# 數學課程大綱

### 哲慧的星級數學課程

### M1 Course Outline

#### Algebra

- Simplifying algebraic expressions
- Solving linear and quadratic equations
- Inequalities and their solutions
- Functions and their representation (graphs, tables, and equations)
- Transformation of functions (reflection, translation, dilation)

#### Coordinate Geometry

- Distance and midpoint formula
- Equations of lines and their slopes
- Intersection of lines and angle between lines
- Circle equations and their properties
- Conic sections (parabolas, ellipses, hyperbolas)

#### Calculus

- Limits and derivatives
- Differentiation rules (power rule, chain rule, product rule)
- Maxima and minima of functions
- Implicit differentiation and related rates
- Integrals and antiderivatives

#### Vectors

- Vector operations (addition, subtraction, scalar multiplication)
- Vector representation (magnitude and direction)
- Dot product and cross product
- Applications of vectors in geometry and physics
- Parametric equations and their representation in space

#### Trigonometry

- Trigonometric functions (sine, cosine, tangent)
- Trigonometric ratios and their relationships
- Inverse trigonometric functions
- Trigonometric identities and equations
- Applications of trigonometry in mathematics and science

#### Statistics

- Data representation (tables, graphs, histograms)
- Measures of central tendency (mean, median, mode)
- Measures of dispersion (range, variance, standard deviation)
- Probability distributions (binomial, normal)
- Hypothesis testing and confidence intervals

### M2 Course Outline

#### Algebra and Calculus

- Polynomial functions and their properties
- Factorization of polynomials and synthetic division
- Rational functions and their graphs
- Indeterminate forms and L’Hôpital’s rule
- Exponential and logarithmic functions
- Differential equations and their solutions

#### Coordinate Geometry

- Three-dimensional coordinate geometry
- Equations of planes and their intersections
- Cylinders, cones, and spheres
- Vector equations of lines and planes
- Parametric equations in three dimensions

#### Trigonometry

- Trigonometry in two dimensions
- Trigonometry in three dimensions
- Solutions of triangles and their applications
- Law of sines and cosines
- Vectors and their applications in trigonometry

#### Statistics and Probability

- Sampling and estimation
- Correlation and regression
- Discrete random variables and their distributions
- Continuous random variables and their distributions
- Joint and marginal distributions
- Conditional distributions and independence

#### Combinatorics and Discrete Mathematics

- Counting principles (permutation and combination)
- Principle of inclusion and exclusion
- Generating functions and recurrence relations
- Graph theory (network flows and Euler paths)
- Matrix algebra and its applications

#### Numerical Methods

- Numerical solutions of equations (bisection, Newton-Raphson)
- Interpolation and extrapolation (Lagrange, Newton)
- Numerical differentiation and integration
- Numerical solutions of differential equations (Euler, Runge-Kutta)
- Error analysis and its impact on numerical methods