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.
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.
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,
to add clear explanations of a handful of very basic things --
are often missing from conventional calculus classes.
more thing deserves some comment. This section of the website is
primarily concerned with ordinary, single-variable calculus.
However, we occasionally invoke Euclidean N-space 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 single-variable calculus is easiest
to understand when the explanation is given in Rn
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
, and if not, perhaps it will be
possible to pick up the context from the explanations as they are
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. Picture-based
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.]
||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
of the chain rule, as well as any oddities which may come up along
on integrals; some illustrations to go with the definition, and a
discussion of why the definition of the Riemann integral is
way it is.
|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
pitfalls of that approach, and present a rigorous proof of the
||A side trip into something
we'll need a little later
||Another side trip into
something else we'll need a little later
Powers of X
|The integrals of x, x2, x3
... These are typically presented as anti-derivatives, but
they're easy enough to derive -- and understand -- as geometric
objects. As fundamental integrals, they're worth understanding
Powers of X
|The derivatives of x, x2, x3
... These are fundamental, so they're worth understanding at a
fundamental level, too.
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.
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.