The recent European Programme Specifications have forced a reassessment of the structure and syllabi of the entire system of Italian higher education, and an ensuing rethinking of the teaching material. Nowadays many lecture courses, especially rudimentary ones, demand that students master a large amount of theoretical and practical knowledge in a span of just few weeks, in order to gain a small number of credits. As a result, instructors face the dilemma of how to present the subject matter. They must make appropriate choices about lecture content, the comprehension level required from the recipients, and which kind of language to use. This textbook is meant to help students acquire the basics of Calculus in curricula where mathematical tools play a crucial part (so Engineering, Physics, Computer Science and the like). The fundamental concepts and methods of Differential and Integral Calculus for functions of one real variable are presented with the primary purpose of letting students assimilate their effective employment, but with critical awareness. The general philosophy inspiring our approach has been to simplify the system of notions available prior to the university reform; at the same time we wished to maintain the rigorous exposition and avoid the trap of compiling a mere formulary of ready-to-use prescriptions.