CalcFirstYear.pdf - First Year Calculus For Students of Mathematics and Related Disciplines Copyright Michael M Dougherty and John Gieringer Preface

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Unformatted text preview: First Year Calculus For Students of Mathematics and Related Disciplines Copyright Michael M. Dougherty and John Gieringer Preface Calculus is notable in that any competent mathematician with at least a masters degree, and many with just a strong bachelors, should be fluent enough in the subject to passably teach the courses, since calculus and calculus-descendant studies form such important parts of their training. Perhaps consequently, there are almost as many opinions about how it should be taught as there are people teaching it. A fuzzy and somewhat artificial division into “traditional” and “reform” camps has been the rage for some fifteen years now, though neither seems able to define their own camp very well, let alone the other camp. Textbooks are often sold labeled as traditional or reform, with a growing number giving homage to both. Actually this is not difficult since the camps seem to really disagree mostly on emphasis. When given a choice, professors pick whatever textbook most closely resembles their own philosophy, and make up for the differences using the lectures. When not given a choice, many professors still give the nearly the same lectures since again, they understand calculus very thoroughly and have their own ideas about how to best make sense of it to their particular students. If it were not for the scale of such a project, in both writing and dealing with the actual publishing aspects, there would surely be many more—and more diverse—calculus textbooks available to reflect these opinions. Into this mix I submit this textbook, hoping it will appeal to like minded instructors. It grew out of my own ideas about what was right, and what was lacking in the textbooks from which I learned and later taught. This text has been in the works conceptually since my own graduate school days, when I was privileged to twice teach summer Calculus I at Purdue University using the very ambitious text by Richard Hunt. When I later, as an assistant professor, found that his second edition would not be published, I searched in vain for an alternative that was of a similar spirit and could find none. Some seven years after teaching those courses at Purdue, and never being totally comfortable with the texts available in the market, I finally began putting my own ideas onto paper.1 In talking to colleagues over the years, I am led to believe several do share visions similar to my own. To them I offer this as at least a step in their direction. I hope that its format will prove consistent with the goals of those colleagues, and that the text will fill their needs. How well we teach, and how well the students learn calculus are partly functions of our own enthusiasms. I hope that the approach is fresh and energizing to some of my fellow calculus instructors, high school as well as college, who have been looking for a textbook with some of the elements offered here. Incidentally, the title of this textbook is not meant to exclude students who are in fields of study other than mathematics. Indeed, it is hoped that anyone who pursues calculus for whatever reason will do so as a “student of mathematics.” It is not uncommon for a gathering of individuals to contain some who may be considered “students of Shakespeare” but never formally 1 This is not to say that this textbook is a clone of Richard Hunt’s. I only claim that his was my first inspiration, and none of the other available texts seemed to me as inspired with a vision as did his. I have heard Hunt’s strategy described as “sneaking in some real analysis.” This seems an apt description of his. I do the same, though to some extent I wonder who sneaked the real analysis out of the calculus. But my mission is not just to return to that, as the rest of the preface here will explain. ii iii completed a Literature or related major. The phrase here simply means that such individuals care enough about the subject to take personal time to examine it thoughtfully and continuingly, and to become respectably articulate in the subject, at least when among one’s peers. On the other hand, one can study the mechanics of, say, calculus without considering its more technical details or its conceptual content. To do so is akin to learning to quote Shakespeare’s plays without actually understanding the themes, or knowing the contexts. With calculus as with Shakespeare such lack of understanding can lead to trouble, in the form of embarrassment in the case of Shakespeare, and perhaps more catastrophic consequences in the case of calculus applied to real-world problems. The more of a “student” one is in a particular subject, the better trouble can be avoided and the more the subject can be enjoyed and enriching. Of course this textbook is intended to be thorough for those whose major field of study is Mathematics. However, it is hoped that Calculus for Students of Mathematics will inspire each reader—whose study may be any field—to become, for a while if not for a lifetime, a true “student of mathematics.” What is different about this text? At the risk of appearing trite, this text is actually meant to be read, perhaps even “curled up with and read.” Many texts are too sparse, and others too concise, in their explanations and it is often up to the lecturer to fill in the details or give alternate explanations more fit for student consumption. Indeed few students can today learn calculus on their own with one of the current calculus textbooks. This text is an attempt to reverse the contemporary roles of mathematics textbook (quick guide) and professor (expander), so that the professor does not have to sprint through the details but can, in good conscience, give the highlights or supplement with his own particular insights, knowing that the students have a complete treatment in the textbook.2 Much effort has been made for the text to be self-contained. A reasonably prepared and dedicated student should be able to learn enough calculus independently with this text to be able to solve all but perhaps the most challenging problems contained here. The text is naturally more verbose than most, and is peppered with cross references and footnotes. This will be a different style for many students, but one which is worth learning how to read. Added depth Concepts and examples are explored deeply and coherently, with an eye towards more advanced topics. Some insights students normally arrive at on their own are explained outright, but there is enough extra depth in the explanations and examples that students should have a more coherent overall understanding, and indeed may arrive at yet deeper insights on their own as they ponder, for instance, the applications of the principles they learn here. Pedagogically Linear Order This is as opposed to theoretically linear order. I have spent a great deal of thought on the order of topics, and have experimented with various orders extensively with my own classes. I have found that a few simple changes can make profound differences in the rate in which material is absorbed. 