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Calculus of a Single Variable

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, 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 encountered Rn, 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.  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.]

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.
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, 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 well.
Derivatives of
Powers of X
The derivatives of x, x2, x3 ... 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.