A high-school AP Calculus textbook by Sullivan and Miranda, AP-CED-aligned (2019+ revisions). Designed for the year-long AP Calculus AB and BC courses, with explicit Big Ideas, Mathematical Practices, and Student Skills mapping. Single-variable scope only.
Prerequisite material from precalculus: functions, library of functions, transformations, inverse functions, exponentials, logarithms, and trigonometry. Assumed knowledge before AP Calculus topics begin.
The foundational limit concept: numerical and graphical estimation, properties of limits, continuity at a point and on an interval, limits involving trig/exp/log functions, infinite limits, and limits at infinity.
Definition of the derivative, derivative as a function, fundamental differentiation rules, derivatives of polynomials/exponentials/trig functions, the chain rule, implicit differentiation, derivatives of inverse trig functions.
Using derivatives to solve problems: related rates, max/min values, MVT, curve analysis with f' and f'', L'Hôpital's Rule, optimization, linearization, Newton's Method, and antiderivatives.
Definite integrals as Riemann sums, the Fundamental Theorem of Calculus, properties of the definite integral, the substitution method, and a split treatment of numerical integration (Trapezoidal Rule + Simpson's Rule).
Using definite integrals to solve geometric and physical problems: area between curves, volumes by slicing/disk/shell, arc length, work, fluid pressure.
Introductory ordinary differential equations: separable DEs, exponential models, and the logistic-growth model.
Advanced integration techniques: integration by parts, trig substitution, partial fractions, improper integrals. Most BC-level content.
Sequences, infinite series, convergence tests, power series, Taylor and Maclaurin series. All BC-level content.
Parametric curves and their calculus, polar coordinates and polar-curve calculus, plane-vector functions and motion. All BC-level content.