Course content

Sets
- The notion of sets
- Operations for sets
- Intervals
Functions
- The notion of function
- Operations for functions
Range
- The range of a function
- Functions and graphs
- Transformations of the axes
- Symmetry of functions
Injectivity
- Injective functions
- The inverse of a function
- Power functions
- Equations and functions
Applications
- Applications of functions
Polynomials
- The notion of polynomial
- Calculating with polynomials
- Division with remainder for polynomials
Linear polynomials
- Linear functions
Quadratic polynomials
- Quadratic functions
- Quadratic equations
- Quadratic inequalities
Factorization of polynomials
- The notions gcd and lcm for polynomials
- Rules of calculation for gcd and lcm of polynomials
- The Euclidean algorithm for polynomials
- Factorization of polynomials
- The Fundameltal Theorem of Algebra
- Polynomial interpolation
- The extended Euclidean algorithm for polynomials
Rational functions
- The notion of rational function
- Normal form for rational functions
- Partial fraction decomposition for rational functions
Applications
- Applications of polynomials and rational functions
Chapter 3: Trigonometric functions
Basics
- Definitions of sin and cos
- Right triangles and trigonometric functions
- Periodicity of trigonometric functions
Calculation
- Special values of trigonometric functions
- Addition formulas for trigonometric functions
- Triangles and trigonometric functions
More trigonometric functions
- Tangent and cotangent
- Inverse trigonometric functions
Applications
- Applications of trigonometric functions
Chapter 4: Exponential and logarithmic functions
Definition exp
- The notion of exponential function
- Rules of calculation for exponential functions
- Equations with exponential functions
Definition log
- The notion of logarithm
- Rules of calculation for logarithms
- Equations with logarithms
Growth
- Exponential growth
Applications
- Applications of exponential and logarithmic functions
Chapter 5: Limits
Definition
- The notion of limit
- The notion of limit and infinity
- Limits of rational functions
- Vertical asymptotes
Rules for calculating limits
- Rules for limits
- Horizontal asymptotes
- Oblique asymptotes
- Squeeze theorem for limits
Exp and gonio
- Limits of exponential functions
- Trigonometric limits
Applications
- Applications of limits
Chapter 6: Sequences and series
Definition
- The notions of sequence and series
- Arithmetic series
- Geometric series
Convergence
- Convergence
- Monotonic sequences
- Divergence
Rules
- Rules for limits and sequences
Power series
- Power series
- Convergence criteria
Length
- Length
Applications
- Applications of sequences and series
Chapter 7: Continuity
Definition of continuity
- The notion of continuity
- Global minimum and maximum
- Continuous extension
Min-max and Intermediate Value Theorem
- The Min-Max Theorem
- Intermediate Value Theorem
Limits
- Limits of continuous functions
- Rules for continuity
Applications
- Applications of continuity
Chapter 8: Differentiation
Definition
- The notion of difference quotient
- The notion of differentiation
- A simple derivative
Simple rules
- The derivative of a sum function
- The derivative of a polynomial
- The product rule for differentiation
- Tangent lines
More rules
- The chain rule for differentiation
- Derivatives of trigonometric functions
- The quotient rule for differentiation
- Derivatives of inverse functions
Exp and log
- The natural logarithm
- Derivatives of exponential and logarithmic functions
Applications
- Applications of differentiation
Chapter 9: Analysis of functions
Minima and maxima
- Local minima and maxima
- The Mean Value Theorem
- Monotonocity
Higher derivatives
- Higher derivatives
Implicit derivatives
- Implicit derivatives
Approximation with polynomials
- Linear approximation
- Taylor series
- Taylor series of some known functions
De L'Hôpital
- The De L'Hôpital rule
Applications
- Applications of analysis of functions
Chapter 10: Integration
Antiderivation
- The notion of an antiderivative
- Antiderivatives of some known functions
- Integration by parts
Area
- Area
Integral
- Riemann sums
- The integral of a function
- Rules of calculation for integrals
Estimates
- Estimates of integrals
- Mean Value Theorem for integrals
The Fundamental Theorem of Calculus
- The Fundamental Theorem of Calculus
Applications
- Applications of integration

Calculus


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