Limits

• Definition of a limit
• Formal definition of a limit
• Properties of limits
• Some important limits
• Some limit calculations
• L' Hospital's Rule
• Limits and the notion of continuity
• Limits at infinity

Differential Calculus

• Differentiation of polynomials and rational functions
• Differentiation of compositions of functions
• Differentiation of trigonometric functions
• The geometry of graphs
• Finding maximum and minimum values
• Implicit differentiation
• Partial derivatives
• Rolle's theorem and the Mean Value theorem
• Derivative formulae

Integral Calculus

• The Fundamental theorem of calculus
• The natural logarithm function
• The exponential function
• Hyperbolic functions
• The definite integral
• Areas under curve
• Rolle's theorem and the Mean Value theorem
• Integration by substitution
• Integration of rational functions
• Integration by parts
• Inverse trigonometric functions
• Integration techniques
• Improper integrals
• Integration formulae tables

Differential Equations

• Growth and decay models
• Separable variables
• First-order linear differential equations

Series

• Infinite series: absolute convergence
• Infinite series (revisited)
• The Ratio Test
• Taylor series
• Taylor's Theorem

Linear Equations and Matrices

• Linear systems
• Matrices
• Gaussian elimination
• Gauss-Jordan reduction
• Inverse of a matrix
• Matrix algebra (supplementary notes)

Determinants

• Determinants
• Cramer's Rule and Laplace expansion for n > 3

Vectors in R² and R³

• Elementary vector analysis
• Lines and planes
• Vectors (supplementary notes)

Root-finding methods

• Bisection method
• Fixed-point iteration method
• Newton's method
• Regula Falsi method
• Secant method

Iterative methods

• Aitken's acceleration formula
• Jacobi & Gauss-Siedel methods

Interpolating polynomials

• Cubic Bezier curves and matrix forms
• Cubic Splines
• Lagrange interpolating polynomials
• Least-squares approximations
• Newton's divided difference method
• Newton-Gregory polynomials
• Piecewise polynomial functions
• Taylor's polynomials

Matrix methods

• Gaussian elimination (with partial pivoting)
• L-U decomposition

Numerical differentiation

• Richardson's interpolation

Numerical integration

• Gaussian quadrature
• Newton-Cotes formulae
• Numerical Integration
• Romberg integration
• Trapezoidal rule
• Simpson's rules

Errors

• Errors: sources and propagation
• Types of errors
• Error formula for Simpson's Rule
• Error formula for the Trapezoidal Rule

Logic and Sets

• Logic and sets
• Relations
• Functions

Proof Techniques

• Direct proof
• Indirect proof - contrapositive
• Proof by cases
• Proof by induction
• Inference

Combinatorics and Graphs

• Permutations and Combinations
• Recurrence relations
• The summation notation
• Inclusion - Exclusion principle
• Directed graphs

Data-handling

• Measurement scales
• Presentation of data
• Descriptive statistics
• Expectation and variance
• Collection of data
• Sampling methods

Probability and Distributions

• Probability theory
• Binomial distribution
• Poisson distribution
• Normal distribution

Statistical Inference

• Estimation
• Hypothesis testing for one-sample mean and proportion
• Hypothesis testing for two-sample means
• Chi-Squared test
• Paired t-test
• Independent-samples t-test
• One-way ANOVA
• Type I and II errors

Reliability and Validity

• Reliability
• A pictorial concept of reliability
• Validity
• Convergent and discriminant validity
• Interpretation of reliability and validity coefficients

Multivariate Data

• Correlation
• Simple linear regression
• Multivariate analysis techniques