Calculus

Chapter 1: Functions

• Sets
  1. The notion of sets
  2. Operations for sets
  3. Intervals
• Functions
  1. The notion of function
  2. Operations for functions
• Range
  1. The range of a function
  2. Functions and graphs
  3. Transformations of the axes
  4. Symmetry of functions
• Injectivity
  1. Injective functions
  2. The inverse of a function
  3. Power functions
  4. Equations and functions
• Applications
  1. Applications of functions

Chapter 2: Polynomials & rational functions

• Polynomials
  1. The notion of polynomial
  2. Calculating with polynomials
  3. Division with remainder for polynomials
• Linear polynomials
  1. Linear functions
• Quadratic polynomials
  1. Quadratic functions
  2. Quadratic equations
  3. Quadratic inequalities
• Factorization of polynomials
  1. The notions gcd and lcm for polynomials
  2. Rules of calculation for gcd and lcm of polynomials
  3. The Euclidean algorithm for polynomials
  4. Factorization of polynomials
  5. The fundameltal theorem of algebra
  6. Polynomial interpolation
  7. The extended Euclidean algorithm for polynomials
• Rational functions
  1. The notion of rational function
  2. Normal form for rational functions
  3. Partial fraction decomposition for rational functions
• Applications
  1. Applications of polynomials and rational functions

Chapter 3: Trigonometric functions

• Basics
  1. Definitions of sin and cos
  2. Right triangles and trigonometric functions
  3. Periodicity of trigonometric functions
• Calculation
  1. Special values of trigonometric functions
  2. Addition formulas for trigonometric functions
  3. Triangles and trigonometric functions
• More trigonometric functions
  1. Tangent and cotangent
  2. Inverse trigonometric functions
• Applications
  1. Applications of trigonometric functions

Chapter 4: Exponential & logarithmic functions

• Definition exp
  1. The notion of exponential function
  2. Rules of calculation for exponential functions
  3. Equations with exponential functions
• Definition log
  1. The notion of logarithm
  2. Rules of calculation for logarithms
  3. Equations with logarithms
• Growth
  1. Exponential growth
• Applications
  1. Applications of exponential and logarithmic functions

Chapter 5: Limits

• Definition
  1. The notion of limit
  2. The notion of limit and infinity
  3. Limits of rational functions
  4. Vertical asymptotes
• Rules for calculating limits
  1. Rules for limits
  2. Horizontal asymptotes
  3. