- The first-derivative test for relative extrema Video
- Concavity and inflection points Video
- Second derivative test for relative extrema Video
- Curve sketching using first and second derivative tests Video
- Limits to infinity and infinite limits Video
- Graphs with asymptotesVideo
- L’Hôpital’s rule Video
- Applications in business, economics and life sciences
- Higher order derivatives Video
- Leibniz rule Video

- Parametric representation of curves and tracing of parametric curves (except lines in \(\mathbb{R}^3\)) Video
- Polar coordinates and tracing of curves in polar coordinates Video
- Techniques of sketching conics Video
- Reflection properties of conics
- Rotation of axes and second-degree equations
- Classification into conics using the discriminant

- Volumes by slicing disks and method of washers
- Volumes by cylindrical shells
- Arc length
- Arc length of parametric curves
- Area of surface of revolution
- Hyperbolic functions
- Reduction formulae

- Introduction to vector functions and their graphs
- Operations with vector functions
- Limits and continuity of vector functions
- Differentiation and integration of vector functions
- Modeling ballistics and planetary motion
- Kepler’s second law
- Unit tangent, Normal and binormal vectors
- Curvature

- Polynomials
- The remainder and factor theorem
- Synthetic division
- Factored form of a polynomial
- Fundamental theorem of algebra
- Relations between the roots and the coefficients of polynomial equations
- Theorems on imaginary, integral and rational roots
- Polar representation of complex numbers
- De Moivre’s theorem for integer and rational indices and their applications
- The nth roots of unity

- Equivalence relations
- Functions
- Composition of functions
- Invertibility and inverse of functions
- One-to-one correspondence and the cardinality of a set

- Well ordering principle
- The division algorithm in ℤ
- Divisibility and the Euclidean algorithm
- Fundamental theorem of arithmetic
- Modular arithmetic and basic properties of congruences
- Principle of mathematical induction

- Systems of linear equations
- Row reduction and echelon forms
- Vector equations
- The matrix equation \(Ax = b\)
- Solution sets of linear systems
- The inverse of a matrix
- Subspaces
- Linear independence
- Basis and dimension
- The rank of a matrix and applications
- Introduction to linear transformations
- The matrix of a linear transformation
- Applications to computer graphics
- Eigenvalues and eigenvectors
- The characteristic equation
- Cayley−Hamilton theorem

- Algebraic and order properties of \(\mathbb{R}\)
- Absolute value of a real number
- Bounded above and bounded below sets
- Supremum and infimum of a nonempty subset of \(\mathbb{R}\)

- The completeness property of \(\mathbb{R}\)
- Archimedean property
- Density of rational numbers in \(\mathbb{R}\)
- Definition and types of intervals
- Nested intervals property
- Neighborhood of a point in \(\mathbb{R}\)
- Open and closed sets in \(\mathbb{R}\)

- Convergent sequence
- Limit of a sequence
- Bounded sequence
- Limit theorems
- Monotone sequences
- Monotone convergence theorem
- Subsequences
- Bolzano−Weierstrass theorem for sequences
- Limit superior and limit inferior for bounded sequence
- Cauchy sequence
- Cauchy’s convergence criterion

- Convergence and divergence of infinite series of real numbers
- Necessary condition for convergence
- Cauchy criterion for convergence
- Tests for convergence of positive term series: Integral test, Basic comparison test, Limit comparison test, D’Alembert’s ratio test, Cauchy’s nth root test
- Alternating series
- Leibniz test
- Absolute and conditional convergence

- Differential equations and mathematical models
- Order and degree of a differential equation
- Exact differential equations and integrating factors of first order differential equations
- Reducible second order differential equations
- Applications of first order differential equations to acceleration-velocity model
- Growth and decay model

- Introduction to compartmental models
- Lake pollution model (with case study of Lake Burley Griffin)
- Drug assimilation into the blood (case of a single cold pill, case of a course of cold pills, case study of alcohol in the bloodstream)
- Exponential growth of population
- Limited growth of population
- Limited growth with harvesting

- General solution of homogeneous equation of second order
- Principle of superposition for a homogeneous equation
- Wronskian, its properties and applications
- Linear homogeneous and non-homogeneous equations of higher order with constant coefficients
- Euler’s equation
- Method of undetermined coefficients
- Method of variation of parameters
- Applications of second order differential equations to mechanical vibrations

