Set Theory

• Logic and Set Notation
• Introduction to Sets
• Set Operations and Venn Diagrams
• Set Identities
• Cartesian Product of Sets
• Binary Relations
• Properties of Relations
• Operations on Relations
• Composition of Relations
• Closures of Relations
• Equivalence Relations
• Equivalence Classes and Partitions
• Partial Orders
• Total Orders
• Hasse Diagrams
• Special Elements of Partially Ordered Sets
• Well Orders
• Lexicographic Orders
• Lattices
• Topological Sorting
• Well Ordering Principle
• Counting Relations
• Relational Databases
• Functions as Relations
• Injection, Surjection, Bijection
• Inverse Functions
• Composition of Functions
• Floor and Ceiling Functions
• Counting Functions
• Pigeonhole Principle
• Cardinality of a Set
• Countable and Uncountable Sets
• Comparing Cardinalities
• Cantor-Schröder-Bernstein Theorem
• Cantor’s Theorem
• Ordinal Numbers
• Cardinal Numbers
• Paradoxes of Set Theory
• Axiomatic Set Theory

Limits and Continuity of Functions

• Definition of Limit of a Function
• Properties of Limits
• Trigonometric Limits
• The Number e
• Natural Logarithms
• Indeterminate Forms
• Use of Infinitesimals
• L’Hopital’s Rule
• Continuity of Functions
• Discontinuous Functions

Differentiation of Functions

• Definition of the Derivative
• Basic Differentiation Rules
• Derivatives of Power Functions
• Product Rule
• Quotient Rule
• The Chain Rule
• Derivatives of Exponential Functions
• Derivatives of Inverse Functions
• Derivatives of Logarithmic Functions
• Derivatives of Trigonometric Functions
• Derivatives of Inverse Trigonometric Functions
• Derivatives of Hyperbolic Functions
• Logarithmic Differentiation
• Implicit Differentiation
• Derivatives of Parametric Functions
• Derivatives of Polar Functions
• Derivatives of Vector-Valued Functions
• Second Derivatives
• Higher-Order Derivatives
• Leibniz Formula
• Table of Derivatives
• Differential of a Function
• Higher-Order Differentials

Applications of the Derivative

• Tangent and Normal Lines
• Rolle’s Theorem
• Lagrange’s Mean Value Theorem
• Cauchy’s Mean Value Theorem
• Newton’s Method
• Related Rates
• Linear Approximation
• Rectilinear Motion
• Planar Motion
• Critical Points
• Increasing and Decreasing Functions
• Local Extrema of Functions
• Global Extrema of Functions
• Convex Functions
• Inflection Points
• Asymptotes
• Curve Sketching
• Classical Inequalities
• Proving of Inequalities
• Curvature and Radius of Curvature
• Evolute and Involute
• Envelope of a Family of Curves
• Osculating Curves
• Optimization Problems in 2D Geometry
• Optimization Problems in 3D Geometry
• Optimization Problems Involving Numbers
• Optimization Problems in Physics
• Optimization Problems in Economics
• Van der Waals Equation

Integration of Functions

• Antiderivatives and Initial Value Problems
• The Indefinite Integral and Basic Rules of Integration
• Integration by Substitution
• Integration by Parts
• Integration by Completing the Square
• Partial Fraction Decomposition
• Integration of Rational Functions
• Integration of Irrational Functions
• Weierstrass Substitution
• Trigonometric Integrals
• Integration of Hyperbolic Functions
• Integrals of Vector-Valued Functions
• Trigonometric and Hyperbolic Substitutions
• Riemann Sums and the Definite Integral
• The Fundamental Theorem of Calculus
• Trapezoidal Rule
• Simpson’s Rule
• Improper Integrals

Applications of Integrals

• Area of a Region Bounded by Curves
• Volume of a Solid with a Known Cross Section
• Volume of a Solid of Revolution: Disks and Washers
• Volume of a Solid of Revolution: Cylindrical Shells
• Area of a Surface of Revolution
• Average Value of a Function
• Probability Density Function
• Arc Length
• Net Change Theorem
• Distance, Velocity and Acceleration
• Force, Work and Energy
• Fluid Pressure
• Mass and Density
• Center of Mass and Moments
• Moment of Inertia
• Pappus’s Theorem
• Integrals in Electric Circuits
• Applications of Integrals in Economics

Infinite Sequences and Series

• Infinite Sequences
• Geometric Series
• Infinite Series
• Comparison Tests
• The Integral Test
• The Ratio and Root Tests
• Alternating Series
• Power Series
• Differentiation and Integration of Power Series
• Taylor and Maclaurin Series

Double Integrals

• Definition and Properties of Double Integrals
• Iterated Integrals
• Double Integrals over Rectangular Regions
• Double Integrals over General Regions
• Change of Variables in Double Integrals
• Double Integrals in Polar Coordinates
• Geometric Applications of Double Integrals
• Physical Applications of Double Integrals

Triple Integrals

• Definition and Properties of Triple Integrals
• Triple Integrals in Cartesian Coordinates
• Change of Variables in Triple Integrals
• Triple Integrals in Cylindrical Coordinates
• Triple Integrals in Spherical Coordinates
• Calculation of Volumes Using Triple Integrals
• Physical Applications of Triple Integrals

Line Integrals

• Line Integrals of Scalar Functions
• Line Integrals of Vector Fields
• Green’s Theorem
• Path Independence of Line Integrals
• Geometric Applications of Line Integrals
• Physical Applications of Line Integrals

Surface Integrals

• Surface Integrals of Scalar Functions
• Surface Integrals of Vector Fields
• The Divergence Theorem
• Stoke’s Theorem
• Geometric Applications of Surface Integrals
• Physical Applications of Surface Integrals

Fourier Series

• Definition of Fourier Series and Typical Examples
• Fourier Series of Functions with an Arbitrary Period
• Even and Odd Extensions
• Complex Form of Fourier Series
• Convergence of Fourier Series
• Bessel’s Inequality and Parseval’s Theorem
• Differentiation and Integration of Fourier Series
• Applications of Fourier Series to Differential Equations
• Orthogonal Polynomials and Generalized Fourier Series