📄 Executive Summary
Active Prelude to Calculus is designed for college students preparing for single-variable calculus, whether in a formal preparatory course or through independent study. Assuming a standard background in high school algebra, the book focuses strictly on the mathematical concepts and reasoning required for calculus. Rather than functioning as a broad survey of general college algebra and trigonometry, the text intentionally narrows its scope to omit topics that do not directly support calculus, concentrating instead on functional behavior, rates of change, and mathematical modeling.
The curriculum progresses systematically across five major areas, grounded in the organizing principle of functions as dynamic input-output processes that represent quantities changing in tandem. Initial chapters establish foundational tools through function transformations, compositions, inverses, and the central concept of the average rate of change. From here, the text explores circular motion and trigonometry, introducing sine and cosine through the unit circle before examining right-triangle relationships, inverse trigonometric functions, and reciprocal identities. Subsequent sections examine exponential and logarithmic functions using the natural base, followed by power, polynomial, and rational functions. The text introduces limit notation to analyze long-term trends, end behavior, and asymptotes, establishing the direct bridge to calculus.
Throughout each topic, the book connects verbal, tabular, graphical, and symbolic perspectives while distinguishing exact values from numerical approximations. Readers develop practical modeling skills by building algebraic formulas for physical systems, including objects under gravitational acceleration, cooling fluids via Newton's Law of Cooling, constrained container geometry, logistic population dynamics, and cubic Bezier curves. Upon completing the book, readers should be able to analyze families of functions, construct equations from applied relationships, calculate rates of change across intervals, and evaluate asymptotic behavior. The book intentionally excludes advanced calculus techniques such as differentiation and integration, as well as peripheral algebra topics that do not serve calculus preparation.
