Author's Review of Advanced Calculus of One Variable

This book was written for the course at Colorado State University "Advanced Calculus of One Variable." It was always intended to be put on the web as a free book---under the heading of "How to Download the Text" I will describe the different versions available to anyone who wants to use it. I have tried to develop the text as I would any book that was to be published---I hope I have succeeded. Our course is a 4 credit course so we have approximately 60 class periods. I wrote the text with the idea of covering one section per day (there are 42 sections). If you look at it, you will see that I didn't (couldn't) stay with that criteria for all sections. I think that there are four or five sections that will clearly take at least two days.

My plan for teaching the course out of this text was to have the students read a section and do the homework problems at the end of the section. I would then begin the class by asking them if they had any questions---not expecting too much at first. I would then be prepared to discuss the True-False questions with them and discuss any proofs or examples that I thought were especially important. My plan (hope) is to teach them how to read mathematics and force them to talk about it in class. I will teach the course in the fall of 09 so we will see.

There are not as many homework problems as most texts. Though I sometime appreciate having many problems to choose from, I tried to choose my problems carefully---for most sections including problems that I wanted them to work. If you need and/or want more problems, I think you know where you can find many of them.

Maybe there is one thing where this text will be a bit different from all the rest is in that I consider this course as Calc 4, a continuation of Calc 1-3. Moreso, I consider the course as the place where you repeat Calc 1 and Calc 2, this time skipping the routine calculations and maybe doing the material the way it should have been done in the first place. I feel that this course, and especially this text, completes the Calc 1 and 2 courses. For this reason I think this course and text is especially good for Mathematics Education students who may some day teach AP Calculus.

The treatment of the material is reasonably standard. I felt that Chapter 1 was difficult to write---it's always difficult to get a course like this started. I decided to use the axiomatic approach to get the reals. I tried to include enough of the material so that they understand and can work with it, but tried also not to bore everyone to death. I tried to make it mathematically rigorous---though for sanity I decided not to prove the very important result that there is only one complete ordered field. I also include some basic material about proofs in Chapter 1. This material is needed for most of our students---it may not be necessary for your students.

I decided that I should put the topology next---because I think that's the most logical place for it. The bad thing about this approach is that the compactness proofs are tough. I considered the fact that it might be nice to not include the compactness results until later when the students might be better at the proofs. They surely can be skipped and returned to when needed.

Chapter 3 is a very common treatment of limits of sequences. I include infinite limits at the end of the chapter. In each case I always try to remember to include infinite limits (and later limits at infinity) but don't hit them too hard. Chapters 4-7 include all of the common results on limits of functions, continuity, differentiation and integration. I have used the traditional epsilon-delta definitions for limits of functions and continuity---but then included the sequential results immediately. I tried to force the students to understand the epsilon-delta approach, and appreciate the ease of the sequential approach.

One topic that is not included in all texts at this level is that of using our theorems on inverse functions to show that the functions x^r, where r is rational, are well-defined, continuous and differentiable. We used the same approach to show that the inverse trig functions, when the domain has been fixed so that they were invertible, were also continuous and differentiable. Then of course, when we got to a place where we could use the integral to define the logarithm, we defined the exponential function and showed that it was continuous and differentiable. This follows the "late exponential" approach of many of the Calc 1 and 2 texts.

And finally we try to treat real series, sequences of functions and series of functions via Taylor polynomials. We develop Taylor polynomials with the idea of approximating functions. The convergence of the sequence of Taylor polynomials leads us to convergence of sequences of functions. Of course the Taylor polynomials look like a series for large n, so we use Taylor polynomials with large n to motivate series---of functions. We decide that we must be able to know whether or not these series converge---pointwise. For this reason we introduce the usual material on convergence of real series. And then of course we decide that we want to know that our series are continuous, and we want to differentiate and integrate our series. Thus we include uniform convergence and the appropriate results. Again I think that though this material is surely at a higher level than in Calc 1 and 2, the material complements the approach taken in Cacl 1 and 2.

I hope the text works for you.