Calculus is a magical subject. A first encounter in school leads to a radical revision of one's ideas of what is mathematics. We are transported from a rather staid enterprise of counting and measuring to an adventure encompassing change, fluctuation, and a vastly increased ability to understand and predict the workings of the world. At the same time, the student encounters “magic” with both its connotations: awe and wonder on the one hand, mystery and a sense of trickery on the other. Calculus can appear to be a bag of tricks that are immensely useful, provided the apprentice wizard can perfectly remember the spells. As the student pursues mathematics further at university, her instructors may use courses in analysis to persuade her that calculus is a science rather than a mystical art. Alas, all too often the student perceives the new instruction as mere hair-splitting which gives no new powers and may even undermine her previous attainments. The first analysis course is for many an experience that makes them regret taking up higher mathematics.

This book is written to support students in this transition from the expectations of school to those of university. It is intended for students who are pursuing undergraduate studies in mathematics or in disciplines like physics and economics where formal mathematics plays a significant role. Its proper use is in a “calculus with proofs” course taught during the first year of university. The goal is to demonstrate to the student that attention to basic concepts and definitions is an investment that pays off in multiple ways. Old calculations can be done again with a fresh understanding that can not only be stimulating but also protects against error. More importantly, one begins to learn how knowledge can be extended to new domains by first questioning it in familiar terrain. For students majoring in mathematics, this book can serve as bridge to real analysis. For others, it can serve as a base from where they can make expeditions to various applications.