Calculus I

This book covers the standard Calculus I sequence: limits and continuity, differentiation, applications of the derivative, indefinite and definite integrals, techniques of integration, and improper integrals.

It opens with a thorough review of the precalculus that the rest of the book assumes. Its guiding principle is that an idea should first make sense and only then be made precise. Limits are introduced through numerical and graphical exploration before the ε–δ definition appears, and when it does, the limit laws are proved rather than asserted. Results are developed geometrically wherever geometry is what explains them, and more than three hundred fully worked examples show the reasoning in full rather than the answer alone.

Optional passages on history, subtle counterexamples, and finer points of theory are set apart, so readers can follow the main argument straight through or pause to go deeper. The result is a text for the student who needs to solve the problem at hand, and for the one who wants to understand why the method works.