Dr. Michael Dougherty's Calculus Textbook Project

(The whole book, as of November 19, 2009, can be had here in a 8,161,739 -byte PDF file.)

The version above has some additions to Chapter 2, which is pre-calculus stuff but might help students to learn to parse functions better. I have been experimenting with "diagramming functions" in a flow-chart manner. Numbered pages 138-145 (add about 20 to get "physical pages") introduce this, and use it for some of the usual "simple" transforms of functions (shifts, reflections, contractions/dilations). I experimented with \ovalnode's in pstricks for these. Some later pages have earlier experimental graphics for more complicated functions (which are combinations of functions). The links below are presently out of date for this topic. I hope to be done with this pre-calculus stuff before too much longer, but it is currently where I am inspired to work, so I better follow that inspiration while it's there.

This page contains excerpts from my calculus textbook project. This is very much a work in progress, though some sections are nearly complete and have been used in my courses. Comments can be sent to me at michael.dougherty@swosu.edu. Check back for updates periodically. Files below are current as of November 30, 2008, reflecting some recent progress. The whole book file will often be more current.

In case you're wondering why this book is different, I'll give a few quick reasons.

1. You can read this book just by reading this book, mostly without a large pad of paper and a lot of pens next to you trying to figure out why what I wrote is true, or for that matter what I meant. At least that's my goal: a book you can relax with to read calculus!

   Well, that's not quite right. You can, for the most part, read the book without a lot of frustration because it's spelled out for you in more detail than usual, though it never hurts to have that paper and pen ready to clarify things for yourself. And of course, exercises are included (not enough yet) and those are usually nontrivial, requiring a creative effort and lots of paper on the part of the reader if the material is to be truly learned.

   The usual technique for a calculus book is the opposite, in my opinion leaving out too much detail--or trying to pack too much into each of the (consequently) insufficiently numerous examples--and thus requiring the student to mull over points for a very long, frustrating time. That's OK and time-tested, particularly for students who have the time and energy to work through the explanations and exercises, often with the help of an instructor or teaching assistant. However, students who don't have the tolerance for incomplete explanations but who might make fine students if more is spelled out for them might find a resource that fits them better here. Furthermore, even the former group of students often leave calculus with some glaring misconceptions despite their persistence, and a more expository presentation should benefit them as well.

2. There is much experimentation with topics and orders, though if it is important enough for most calculus texts, it will be here too, eventually anyhow.
3. The whole project is done in "black and white," with no color illustrations (though plenty of figures are included), and no margin notes (though I make liberal use of footnotes), and the only use of multicolumns is in the exercises, to set them apart.
4. You can have these beta versions for free! In fact, if you are an instructor and have any use for anything here, feel free to print it up and distribute it, as long as you (a) do not profit from it (through, for instance, sales), (b) give me credit, and (c) don't try to include it in anything you're writing without first asking me and of course giving me credit. I'd be very pleased if any of my ideas here are helpful to anyone, and I'll be the first to give my permission for that. While I do hope it will eventually make it into hardback print, which I intend to be much cheaper than present offerings, I don't mind a bit of free advertising. In fact, if you have any ideas to share I'll put them in there and give you credit, assuming I like them too, and they seem to be an appropriate fit.

Note that the text is first formatted in LaTeX, which generates a DVI file, which is then converted to Postscript, and then to PDF with ps2pdf. A good program for rendering the PDF files for the screen is Adobe's Acrobat Reader (see link at bottom), though there are others, particularly gv and evince for Linux.

Graphics are handled almost entirely through the PSTricks package under LaTeX. Thanks so much to that community, particularly Herbert Voss for all the great work done on Timothy Van Zandt's original PSTricks package. See also notes at the bottom of the Chapter 12 commentary, at the bottom of this page.

Actually, after a long attempt in April, 2009 I gave up and borrowed some graphics from "wikimedia" for conic sections at the front end of Chapter 12. Sigh. --MMD

• Front Matter: title page, preface, table of contents. Thanks to Dr. Michael Green for advice on "toning down" my criticisms of well-established approaches to Calculus texts and instruction. 159,372 bytes.
• Chapter 1: Symbolic Logic. Was fairly complete, but got to be too large. I am revising and cutting it extensively, informed by some experience from using it in my classes. I do want students to see the quantifiers, since we use them in epsilon-delta definitions and proofs. 539,406 bytes.
• Chapter 2: Real Numbers, Algebra and Functions. Very incomplete and needing revision. I am trying to have the reader revisit algebra and functions in a different and sophisticated way, to encourage him (or her, I use it with "gender-neutral" intent) to think more rather than just "go through the motions." You can have a look at my first attempts to diagram functions. If there's a better system of doing so out there, let me know! 2,170,823 bytes.
• Chapter 3: Limits and Continuity. Here's where I made the most progress at the outset. Hopefully a more complete treatment of continuity and limits than in other textbooks. I concentrate heavily on "forms," both determinate and indeterminate. Except for the sequence sections I'm still mulling whether to include, I have used this in courses three times. My best students think this is hard, so I must be doing something right! Limits are hard, but after this they should be able to hand-wave more of the kind of limits you get with wacky improper integrals and ratio/root tests. 936,210 bytes.
• Chapter 4: Computing derivatives. Fairly complete, but I want to sneak in more graphing so I don't have to make a big deal about it in a single section later. I am very big on Leibniz notation! I am presently needing more and revised exercises. Note that I do ALL differentiation rules, with the exception of hyperbolic functions, here. So the logs, exponentials, arc-trig's and (f(x))^g(x) (check out my explanation for a general formula for that derivative!) are all done here. Hardly any applications are included yet, though I will include more graphing problems within the text here. The purists will note that I do graphing before the Mean Value Theorem, so I let students play with increasing/decreasing before I prove f '>0 implies f increasing, for instance. This is done for pedagogical reasons. It seems intuitive enough, and the proof can occur later when the fact is well-learned. I still have to find a way to work second derivatives into the text better. The next chapter will include the MVT, its applications, and the other usual applications, including more graphing, when all is done. 1,164,894 bytes.
• Chapter 5: Analyzing Functions with Derivatives. Still in infancy. Here I do get to use the logic more, since the statements, proofs and applications of theorems are often require logically equivalent forms of the basic results.

  One thing I plan to have different about my approach is that I will mix some graphing problems in with Chapters 3 and 4, so that I do not have to have omnibus sections of this chapter devoted to graphing. Graphs reflecting the sign of f '(x) do not have to wait until this chapter, though the proof that positive derivative means increasing function will be done here. This will have related rates, MVT applications (f ' in some range, what is the range in which we can find f given a starting point?), max/min problems, etc. For now, don't expect much here. 328,240 bytes.

• Chapter 6: Basic Integration. Still in infancy. The usual stuff at the moment. I plan to show why the Fundamental Theorem is more obvious, by exploring position/velocity.

  At the moment there is an "artifact" hyperbolic section, that is just beginning, and likely to end up somewhere else.

  One key here is that I am very careful in choosing the method for substitution. The short-cut, completing-the-chain-rule-derivative method is avoided here because it is a dead end, and nearly useless for advanced methods of the next chapter. Why authors continue to advance this as the method to use here is beyond me. Instead, my method here (and of course I'm not claiming I invented this but that I'm choosing this mehtod) is to replace everything in the original integral with u and du terms. I show the reader the other methods but focus on the method I believe has more use in their future studies, such as trigonometric and "miscellaneous" substitution.

  See also notes on the next two chapters. 450,523 bytes.

• Chapter 7: Advanced Integration Techniques.

  Note: this is getting a lot of updates lately because I'm using it in a class I'm teaching, so see the file at the top (whole book) for the latest.