2 Perhaps only in mathematics are the professor’s lectures traditionally more complete, for the key topics at least, than the text. This is especially true if we include the question and answer dialogues with the students. In contrast, imagine a biology or history professor giving quantitatively more details in lecture than contained in the readings! iv PREFACE While this text is more theoretical than most, it was written with an awareness that there is a momentum to learning. Too many starts and stops in the development can dissipate energy from a calculus class. For that reason, it is sometimes better to show the final, “working” theory than risk bogging down in the preliminary theorems, with or without proofs. For instance, many texts will develop the natural logarithm as a definite integral, show that it works like a logarithm should and therefore must be a logarithm of some kind, and then call some theorem on inverse functions—a topic often painfully developed in its own, barely motivated section—to finally derive the function ex and its algebraic and calculus properties. I am in good company in deferring the theoretical development—until the reader is well-practiced with both these functions—and then giving the axiomatic theoretical development for completeness. The speed in which the computational skills are developed is greater, and the theoretical development is better-appreciated. I also develop all of the derivative rules in the same chapter (Chapter 4). A reasonable argument can be made that the exponential, logarithmic and arc-trigonometric functions should be introduced later, in between other calculus topics, so students can first further digest the earlier differentiation rules through applications. Though that is a standard pedagogical technique, instead I attempt to exploit the momentum of learning the differentiation rules so that they can be completely dispatched, and then reinforced through use in the chapters on applications. Of course the professor is welcome to break up the material, perhaps to have an exam after the first few sections if it seems appropriate for the particular class. Similarly, after all the differentiation rules are developed, and a chapter has been devoted to applications of derivatives, I devote two chapters on indefinite integrals before using them in Riemann Sum-motivated applications. The first of these integration chapters exhausts all the functions introduced earlier, in substitution-type settings. The second is my advanced integration technique chapter which builds upon the momentum of the first integration chapter. After these two chapters comes the chapter on Riemann Sums and applications of definite integrals. This approach allows the text to maintain the momentum from the derivative chapters, uninterrupted by Riemann Sums until they can be immediately motivated by the applications, and the student should be able to handle any integral which might arise, since by then the student has accomplished a considerable amount of integration. While this approach is actually less “gentle” for the development of antidifferentiation techniques, it has less stops and starts, and should help the student retain those skills throughout the applications. Continuity before limits. One reason I feel comfortable developing a topic completely—without interrupting to allow the reader to “sleep on it”—is that I front-load the text with rigor. Especially in the topics of limits and continuity, my path is perhaps not the quickest through these topics, but rather the path that will give the best hope for a comprehensive understanding. It is coincidentally also the most linear for the theoretical development. In particular I put continuity before limits, defining both in their own rights, using ε-δ definition. I strongly believe that Calculus loses much rigor when we omit ε-δ (even if students do not always understand these proofs as much as we would like), and that this omission causes much ad hoc explanation in the rest of our limit discussions (which can then barely be called “developments”). However, I realize that this is not a real analysis text, and so I only require the student to give ε-δ proofs for the first continuity section where I think it is best motivated (for instance by reference to tolerances), after which theorems ensure we never need to use them again in the exercises. My section on continuity on intervals has a couple of intuitive topological theorems3 on the images of intervals under continuous functions, from which I can easily state 3 Topological proofs are omitted to avoid the need to define connectedness and compactness. v the Intermediate Value Theorem (IVT), and the Extreme Value Theorem (EVT), using the former to give a method for solving polynomial and rational inequalities. I then have several limit sections to take care of all the first semester techniques, including separate sections for vertical and horizontal asymptote phenomena. Compared to other texts, the extensiveness of this particular chapter is perhaps the most innovative feature of the textbook. It is my sincere hope that it will help solve many of the difficulties associated with teaching these two topics. Symbolic logic included. To help with the rigor and communication of ideas, I include an early introduction to symbolic logic, which I then mix into the prose throughout the rest of the text. This is done for many reasons. First, it adds clarity through precision of the arguments. Second, the symbols naturally illustrate the logical “flow” of the arguments. Finally, it is my hope that this will be a hook for many students who have had difficulty relating abstract mathematics to everyday life, since the symbolic logic arguments have common sense appeal. Learning about logical equivalence is particularly useful in calculus since many theorems are stated in one form, used in another equivalent form, and possibly proved in still another form. Without some logical sophistication, such a discussion can be very confusing for calculus students. In particular, the contrapositive and the difference between implication and equivalence are stressed, as these can be problematic throughout one’s college