高中預科微積分/三角學導師

Precalculus serves as a bridge between Algebra, Geometry, and Calculus. It covers a broad range of topics that help students develop the skills needed for success in Calculus. Here are the standard topics typically covered in a Precalculus course:

1. Functions and Their Properties
Types of Functions: Linear, quadratic, polynomial, rational, exponential, logarithmic, trigonometric, inverse, and piecewise functions
Domain and Range: Determining the domain and range of functions
Operations on Functions: Addition, subtraction, multiplication, division, and composition of functions
Inverse Functions: Definition and finding inverse functions
Graphing Functions: Understanding transformations (translations, reflections, stretching, shrinking)
2. Polynomial and Rational Functions
Polynomial Functions: Long division, synthetic division, zeros of polynomials (factoring, Rational Root Theorem)
Graphing Polynomial Functions: End behavior, turning points, and identifying key characteristics of graphs
Rational Functions: Asymptotes (vertical, horizontal, oblique), holes, and graphing rational functions
Solving Polynomial and Rational Equations: Techniques for solving algebraically
3. Exponential and Logarithmic Functions
Properties of Exponents: Growth and decay models
Exponential Functions: Graphing and solving exponential equations
Logarithmic Functions: Definition of logarithms, properties (product, quotient, power rules), and solving logarithmic equations
Exponential and Logarithmic Applications: Compound interest, population growth, and decay models
4. Trigonometry
Angles and Radian Measure: Degrees vs. radians, converting between them
Trigonometric Functions: Definitions using the unit circle, sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent)
Graphing Trigonometric Functions: Amplitude, period, phase shift, and vertical shift
Inverse Trigonometric Functions: Understanding and solving inverse trig functions
Trigonometric Identities: Pythagorean, sum and difference identities, double-angle, half-angle identities
Solving Trigonometric Equations: Using identities and inverse functions
Law of Sines and Cosines: Applications to solving triangles
5. Complex Numbers
Definition and Operations: Addition, subtraction, multiplication, division of complex numbers
Polar Form of Complex Numbers: Conversion between rectangular and polar forms, De Moivre's Theorem
6. Vectors
Vector Operations: Addition, subtraction, dot product
Magnitude and Direction: Representing vectors graphically, finding the magnitude and direction
Applications of Vectors: Solving physics-related problems involving force, velocity, etc.
7. Sequences and Series
Arithmetic Sequences and Series: Finding terms, sum formulas
Geometric Sequences and Series: Finding terms, sum formulas, including infinite geometric series
Mathematical Induction: Proving statements involving sequences or inequalities
8. Conic Sections
Parabolas: Equations, vertex, focus, and directrix
Ellipses: Equations, foci, major and minor axes
Hyperbolas: Equations, foci, asymptotes
Applications of Conic Sections: Real-world problems involving parabolas, ellipses, and hyperbolas
9. Polar Coordinates and Parametric Equations
Polar Coordinates: Plotting points in the polar plane, converting between polar and rectangular coordinates
Polar Equations: Graphing polar equations, finding symmetries, and analyzing polar curves
Parametric Equations: Graphing and eliminating the parameter, applications to motion and curves
10. Limits (Introduction to Calculus)
Concept of a Limit: Understanding limits intuitively, graphically, and algebraically
Limits at Infinity: End behavior of functions and asymptotes
Continuity: Defining continuity at a point, types of discontinuities
Limit Laws: Basic properties and techniques for calculating limits
11. Additional Topics (Optional)
Binomial Theorem: Expanding binomials, Pascal's Triangle
Probability and Combinatorics: Permutations, combinations, and basic probability