  I'm maybe 3/4-finished here with the examples and explanations. Parts, partial fractions and trig integrals just need polishing and problems. Trig substitution is a new section that has seen some improvement over the last year but should probably be augmented. And they all need more problems! Tables and other techniques not yet begun. Also some techniques for more general, uglier rational integrands in the spirit of Edwards & Penney will hopefully appear here. 496,134 bytes.

• Chapter 8: Applications of Definite Integrals. Barely started (less than two pages!) But I feel inspiration is coming. Maybe a Summer project.

  Hmm, Summer is now over and I haven't done anything yet. Maybe a Fall project.

  Hmm, it seems to be almost the Spring Semester. We'll see.

  After consulting with respected colleagues, I finalized (I think) the decision to keep this after the advanced integration techniques, presently Chapter 7. My one-time chairman Dr. Gerard East in particular made the case that students just don't get enough integration practice immediately after the advanced techniques until multivariable stuff or differential equations otherwise. In his experience, when the applications follow the techniques, students "know how to integrate" when they do applications and then later multivariables, whereas they forget otherwise. In other words, the applications reinforce the techniques.

  Now any good author who does applications before advanced techniques also scatters some new applications throughout the advanced techniques chapter, but realistically, the instructors and students are concentrating on the techniques and, while any applied problems assigned might reinforce the applications well, and the techniques somewhat, the techniques themselves are already a "black hole" of time and energy, for which further applications are often a distraction (therefore discarded) for students and instructors who are mortal. Presently 26,579 bytes.

• Chapter 9: Advanced Limit Techniques and Improper Integrals: This got a big boost from my Calc 2 course in Spring 2008. I haven't looked at it lately so it's tough to comment, but I liked what was emerging so far. 606,583 bytes.
• Chapter 10: Series of Constants. This also got a boost from my Spring 2008 Calc 2 course, so it is much bigger than my last description posted here.

  I'm not 100% sure that I will follow the usual outline. Many professors have indicated they would like to see power series earlier, and later worry about convergence in that context. I may experiment to see if this and Chapter 12 can be consolidated in a respectable manner.

  But so far I seem to be following the standard outline, more or less. 373,755 bytes.

• Chapter 11: Taylor polynomial/series stuff. A few errors (of omission) were recently corrected.

  This was actually my first chapter written, as I wanted something like it for a course I was teaching. The style is not as refined (perhaps). I recently replaced all graphics (formerly Mathematica^TM and LaTeX Picture) with pstricks-generated graphics.

  There is a good chance large parts of this will be overhauled. Or else earlier parts of the book will need to be adjusted so students are prepared for the spirit of the early presentation here, such as earlier introduction of the idea that if f(x) and g(x) are approximately equal, so to some extent will be their definite integrals (depending upon the intervals, etc.). There is some good, even original stuff here, but some of those parts read more like a seminar for undergraduates in alternative perspectives on Taylor poynomials.

  To be sure, the usual, "let's algebraically find a polynomial that matches all the derivatives of f at x=a" is perhaps too far removed from normalcy for most novice calculus students. I like to think my method of working from more and more refined hypothetical assumptions placed on the function f (f constant, then instead f ' constant, then f ' ' constant, etc.) and deriving polynomials by integration--which also gives good intuition into the reasonableness of the error terms--more motivating to students. 739,413 bytes.

• Chapter 12. I thought I'd at least get started on some analytic geometry, inluding vectors and polar/cylindrical/spherical coordinates. I got a few more pages written but it's still in its infancy. I also included the "layout" pages for any LaTeX folks who are wondering. I should point out that, at the time anyhow, it seemed simpler to just fool the program into thinking I wanted A4 paper, so it wouldn't complain so much when I wanted to use more of the page. If I recall correctly, I was using an original print of Angus Taylor's General Theory of Functions and Integration to figure out the dimensions I wanted. I've been told by folks who know better that my lines of text may be a bit long, and I may need another approach, though I'm not sure what. I don't like how most calculus texts leave wide margins for occasional figures, which means a lot of wasted space. I'm finding out that the LaTeX figures in the book class do too, so I'm trying to minimize that. Anyhow, this last file is 212,126 bytes.