Our text differs significantly from standard treatments. Here are some of these differences, together with some of our assumptions, goals, and strategies.
Combining graphical, numerical, and algebraic viewpoints. Throughout the text we insist that students manipulate and compare graphical, numerical, and algebraic representations of mathematical objects. In studying functions, for example, students manipulate not only elementary functions but also functions presented graphically and tabularly. In the context of formal differentiation, exercises ask students to apply the chain rule to combinations of functions presented in various ways. Graphical and numerical techniques, with error estimates, complement algebraic antidifferentiation methods. For series, routine convergence tests are emphasized less than finding-and defending-numerical limit estimates.
A mainstream course. Our main strategy-combining graphical, numerical, and algebraic viewpoints to clarify concepts and make them concrete-aims explicitly at mainstream students, who especially need such help. We regard a more conceptual calculus as also more applicable. To use calculus ideas and techniques, students must know what they are doing and why, not merely how. Thus, our text is appropriate for a general audience: mathematics majors, science and engineering majors, and non-science majors.
Concepts vs. rigor. Proving theorems in full generality is less valuable, we think, than helping students understand concretely what theorems say, why they're reasonable, and why they matter. Too often, fully rigorous proofs address questions that students are unprepared to ask.
Still, we believe that introducing calculus students to the idea of proof-and to some especially important classical proofs-is essential. We prove major results, but emphasize only those that we believe contribute significantly to understanding calculus concepts. In examples and problems, too, we pay attention to developing analytic skills and synthesizing mathematical ideas.
More varied exercises. However clear its exposition, a textbook's problems generate most of students' mathematical activity and occupy most of their time. Through the problems we assign, we tell students concretely what we think they should know and do.
Routine drills and challenging theoretical problems are standard in texts; ours includes some of both. More distinctive, though, are problems that fall between these poles:
Strategies for better problem solving. Emphasizing problem solving is nowadays de rigeur in calculus textbooks; ours is no exception. What, concretely, does such an emphasis entail in content and strategy?
Conceptual understanding, we believe, is the weakest link in students' ability to solve nontrivial problems. Thus, the goal of better problem solving skills is implicit in the goal of deeper conceptual understanding. To that end: