Dr. Michael Dougherty's Calculus Textbook Project
This page contains excerpts from my calculus textbook project, written
in collaboration with John Gieringer of Alvernia University,
Reading, Pennsylvania. It can be downloaded in its entirety
(9,662,820 bytes, updated November 20, 2012) here:
in collaboration with John Gieringer of Alvernia University,
The "big file" above will always be the most recent available version.
Older versions of the individual chapters are given below, for smaller
downloads. Those files may be very old!
(OK, I updated them November 20, 2012.)
Advice for Other Authors
If you are intersted in using your own textbook manuscript in your
classroom, I have some advice for you. For a little paper
on my own personal experiences, check out
Teaching from Your Textbook Manuscript. While much
of it probably seems obvious enough, one strategy which worked
especially well for me
was to have students turn in "binders" for grades, collated by
topic. So for my Section 1.1, they would first have
the pages from my textbook project (2-sided, pre-punched),
then my classroom notes in their own hands,
then the graded homework and possibly quizzes.
Next would be the same for Section 1.2, and so on,
until it was time for an exam. They turned in their binders the
day they took the
exam, and when the exam was returned they were expected to put it
into the binder after all sections it covered, along with the key.
Then it would all start over again. This really helped them to see
how to organize things. In fact by the end of the semester many had
two or three different binders because it piled up so much,
but being organized as it was, on the second (or so) time through
it made for an easy read.
As simple as it sounds, I had many
students tell me they started doing it in their other courses,
even though it was not required, and how amazed they were how
effective it was to help them study. Before I had them do this for
a grade, they had the handouts scattered throughout their "stuff,"
causing predictable problems.
The paper above describes more of the hows and whys of teaching from
your own book. Maybe there's something interesting and useful to be found
in the paper. I sure enjoyed writing it and presenting it at an MAA
All Web Updates of the Book are Frozen Until Further Notice
This project is getting closer to completion, and we are hoping
to work harder on a publishable form, and for copyright
reasons it might not appear here when completed. Also, there are some
new innovations, and admittedly we don't want to give away
any more of the store before publication.
Also some courses are
classroom-testing it in its present form, and it could goof
them up if we keep changing the content.
A big thank you and shout-out to Professor Sarah Koskie of IUPUI for some
suggestions for improvement, and for "field-testing" some of the material
in one of her classes.
Section 5.1 has had a lot of changes and additions. It's close to complete.
We introduce higher-order derivatives, explain the meaning of f'', and
immediately set off to graphing. Most professors would probably want
to make it a two-lecture section.
Chapter 11 had lots of changes. A few minor tweaks will happen in the
next day or so but otherwise it will be frozen for a while.
Chapter 5 is now on the editing block.
Our textbook is meant to be read--curled up with and read! We go
into much detail, with more examples than most in the topics
which are the most confusing. If you currently are using
one of the standard textbooks, you can use ours as a supplement.
If you want to learn calculus on your own, ours can be your
main text (though the standard ones have more exercises).
We occasionally hear from folks around the world
who happened upon this site
that they found it useful, and we hope you agree.
A few "differences" might turn off some readers, particularly regarding
the first two chapters. We
spend a chapter on symbolic logic so we can use it elsewhere,
including a rather in-depth (for a calculus book) development
of epsilon-delta proofs. If you just want more complete
examples for derivative and integral problems, you can
go to those sections (preferably from the current "whole book"
file linked above).
If you want a much deeper, and more useful and "form-based"
discussion of limits (to better prepare students
for things like convergence tests and improper integrals), check those out.
Another difference is that there is a very expanded Chapter 2, which
is still preliminary but has some good algebraic stuff which is easier
to accomplish after raising the sophistication level, made possible
by a study of Chapter 1's logic.
While this is a work in progress, some chapters and more
sections are basically complete
and have been successfully used in courses. We're hoping this is the year we
can really push it towards completion.
If you have any comments please send them to
If you found this material useful, we would like to know.
You are free to use this pre-published material in your classes
or for your own studying, but if you use printed versions
we would appreciate credit, and you letting us know.
We are most encouraged to work longer hours
to complete the book when we hear from readers who
found it useful. If we are doing something wrong we'd like to know
Check back for updates periodically.
Most Recent Changes
John Gieringer (of Alvernia College in Reading, Pennsylvania)
is the second half of the "we" and "us" mentioned above, and
has contributed many and varied fine examples, and caught
numerous mistakes, and made the original author need
to learn to better draw circuits! (Thanks a lot, John, ;-))
These are spread all over the first five chapters.
