com RQWKH 6+28/'(56 RI *,$176 www.com RQWKH 6+28/'(56 RI *,$176 6,1*/( $&2856(,1 9$5,$%/( &$/&8/86 *+60,7+*-0F/(//$1' UNSW PRESS www.com A UNSW Press book Published by University of New South Wales Press Ltd University of New South Wales UNSW Sydney NSW 2052 AUSTRALIA www.au © GH Smith and GJ McLelland First published 2003 This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright Act, no part may be reproduced by any process without written permission. Inquiries should be addressed to the publisher. National Library of Australia Cataloguing-in-Publication entry: Smith, Geoff, 1953–.
On the shoulders of giants: a course in single variable calculus. Calculus of variations.64 Printer BPA Illustrations pages 2, 5, 8 and 195 Anita Howard Cover design Di Quick www.com CONTENTS Preface v 1 Terror, tragedy and bad vibrations 1 1.2 The Tower of Terror .3 Into thin air .4 Music and the bridge .6 Rules of calculation .1 Rules of calculation .2 Intervals on the real line .3 Graphs of functions .4 Examples of functions. 20 3 Continuity and smoothness 27 3.2 Rules for differentiation .3 Velocity, acceleration and rates of change .1 The Tower of Terror .2 Solving differential equations .4 Increasing and decreasing functions. 70 6 Series and the exponential function 75 6.1 The air pressure problem .2 Infinite series .3 Convergence of series .4 Radius of convergence .com ii CONTENTS 6.5 Differentiation of power series .6 The chain rule .7 Properties of the exponential function .8 Solution of the air pressure problem .1 Vibrating strings and cables .3 More on the sine and cosine functions .4 Triangles, circles and the number .5 Exact values of the sine and cosine functions .6 Other trigonometric functions .1 Second order linear differential equations .4 Complex roots of the auxiliary equation .5 Simple harmonic motion and damping .1 Another problem on the Tower of Terror .2 More on air pressure .3 Integrals and primitive functions .4 Areas under curves .7 Evaluation of integrals .8 The fundamental theorem of the calculus .9 The logarithm function .1 The existence of inverses .2 Calculating function values for inverses .3 The oscillation problem again .4 Inverse trigonometric functions .5 Other inverse trigonometric functions .2 Properties of the hyperbolic functions .3 Inverse hyperbolic functions .com CONTENTS iii 12 Methods of integration 235 12.2 Calculation of definite integrals .3 Integration by substitution .4 Integration by parts .5 The method of partial fractions .6 Integrals with a quadratic denominator.
249 13 A nonlinear differential equation 251 13.1 The energy equation. 259 Answers 261 Index 281 www.com This page intentionally left blank www.com PREFACE If I have seen further it is by standing on the shoulders of Giants. Sir Isaac Newton, 1675. This book presents an innovative treatment of single variable calculus designed as an introductory mathematics textbook for engineering and science students.
The subject material is developed by modelling physical problems, some of which would normally be encountered by students as experi- ments in a first year physics course. The solutions of these problems provide a means of introducing mathematical concepts as they are needed. The book presents all of the material from a traditional first year calculus course, but it will appear for different purposes and in a different order from standard treatments. The rationale of the book is that the mathematics should be introduced in a context tailored to the needs of the audience.
Each mathematical concept is introduced only when it is needed to solve a particular practical problem, so at all stages, the student should be able to connect the mathematical concept with a particular physical idea or problem. For various reasons, notions such as relevance or just in time mathematics are common catchcries. We have responded to these in a way which maintains the professional integrity of the courses we teach. The book begins with a collection of problems.
A discussion of these problems leads to the idea of a function, which in the first instance will be regarded as a rule for numerical calculation. In some cases, real or hypothetical results will be presented, from which the function can be deduced. Part of the purpose of the book is to assist students in learning how to define the rules for calculating functions and to understand why such rules are needed. The most common way of expressing a rule is by means of an algebraic formula and this is the way in which most students first encounter functions.
Unfortunately, many of them are unable to progress beyond the functions as formulas concept. Our stance in this book is that functions are rules for numerical calculation and so must be presented in a form which allows function values to be calculated in decimal form to an arbitrary degree of accuracy. For this reason, trigonometric functions first appear as power series solutions to differential equations, rather than through the common definitions in terms of triangles. The latter definitions may be intuitively simpler, but they are of little use in calculating function values or preparing the student for later work.
We begin with simple functions defined by algebraic formulas and move on to functions defined by power series and integrals. As we progress through the book, different physical problems give rise to various functions and if the calculation of function values requires the numerical evaluation of an integral, then this simply has to be accepted as an inconvenient but unavoidable property of the problem. We would like students to appreciate the fact that some problems, such as the nonlinear pendulum, require sophisticated mathematical methods for their analysis and difficult mathematics is unavoidable if we wish to solve the problem. It is not introduced simply to provide an www.com vi PREFACE intellectual challenge or to filter out the weaker students.
Our attitude to proofs and rigour is that we believe that all results should be correctly stated, but not all of them need formal proof. Most of all, we do not believe that students should be presented with handwaving arguments masquerading as proofs. If we feel that a proof is accessible and that there is something useful to be learned from the proof, then we provide it. Otherwise, we state the result and move on.
Students are quite capable of using the results on term-by-term differentiation of a power series for instance, even if they have not seen the proof. However, we think that it is important to emphasise that a power series can be differentiated in this way only within the interior of its interval of convergence. By this means we can take the applications in this book beyond the artificial examples often seen in standard texts. We discuss continuity and differentiation in terms of convergence of sequences.
We think that this is intuitively more accessible than the usual approach of considering limits of functions. If limits are treated with the full rigour of the - approach, then they are too difficult for the average beginning student, while a non-rigorous treatment simply leads to confusion. The remainder of this preface summarises the content of this book. Our list of physical problems includes the vertical motion of a projectile, the variation of atmospheric pressure with height, the mo- tion of a body in simple harmonic motion, underdamped and overdamped oscillations, forced damped oscillations and the nonlinear pendulum.
In each case the solution is a function which relates two vari- ables. An appeal to the student’s physical intuition suggests that the graphs of these functions should have certain properties. Closer analysis of these intuitive ideas leads to the concepts of continuity and differentiability. Modelling the problems leads to differential equations for the desired functions and in solving these equations we discuss power series, radius of convergence and term-by-term differen- tiation.
In discussing oscillation we have to consider the case where the auxiliary equation may have non-real roots and it is at this point that we introduce complex numbers. Not all differential equations are amenable to a solution by power series and integration is developed as a method to deal with these cases. Along the way it is necessary to use the chain rule, to define functions by integrals and to define inverse functions. Methods of integration are introduced as a practical alternative to numerical methods for evaluating integrals if a primitive function can be found.
We also need to know whether a function defined by an integral is new or whether it is a known elementary function in another form. We do not go very deeply into this topic. With the advent of symbolic manipulation packages such as Mathematica, there seems to be little need for science and engineering students to spend time evalu- ating anything but the simplest of integrals by hand. The book concludes with a capstone discussion of the nonlinear equation of motion of the simple pendulum.
Our purpose here is to demonstrate the fact that there are physical problems which absolutely need the mathematics developed in this book. Various ad hoc procedures which might have sufficed for some of the earlier problems are no longer useful. The use of Mathematica makes plotting of elliptic functions and finding their values no more difficult than is the case with any of the common functions. We would like to thank Tim Langtry for help with LATEX.
Tim Langtry and Graeme Cohen read the text of the preliminary edition of this book with meticulous attention and made numerous suggestions, comments and corrections. Other useful suggestions, contributions and corrections came from Mary Coupland and Leigh Wood.com CHAPTER 1 TERROR, TRAGEDY AND BAD VIBRATIONS 1.1 INTRODUCTION Mathematics is almost universally regarded as a useful subject, but the truth of the matter is that mathematics beyond the middle levels of high school is almost never used by the ordinary person. Certainly, simple arithmetic is needed to live a normal life in developed societies, but when would we ever use algebra or calculus? In mathematics, as in many other areas of knowledge, we can often get by with a less than complete understanding of the processes. People do not have to understand how a car, a computer or a mobile phone works in order to make use of them.
However, some people do have to understand the underlying principles of such devices in order to invent them in the first place, to improve their design or to repair them. Most people do not need to know how to organise the Olympic Games, schedule baggage handlers for an international airline or analyse traffic flow in a communications network, but once again, someone must design the systems which enable these activities to be carried out. The complex technical, social and financial systems used by our modern society all rely on mathematics to a greater or lesser extent and we need skilled people such as engineers, scientists and economists to manage them. Mathematics is widely used, but this use is not always evident.
Part of the purpose of this book is to demonstrate the way that mathematics pervades many aspects of our lives. To do this, we shall make use of three easily understood and obviously relevant problems. By exploring each of these in increasing detail we will find it necessary to introduce a large number of mathematical techniques in order to obtain solutions to the problems. As we become more familiar with the mathematics we develop, we shall find that it is not limited to the original problems, but is applicable to many other situations.
In this chapter, we will consider three problems: an amusement park ride known as the Tower of Terror, the disastrous consequences that occurred when an aircraft cargo door flew open in mid-air and an unexpected noise pollution problem on a new bridge. These problems will be used as the basis for introducing new mathematical ideas and in later chapters we will apply these ideas to the solution of other problems.