Differential Calculus: From Practice to Theory
covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the foundational ideas (limits, continuity) are developed to replace infinitesimals, first intuitively then rigorously. This approach is more historically accurate than the usual development of calculus and, more importantly, it is pedagogically sound. The text also incorporates curated activities from the TRansforming Instruction in Undergraduate Mathematics Instruction via Primary Historical Sources (TRIUMPHS) project to provide students with ample opportunities to develop relevant competencies.