📄 Executive Summary
Active Calculus is an introductory single-variable calculus textbook intended for college undergraduates and self-directed learners seeking a conceptual, activity-driven approach to the discipline. The text assumes a standard precalculus foundation, including high-school algebra, function behavior, and trigonometry. Rather than relying on traditional lecture expositions or passive collections of worked examples, the book uses an inquiry-oriented structure where students investigate contextual problems, observe patterns, and construct mathematical meaning. Formal proofs are largely replaced with plausibility arguments, geometric reasoning, and physical intuition.
The curriculum progresses from foundational rates of change through integration, differential equations, and infinite series approximations. It begins by investigating straight-line motion and average velocity to introduce the limit concept and the derivative. After establishing continuity, differentiability, the second derivative, and local linearization, the text systematically presents standard differentiation techniques for power, exponential, trigonometric, inverse, and composite functions, along with implicit differentiation. These computational rules are applied directly to related rates problems, indeterminate limits via L'Hôpital's Rule, curve sketching for parameter-dependent families of functions, and single-variable optimization.
From rate of change, the text transitions to accumulation, using the relationship between velocity and distance to define Riemann sums and the definite integral. Readers study both forms of the Fundamental Theorem of Calculus and develop symbolic integration techniques, including substitution, integration by parts, and partial fractions, while also learning computer algebra usage and numerical approximations such as the Midpoint, Trapezoid, and Simpson's rules. The text applies integration to calculate planar area, arc length, volumes of solids of revolution, mass distribution, center of mass, work, and hydrostatic force. The final chapters extend these concepts to first-order differential equations—covering slope fields, Euler's method, and separable models like logistic population growth—and conclude with Taylor polynomials, geometric series, power series operations, and error estimation using the Alternating Series Estimation Theorem and the Lagrange Error Bound.
After completing the book, a reader will be able to construct and interpret mathematical models of changing quantities, determine derivatives and integrals through algebraic and numerical methods, analyze dynamic systems qualitatively and quantitatively, and approximate complex functions with bounded error. The text intentionally restricts its scope to single-variable calculus, setting aside multivariable calculus, vector fields, and formal epsilon-delta real analysis.
