1 Limits of Functions Defining function limits via epsilon-delta and proving limit laws and squeeze theorem 2 Continuous Functions Definition and properties of continuous functions: extreme and intermediate value theorems 3 Limits of Sequences Convergence of sequences, limit laws, standard limits and e, monotone convergence 4 Infinite Series Partial sums, convergence, geometric and p-series, convergence tests, absolute and conditional convergence 5 Power Series Power series, radius of convergence, elementary function expansions, and analytic functions 6 Differentiation and Derivatives Definition of the derivative, differentiability and continuity, derivatives and higher-order derivatives 7 Differentiation Termwise power series differentiation, elementary function derivatives, product, quotient, and chain rules 8 Mean Value Theorem Mean value theorem and applications: monotonicity, extrema, convexity tests, L’Hospital’s rule, optimization 9 Taylor’s Theorem Taylor polynomials, Lagrange remainder, Maclaurin expansions, approximation and limits 10 Integration Antiderivatives, indefinite integrals, Riemann sums, definite integrals, properties, and mean value theorem 11 The Fundamental Theorem of Calculus Fundamental theorem, existence of antiderivatives, Leibniz rule, termwise integration of power series 12 Improper Integrals Improper integrals over infinite intervals, singular integrals, comparison test, absolute convergence 13 Curves and Vector-Valued Functions Vector-valued functions, parametric curves, velocity, tangents, arc length, curvature, and acceleration dec... 14 Functions of Several Variables and Partial Derivatives Partial derivatives, gradient, differentiability, multivariable chain rule, extrema 15 Multiple Integrals Multiple integrals, Fubini’s theorem, change of variables, and the Jacobian determinant 16 Vector Fields Vector fields, gradient and conservative fields, potentials, divergence and curl, differential identities 17 Line Integrals Scalar and vector line integrals, work, fundamental theorem, path independence, conservative fields 18 Green’s Theorem Green’s theorem, area formulas, circulation and divergence forms, simple connectivity, conservative fields 19 Surface Integrals and Flux Parametric surfaces, normal vectors, surface area, scalar surface integrals, flux 20 The Divergence Theorem and Stokes’ Theorem Divergence theorem, Stokes’ theorem, irrotational and conservative fields, unification of integral theorems