi Contents Solutions to exercises 385 Program index 415 Index 417 xii