Space and shape have special importance in science and technology, as they underly an impressive range of realworld structures and phenomena. Thanks to concepts and methods from calculus, it becomes possible to approximate functions, curves, surfaces, and vector fields in terms of respective first-order (tangent) approximations obtained by using the Taylor series. By providing an effective approximation, this approach has been extensively adopted not only in theoretical studies (e.g. linear approximations of non-linear dynamics), but also in the most diverse types of applications. The present work is aimed at providing an accessible introduction to first-order approximation of several mathematical mappings and structures, with emphasis on the possibility of using these approximations as a means to define local coordinate systems that adapt effectively to the studied mathematical entity. After providing a motivation about first-order approximations respectively to single-variable functions, we proceed to parametric curves, multivariate functions, implicit functions and surfaces, as well as vector fields. Several numeric case-examples and illustrations have been provided to help the consolidation of the presented topics.