- Interacting population models
- Epidemic model of influenza and its analysis
- Predator-prey model and its analysis
- Equilibrium points
- Interpretation of the phase plane
- Battle model and its analysis

- Limits of functions (ε δ − approach)
- Sequential criterion for limits
- Divergence criteria
- Limit theorems
- One-sided limits
- Infinite limits and limits at infinity

- Continuous functions
- Sequential criterion for continuity and discontinuity
- Algebra of continuous functions
- Properties of continuous functions on closed and bounded intervals
- Uniform continuity
- Non-uniform continuity criteria
- Uniform continuity theorem

- Differentiability of a function
- Algebra of differentiable functions
- Carathéodory’s theorem
- Chain rule
- Relative extrema
- Interior extremum theorem
- Rolle’s theorem
- Mean-value theorem and applications
- Intermediate value property of derivatives
- Darboux’s theorem

- Taylor polynomial
- Taylor’s theorem with Lagrange form of remainder
- Application of Taylor’s theorem in error estimation
- Relative extrema
- Criterion for convexity
- Taylor’s series expansions of e^x, sin x, and cos x

- Functions of several variables
- Level curves and surfaces
- Limits and continuity
- Partial differentiation
- Higher order partial derivative
- Tangent planes
- Total differential and differentiability
- Chain rule
- Directional derivatives
- The gradient
- Maximal and normal property of the gradient
- Tangent planes and normal lines

- Extrema of functions of two variables
- Method of Lagrange multipliers
- Constrained optimization problems
- Definition of vector field
- Divergence and curl

- Double integration over rectangular and nonrectangular regions
- Double integrals in polar coordinates
- Triple integral over a parallelopiped and solid regions
- Volume by triple integrals
- Triple integration in cylindrical and spherical coordinates
- Change of variables in double and triple integrals

- Line integrals
- Applications of line integrals: Mass and Work
- Fundamental theorem for line integrals
- Conservative vector fields
- Green’s theorem
- Area as a line integral
- Surface integrals
- Stokes’ theorem
- Gauss divergence theorem

- Introduction
- Classification
- Construction and geometrical interpretation of first order partial differential equations (PDE)
- Method of characteristic and general solution of first order PDE
- Canonical form of first order PDE
- Method of separation of variables for first order PDE

- Gravitational potential
- Conservation laws and Burger’s equations
- Classification of second order PDE
- Reduction to canonical forms
- Equations with constant coefficients
- General solution

- Mathematical modeling of vibrating string and vibrating membrane
- Cauchy problem for second order PDE
- Homogeneous wave equation
- Initial boundary value problems
- Nonhomogeneous boundary conditions
- Finite strings with fixed ends
- Non-homogeneous wave equation
- Goursat problem

- Method of separation of variables for second order PDE
- Vibrating string problem
- Existence and uniqueness of solution of vibrating string problem
- Heat conduction problem
- Existence and uniqueness of solution of heat conduction problem
- Non-homogeneous problem

- Definition of Riemann integration
- Inequalities for upper and lower Darboux sums
- Necessary and sufficient conditions for Riemann integrability
- Definition of Riemann integration by Riemann sum and equivalence of the two definitions
- Riemann integrability of monotone functions and continuous functions
- Properties of Riemann integrable functions
- Definitions of piecewise continuous and piecewise monotone functions and their Riemann integrability
- Intermediate value theorem for integrals
- Fundamental theorems (I and II) of calculus
- Integration by parts

- Improper integrals of Type-I, Type-II and mixed type
- Convergence of beta and gamma functions, and their properties

- Pointwise and uniform convergence of sequence of functions
- Theorem on the continuity of the limit function of a sequence of functions
- Theorems on the interchange of the limit and derivative, and the interchange of the limit and integrability of a sequence of functions
- Pointwise and uniform convergence of series of functions
- Theorems on the continuity, derivability and integrability of the sum function of a series of functions
- Cauchy criterion and the Weierstrass M-test for uniform convergence

- Definition of a power series
- Radius of convergence
- Absolute convergence (Cauchy−Hadamard theorem)
- Uniform convergence
- Differentiation and integration of power series
- Abel’s theorem

- Definition and examples of rings
- Properties of rings
- Subrings
- Integral domains and fields
- Characteristic of a ring
- Ideals, Ideal generated by a subset of a ring
- Factor rings
- Operations on ideals
- Prime and maximal ideals