studies and beyond. Of course it is hoped that the discussion of logic will help the dedicated student sharpen his or her own analytical skills in all disciplines, mathematical or otherwise, where logical argument is required. Studying symbolic logic has several other advantages. For instance, college calculus courses are often populated by a mix of students who had some exposure to calculus in high school, while the rest had none. This often leads to overconfidence in the former group and anxiety in the latter. Beginning with symbolic logic evens the playing field at the start, and sends a clear message to those who had calculus before that college calculus will be different, while giving both the novice and the former high school calculus student an opportunity to build the momentum to study calculus at a college level. The logic also sets a tone for a generally more abstract text than most. I feel justified in this since, after all, the underlying principles are abstract and understanding these is crucial for proper application. In this spirit I include, for instance, the axiomatic definition of the real numbers (though again, I am aware this is not supposed to be a real analysis text), in order that correct algebraic operations can be discussed in more exalted language. The discussion includes the least upper bound property so that, much later, convergence of sequences and series will not need to be explained in an ad hoc manner. Using notation from logic, I give a somewhat different review of algebra and trigonometry than what students may be used to, again to get them thinking about these things from a more sophisticated and hopefully fresher perspective. Applications. On a visit to Singapore in 2005, I was twice asked casually why students should study calculus. What was a bit shocking was that this question came from two young, successful Sinaporean adults who had actually studied calculus! Fifteen years earlier, as a newly minted graduate teaching assistant I could rattle off what are probably standard answers: it is useful in engineering, all sorts of sciences, economics, and so on because it allows for deep analysis and computation, impossible without calculus, regarding among other things how quantities change and how those changes accumulate. But by the summer of 2005, either from growing tired of vi PREFACE repeating myself or (I would like to think) a more mature understanding of the subject, my instinct was to pause, look around and take in what I could see as calculus problems everywhere. It happened that both times I was asked this, we were riding on public transportation, so I could imagine applications of calculus in the mechanics of moving the bus or train I was on at the time, in the rate of absorption of the sun’s rays on surfaces set at different angles from those rays, in the centrifugal/centripetal forces generated by all manner of spinning objects (wheels, motor internals), and plenty of other examples if I wanted to indulge my revelrie further. I then tried to explain how eventually, as with all education, once one accumulates a kind of critical mass of it, one starts to “see things” differently, and indeed more deeply. However these two very smart individuals both apparently passed on opportunities to explore, even in their imaginations, the analytic power of the calculus. It is true that I hope the reader would appreciate that all ideas presented here have relevance in either possible applications of the ideas themselves, or at least in their understanding of how the world works. However, it is still somewhat up to the student to be open to the relevance of whatever topic he or she is studying, and to use his or her own imagination as to relevance each time a new idea is explored. There is a kind of truism in dialogue form which has been promulgated by mathemtics educators, which reads as follows: “You can lead a horse to water but you can’t make him drink.” “Yes, but you can sure salt his oats!” My hope is that students will find novel (to them) relevance to how they view the world in applications of even the most abstract treatment of a topic, and that this exercise inspires students to rethink and enhance how they view mathematics and its role in understanding the world. For applications I stick more to physics examples, and only occasionally inject biological or social scientific examples. I believe physics has the clearest connection to calculus, and offers the best motivations for its study. In fact, I do not use the tangent line slope as my introductory motivation for the derivative, but instead use velocity (vis-`a-vis position). I believe velocity is initially more intuitive to more students. The fact that the derivative is graphically the slope of a tangent line is a very convenient device of course, and I exploit it extensively, but too many students walk away uninspired from calculus thinking it is all about tangent lines, and not instead about change (instantaneous and cumulative). Other differences. Also different is the fact that this text is in black and white, further reinforcing a more abstract spirit. This may be more a matter of taste, but I believe there is a place for such a text and that fancy, four-color illustrations can be distracting from the main themes, not to mention far more expensive to produce, a fact not unnoticed by cash-strapped students.4 The entire textbook is typeset in LATEXby the author, using the LATEX book style, with graphics handled by the LATEX pstricks package. Several other LATEX packages were also used, mainly for modifying the format. No graphics were imported but are all generated using LATEX code from these packages. Acknowledgments First I would like to thank anyone who reads any part of this book. I wrote it for you! Even if you do not read it cover-to-cover, I very much appreciate your interest. And I would like very much to hear back, regardless of your opinion. 4 It has been pointed out that many middle school history textbooks use very sophisticated, ostensibly attractive designs, and yet middle school students are not likely to be found under the blankets with a flashlight and their history books. Contrast this with the sparsely illustrated Harry Potter books. Granted, fiction can be more “fun,” but a good telling of the exploits of Julius Caesar might be more compelling than a chart or graph. vii For helping to make this work possible, I am most grateful to my wife of sixteen years, Hung-Chieh Chang (a.k.a. Joy Dougherty), who never showed me any doubt in her mind that this work would eventually be finished, and who put up with the seemingly countless hours I was bonding with several compu...
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