Adams and Essex's calculus book, covering Calc 1, 2, and 3. Simpson's Rule gets its own section here, which most books don't do.
Review of real numbers, coordinate geometry, functions, polynomials, rational functions, and trigonometric functions used throughout calculus.
Examples of velocity and area; limits of functions; limits at infinity and infinite limits; continuity; the formal epsilon-delta definition of limit.
Tangent lines and the derivative; power, product, quotient, and chain rules; trig and higher-order derivatives; differentials, the Mean Value Theorem; implicit differentiation; antiderivatives, velocity and acceleration.
Inverse functions; natural logarithm and exponential; growth and decay; inverse trig functions; hyperbolic functions. Adams uniquely places intro to second-order linear DEs in this chapter.
Related rates, root finding, indeterminate forms, extreme values, concavity, curve sketching, applied optimization, linear approximations, Taylor polynomials.
Sums and sigma notation; areas as Riemann sums; the definite integral and its properties; the Fundamental Theorem of Calculus; substitution; areas of plane regions.
Integration by parts, partial fractions, trig substitution and other inverse substitutions, additional methods (tables/CAS), improper integrals, trapezoid and midpoint rules, Simpson's rule. Adams 10e is unique in natively separating Trapezoid+Midpoint (§6.6) from Simpson's (§6.7) as distinct sections.
Volumes of solids of revolution; arc length and surface area; physical applications (work, fluid pressure); first-order separable differential equations and their applications.
Parametric curves; their slopes, arc lengths, and areas; polar coordinates; polar curves and their geometry.
Sequences; infinite series; convergence tests; absolute and conditional convergence; power series; Taylor and Maclaurin series and their applications.
Three-dimensional analytic geometry; vectors and operations; cross product; lines and planes; quadric surfaces; cylindrical and spherical coordinates.
Vector-valued functions of one variable; calculus of vector functions; curvature, torsion, and the Frenet frame.
Functions of several variables; limits and continuity; partial and higher-order derivatives; the chain rule; linear approximations and differentiability; gradients and directional derivatives.
Extreme values; optimization on restricted domains; Lagrange multipliers. Adams 13.4-13.8 cover advanced/niche topics (n-space Lagrange, least squares, parametric problems, Newton's method for systems, calculus of variations) deferred per single-book candidate gaps.
Double and triple integrals over regions; iterated integrals; integration in polar, cylindrical, and spherical coordinates; change of variables.