More on why this book is different
In case you're wondering why this book is different, here are a few
Special Thank-Yous (Michael Dougherty):
I have had a couple of people send me
edits that I have not had time to incorporate. I have them saved though,
and will include you in my acknowledgments when the time comes.
If you've sent me feedback regarding this project, this means
- 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 we wrote is true,
or for that matter what was meant. At least that's the goal:
a book you can relax with to read calculus!
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 but getting there) 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,
arguably 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
- 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.
- The whole project is done in "black and white," with
no color illustrations, though plenty of figures are included.
Well, actually one colored graphic was stolen from Wikipedia.
There are no margin notes (though we make liberal use of footnotes),
and we only use multicolumns in exercises and when it can
save a tremendous amount of space, or when a small figure
or group of equations can be wrapped by the explanation.
- 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 us credit, and (c) don't
try to include it in anything you're writing without first
asking us and of course giving us credit. We'd be very pleased
if any of our ideas here are helpful to anyone, and we'll
be the first to give our permission for that. While we do
hope it will eventually make it into hardback print,
which we intend to be much cheaper than present
offerings, we don't mind a bit of free advertising.
In fact, if you have any ideas to share we'll put them
in there and give you credit, assuming we 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.
Individual chapters, current as of November 20, 2012 (see top link for
whole book as of that date)
Please note that all book pages are copyright Michael M. Dougherty.
- 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. 168,289 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. Set theory stuff remains, though not as
sophisticated as it was at the start of this project.
- Chapter 2: Real Numbers, Algebra and
Functions. Getting more complete, but probably needing
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 latest, and probalby last, attempts to diagram functions
to help students visualize complicated functions as multistage
processes, or "machines."
If there's a better
system of doing so out there, let me know! 2,248,287 bytes.
- Chapter 3: Limits and Continuity.
This started with the standard treatment, and then it just
occured to me that there are better ways, so it got a major
overhaul, probably around 2003, and has stayed in the same
spirit ever since. My students find it hard but they ask
much more sophisticated limit questions than before. 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. Even 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. 1,208,453 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
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.
- 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 we plan to have different about the approach is that
we will mix some graphing problems in with Chapters 3 and 4, so that
we 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. 872,372 bytes.
- Chapter 6:
Basic Integration. Still in alpha stage, if that. The usual
stuff at the moment. One plan is 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 we are 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, our method here (and of course we're not claiming to have
but that we're choosing this method) is to replace everything in
the original integral with u and du terms. We show
the reader the other methods but focus on the method we believe
has more use in their future studies, such as trigonometric
and "miscellaneous" substitution.
See also notes on the next
two chapters. 800,431 bytes.
- Chapter 7: Advanced Integration Techniques.
This has had some good update in the last few years because
I taught the course, and freshened up this chapter, which
was already pretty well developed.
I'm maybe 7/8-finished here with the examples and explanations.
Parts, partial fractions and
trig integrals just need polishing and problems. Trig
is a new section that has seen some improvement over the
recent years 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. 557,943 bytes.
- Chapter 8: Applications of Definite Integrals.
Barely started (less than two pages!) It has definitely been
on the back burner awaiting inspiration.
After consulting with respected colleagues, I finalized (I
decision to keep this chapter placed 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
Presently 62,145 bytes
- Chapter 9: Advanced Limit Techniques
and Improper Integrals: This got a big boost
from my Calc 2 courses in Spring 2008 and 2009.
I had put it aside for years, but once I got started it
really was kind of fun, particularly using PSTricks
to shade under curves for improper integrals.
- Chapter 10: Series of Constants.
This also got a boost from my Spring 2008 and 2009 Calc 2 courses.
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. 453,511 bytes.
- Chapter 11: Taylor polynomial/series
This has been given a preliminary overhaul, with the original
(plodding) introduction replaced with a summary of what
will be developed and why we should care, an early introduction
of Taylor Polynomials followed by a shorter (truncated)
development from first principles, then the usual stuff,
plus a section which discusses when things go wrong:
with piecewise-defined functions, and functions where the
singularity doesn't lie on the real line, but is found
in the complex plane. There is some introduction to
complex variables, hopefully just enough to whet the
appetite but not enough to scare anyone away from them.
I will (soon?) make another pass over this chapter since
this major overhaul to be sure it works the way I want it
to. 817,070 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 (waste space), so I'm trying to minimize that.
Anyhow, this last file is
927,475 bytes, which is up quite a bit from last time
I updated all the chapter links, so I must have done
something in the meantime.
Michael M. Dougherty,
home page here.