IB Math (Higher Level)

IB Math (Higher Level)

Topic 1: Mathematics as the science

Topic 1: Mathematics as the science

- Number patterns: sequences, series and sigma notation
- Arithmetic sequences and series
- Geometric sequences and series
- Conjectures and proofs
- Mathematical induction
- Counting methods
- The binomial theorem

**Topic 2:**

**Mathematics as a language**

- Relations and functions
- Special functions and their graphs
- Operations with functions
- Transformations of graphs of functions

**Topic 3: The long journey of mathematics**

- Introduction to complex numbers
- Operations with complex numbers
- Polynomial functions: graphs and operations
- Polynomial functions: zeros, sum and product
- Polynomial equations and inequalities
- Solving systems of equations

**Topic 4: Modeling the real world**

- Limits, continuity and convergence
- The derivative of a function
- Differentiation rules
- Exploring relationships between
*f*,*f*’ and*f*’’ - Applications of differential calculus economics
- Optimization and modeling
- Differentiation of implicit functions
- Related rates

**Topic 5: Aesthetics in mathematics**

- Recursive functions
- Properties of exponents and logarithms
- Euler’s number and exponential functions
- Invariance and the exponential function- a different approach to Euler’s number
- Logarithms and bases
- Logarithmic functions and their behavior
- Derivatives of exponential and logarithmic functions
- Angles, arcs and areas

**Topic 6: Exploring randomness**

- Classification and representation of statistical data
- Measures of central tendency
- Measures of dispersion
- Theoretical probability
- Probability properties
- Experimental probability
- Conditional probability
- Independent events
- Probability tree diagrams
- Bayes’ theorem

**Topic 7: The evolution of calculus**

- Integration as anti-differentiation
- Definite integration
- Geometric significance of the definite integral

**Topic 8: Ancient mathematics and modern methods**

- The right-angled triangle and trigonometric ratios
- The unit circle and trigonometric ratios
- Compound angle identities
- Double angle identities
- Graphs of trigonometric functions
- The inverse trigonometric functions
- Solving trigonometric equations
- The cosine rule
- The sine rule
- Area of a triangle

**Topic 9: The power of calculus**

- Derivatives of trigonometric functions
- Related rates of change with trigonometric expressions
- Integration of trigonometric functions
- Integration by substitution
- Integration by parts
- Special substitution
- Applications and modeling

**Topic 10: Modeling randomness**

- Discrete random variables and distributions
- Binomial distribution
- Poisson distribution
- Continuous random variables
- Normal distribution
- Modeling and problem solving

**Topic 11: Inspiration and formalism**

- Geometric vectors and basic operations
- Introduction to vector algebra
- Vectors, points and equations of lines
- Scalar product
- Vector (cross) product and properties
- Vectors and equations of planes
- Angles, distances and intersections
- Modeling and problem solving

**Topic 12: Multiple perspectives in mathematics**

- Complex numbers as vectors
- Complex plane and polar form
- Operations with complex numbers in modulus-argument form
- Powers and roots of complex numbers: De Moivre’s theorem and applications
- Mathematical connections

**Topic 13: Exploration**

- About the exploration
- Internal assessment criteria
- How the exploration is marked
- Academic honesty
- Record keeping
- Choosing a topic
- Getting started