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Philosophy

Several common threads run through Toward a Lean and Lively Calculus, Calculus for a New Century, discussions at many professional meetings, the report of the NCTM/MAA Joint Task Force on Curriculum for Grades 11-13, and the NCTM Curriculum and Evaluation Standards for School Mathematics. First, there is a consistent call for leaner and more conceptual courses, driven by and focused on central ideas. Second, there is a realization that courses should reflect modern technology both in content and in pedagogy.

A diagnosis. Many calculus courses, we believe, slight the conceptual foundations of the subject and overemphasize routine techniques-formal differentiation, antidifferentiation, convergence testing, etc. Analytic objects (integral, derivative, convergence, etc.) are represented and manipulated only algebraically (i.e., via symbolic manipulation of explicit elementary functions). For example, textbooks often treat limits, derivatives, and integrals-all analytic objects-only as algebraic operations on algebraic functions. We try to take a broader view.

Whether one views calculus as an introduction to pure mathematics or as a foundation for applications (or both!), the conclusion is the same-concepts, not techniques, are truly fundamental to the course. Whatever uses they make of the calculus, students need more than a compendium of manipulative techniques. The sine qua non for a useful command of the calculus is a conceptual understanding that is deep and flexible enough to accommodate diverse applications.

A prescription. Our key strategy for improving conceptual understanding is combining, comparing, and moving among graphical, numerical, and algebraic ``representations'' of central concepts. This strategy pervades and unifies our exposition. Bringing graphical and numerical, as well as algebraic, viewpoints to bear on calculus ideas is the philosophical foundation of our text. By representing and manipulating calculus ideas and objects graphically, numerically, and algebraically, we believe that students gain a better, deeper, and more useful understanding.

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