- Ring homomorphisms
- Properties of ring homomorphisms
- First, Second and Third Isomorphism theorems for rings
- The Field of quotients

- Vector spaces
- Subspaces
- Algebra of subspaces
- Quotient spaces
- Linear combination of vectors
- Linear span
- Linear independence
- Basis and dimension
- Dimension of subspaces

- Linear transformations
- Null space
- Range
- Rank and nullity of a linear transformation
- Matrix representation of a linear transformation
- Algebra of linear transformations
- Isomorphisms
- Isomorphism theorems
- Invertibility and the change of coordinate matrix

- Definition and examples of metric spaces
- Sequences in metric spaces
- Cauchy sequences
- Complete metric space

- Open and closed ball
- Neighborhood
- Open set
- Interior of a set
- Limit point of a set
- Derived set
- Closed set
- Closure of a set
- Diameter of a set
- Cantor’s theorem
- Subspaces
- Dense set

- Continuous mappings
- Sequential criterion and other characterizations of continuity
- Uniform continuity
- Homeomorphism
- Contraction mapping
- Banach fixed point theorem

- Connectedness
- Connected subsets of ℝ
- Connectedness and continuous mappings
- Compactness
- Compactness and boundedness
- Continuous functions on compact spaces

- Automorphism
- Inner automorphism
- Automorphism groups
- Automorphism groups of finite and infinite cyclic groups
- Characteristic subgroups
- Commutator subgroup and its properties
- Applications of factor groups to automorphism groups

- External direct products of groups and its properties
- The group of units modulo ( n ) as an external direct product
- Applications to data security and electric circuits
- Internal direct products
- Classification of groups of order ( p^n ), where ( p ) is a prime
- Fundamental theorem of finite abelian groups and its isomorphism classes

- Group actions and permutation representations
- Stabilizers and kernels of group actions
- Groups acting on themselves by left multiplication and consequences
- Conjugacy in ( S_n )

- Conjugacy classes
- Class equation
- ( p )-groups
- Sylow theorems and consequences
- Applications of Sylow theorems
- Finite simple groups
- Nonsimplicity tests
- Generalized Cayley’s theorem
- Index theorem
- Embedding theorem and applications
- Simplicity of ( A_n )

- Functions of complex variable
- Mappings by the exponential function
- Limits
- Theorems on limits
- Limits involving the point at infinity
- Continuity
- Derivatives
- Differentiation formulae
- Cauchy-Riemann equations
- Sufficient conditions for differentiability
- Analytic functions and their examples

- Exponential function
- Logarithmic function
- Branches and derivatives of logarithms
- Trigonometric function
- Derivatives of functions
- Definite integrals of functions
- Contours
- Contour integrals and its examples
- Upper bounds for moduli of contour integrals

- Antiderivatives
- Proof of antiderivative theorem
- Cauchy-Goursat theorem
- Cauchy integral formula
- An extension of Cauchy integral formula
- Consequences of Cauchy integral formula
- Liouville’s theorem
- Fundamental theorem of algebra

- Convergence of sequences and series
- Taylor series and its examples
- Laurent series and its examples
- Absolute and uniform convergence of power series
- Uniqueness of series representations of power series
- Isolated singular points
- Residues
- Cauchy’s residue theorem
- Residue at infinity
- Types of isolated singular points
- Residues at poles and its examples

- Polynomial rings over commutative rings
- Division algorithm and consequences
- Principal ideal domains
- Factorization of polynomials
- Reducibility tests
- Irreducibility tests
- Eisenstein’s criterion
- Unique factorization in ( \mathbb{Z}[x] )
- Divisibility in integral domains
- Irreducibles
- Primes
- Unique factorization domains
- Euclidean domains

- Dual spaces
- Double dual
- Dual basis
- Transpose of a linear transformation and its matrix in the dual basis
- Annihilators
- Eigenvalues
- Eigenvectors
- Eigenspaces and characteristic polynomial of a linear operator
- Diagonalizability
- Invariant subspaces
- Cayley-Hamilton theorem
- Minimal polynomial for a linear operator

- Inner product spaces and norms
- Orthonormal basis
- Gram-Schmidt orthogonalization process
- Orthogonal complements
- Bessel’s inequality

- Adjoint of a linear operator
- Least squares approximation
- Minimal solutions to systems of linear equations
- Normal, self-adjoint, unitary and orthogonal operators and their properties