This book is a compendium of joint work produced by the authors during the period 1998-2012, most of it published in the American Mathematical Monthly, Math Horizons, Mathematics Magazine, and The Mathematical Gazette. The published papers have been edited, augmented, and rearranged into 15 chapters. Each chapter is preceded by a sample of problems that can be solved by the methods developed in that chapter. Each opening page contains a brief abstract of the chapter's contents.

Chapter 1, entitled “Mamikon's Sweeping-Tangent Theorem,” was the starting point of this collaboration. It describes an innovative and visual approach for solving many standard calculus problems by a geometric method that makes little or no use of formulas. The method was conceived in 1959 by my co-author (who prefers to be called Mamikon), when he was an undergraduate student at Yerevan University in Armenia. When young Mamikon showed his method to Soviet mathematicians they dismissed it out of hand and said “It can't be right. You can't solve calculus problems that easily.”

Mamikon went on to get a Ph.D. in physics, was appointed a professor of astrophysics at the University of Yerevan, and became an international expert in radiative transfer theory, all the while continuing to develop his powerful geometric methods. Mamikon eventually published a paper outlining them in 1981, but it seems to have escaped notice, probably because it appeared in Russian in an Armenian journal with limited circulation. (Reference [59] in the Bibliography.)