This section of the website is dedicated to all those who've ever found
themselves struggling just to grasp the basics of elementary calculus,
as a result of unclear texts and opaque explanations.
Elementary
calculus is just geometry wearing a tuxedo, with a little algebra for
decoration. Nearly all the basics can be explained with pictures.
"Epsilonics", while useful, powerful, and vital for use in
analysis, just tends to get in the way of obtaining an intuitive
understanding of what it's all about at the introductory level.
Throughout,
I will generally be assuming some familiarity with the subject. I
expect that readers either will have taken a calculus course, or will
at least have access to a conventional calculus textbook. This
section is certainly not complete enough to cover the whole subject of
calculus from the ground up  Thomas's ninth edition, for example,
weighs just over
five pounds, and it's not because old George B.
was a particularly verbose writer! What I'm attempting to do here,
rather, is
to add clear explanations of a handful of very basic things 
explanations which
are often missing from conventional calculus classes.
One
more thing deserves some comment. This section of the website is
primarily concerned with ordinary, singlevariable calculus.
However, we occasionally invoke Euclidean Nspace in our
explanations, despite the fact that the concept of higher dimension
spaces is often not encountered until linear algebra, which is most
often taken
after elementary calculus (so it's cheating to use
it when explaining the basics). It's an annoying fact that that a
lot of what's going on in ordinary singlevariable calculus is easiest
to understand when the explanation is given in
R^{n}
rather than keeping within the "flat" space of the blackboard  and
very little in linear algebra actually requires calculus. I will
simply hope that most people reading these pages will already have
encountered
R^{n}, and if not, perhaps it will be
possible to pick up the context from the explanations as they are
given.
[Finally,
please be aware that, as of 11/04/2007, these pages are a work in
progress. There is more material to come, and I haven't yet gone
back over all of the initial material to see where my explanations are
murky, or I've introduced concepts or notation without
explanation, or just plain made mistakes. Picturebased
explanations and proofs tend to take substantial time to produce, so
the pace at which additions are made may be slow, even though the
"pending" material has already been written up ... on paper.]
Introduction 
In which we are introduced
to the subject, with some illustrations of what I mean by opaque
explanations, along with some basic definitions and a brief
discussion
of the chain rule, as well as any oddities which may come up along
the
way. 
More
on Integrals 
More
on integrals; some illustrations to go with the definition, and a
brief
discussion of why the definition of the Riemann integral is
phrased the
way it is. 
Fundamental
Theorem 
Derivatives and integrals
are inverse operations; we motivate that fact on this page.
In addition, we discuss the use of dx
and "sloppy infinitesimals" at some length, and then talk a bit
about
pitfalls of that approach, and present a rigorous proof of the
chain
rule. 
Hypercubes 
A side trip into something
we'll need a little later 
Pyramids 
Another side trip into
something else we'll need a little later 
Integrals
of
Powers of X 
The integrals of x, x^{2}, x^{3}
... These are typically presented as antiderivatives, but
they're easy enough to derive  and understand  as geometric
objects. As fundamental integrals, they're worth understanding
well. 
Derivatives of
Powers of X 
The derivatives of x, x^{2}, x^{3}
... These are fundamental, so they're worth understanding at a
fundamental level, too.

Derivatives
of
Sin and Cos 
The derivatives of the trig
functions are simple and obvious, if we work
from a picture. All too often, calculus texts derive them using
opaque calculations involving explicit limits.

Product Rule and
Integration by Parts 
These two are intimately
linked. On this page we provide graphical proofs  or, at
least, motivations  for each of them. 
Page
created on 11/04/2007. Added derivatives of x^n on 9/19/2013
(just five years later), and a couple days later we added the
derivative of the sine. Leibniz arrived on 3/16/2018.