From the Preface: We want you to reason with mathematics. We are not trying to get everyone to give formalized proofs in the sense of contemporary mathematics; 'proof' in this course means 'convincing argument. ' We expect you to use correct reasoning and to give careful explanations. The projects bring out these issues in the way we nd best for most students, but the pure mathematical questions also interest some students. This book of mathematical \background" shows how to ll in the mathematical details of the main topics from the course. These proofs are completely rigorous in the sense of modern mathematics {technically bulletproof. We wrote this book of foundations in part to provide a convenient reference for a student who might like to see the \theorem - proof" approach to calculus. We also wrote it for the interested instructor. In re-thinking the presentation of beginning calculus, we found that a simpler basis for the theory was both possible and desirable. The pointwise approach most books give to the theory of derivatives spoils the subject. Clear simple arguments like the proof of the Fundamental Theorem at the start of Chapter 5 below are not possible in that approach. The result of the pointwise approach is that instructors feel they have to either be dishonest with students or disclaim good intuitive approximations.