A general single-variable and multivariable Early Transcendentals calculus textbook by Hughes-Hallett, Gleason, McCallum et al. 8e (2021). Wiley Calculus Consortium / Rule-of-Four (graphical, numerical, symbolic, verbal) pedagogy. Covers Calc 1, Calc 2, and Calc 3 across Ch 1-16 plus selected motion/parameterization sections of Ch 17.
A library of functions used throughout calculus: linear, exponential, power, polynomial, rational, trigonometric, logarithmic; and a first look at limits and continuity.
Average and instantaneous rates of change; the derivative at a point; the derivative as a function; interpretations and differentiability.
Differentiation rules for elementary functions; product, quotient, chain, implicit; hyperbolic and inverse functions; linear approximation.
Optimization, modeling, related rates, l-Hopital rule, parametric equations, and graph analysis using first and second derivatives.
The definite integral as a limit of Riemann sums; the Fundamental Theorem of Calculus and properties of integrals.
Antiderivatives graphically, numerically, and analytically; the Second FTC; differential equations and motion.
Techniques of integration: substitution, integration by parts, tables, trigonometric substitution, partial fractions, numerical methods, improper integrals.
Applications of the definite integral: areas and volumes, arc length, polar curves, applications to physics.
Sequences, geometric series, series convergence and divergence, tests for convergence, and power series.
Taylor polynomials and Taylor series; finding and using Taylor series; error estimation.
Introduction to differential equations; separable equations and growth/decay are kept.
Multivariable functions; graphs and surfaces; contour diagrams; linear functions of several variables; limits and continuity.
Vector concepts: displacement vectors, general vectors, dot product, cross product. Lines and planes in 3-space are folded into the cross-product section per Hughes-Hallett structure.
Partial derivatives, gradients, directional derivatives, the chain rule, second-order partials, and differentiability for multivariable functions.
Critical points, saddle points, local and global extrema for multivariable functions; constrained optimization with Lagrange multipliers.
Definite integrals of multivariable functions: iterated, triple, polar, cylindrical, and spherical.
Parameterized curves and motion in 2- and 3-space (kept; DCM Calc 3 vector-valued-function coverage).