A high-school AP Calculus textbook organized around the "Rule of Four" (graphical, numerical, algebraic, verbal) approach. Aligned to the College Board AP Calculus Course and Exam Description (AB and BC).
A review of essential precalculus material: lines, functions and graphs, exponential functions, parametric equations, inverses and logarithms, and the trigonometric functions used throughout calculus.
The concept of a limit introduced graphically, numerically, and algebraically; one-sided limits; limits involving infinity; continuity at a point and on an interval; and an introduction to the tangent line problem.
The derivative defined as a limit; differentiability and continuity; differentiation rules including product, quotient, and higher-order derivatives; rates of change; and derivatives of trigonometric functions.
Advanced differentiation techniques: the chain rule, implicit differentiation, derivatives of inverse trigonometric functions, and derivatives of exponential and logarithmic functions.
Applying differentiation to extreme values, the Mean Value Theorem, curve sketching, modeling and optimization, linearization and differentials, and related rates problems.
Estimating with finite sums, the definite integral as a limit of Riemann sums, the relationship between integrals and antiderivatives, the Fundamental Theorem of Calculus, and numerical integration.
Antidifferentiation by substitution and by parts, exponential growth and decay modeling, separable differential equations, and the logistic growth model.
Using the definite integral as a model for net change, geometric applications including areas between curves and volumes by disks/shells, arc length, and applications drawn from physics and statistics.
Sequences and their convergence, L'Hospital's Rule for indeterminate forms, and improper integrals over infinite intervals or with unbounded integrands.
Power series and Taylor series, Taylor's theorem with remainder, finding the radius of convergence, and testing convergence at the endpoints of the interval of convergence.
Calculus on parametrically defined curves, planar vector functions and motion in the plane, and the calculus of polar functions including area and arc length.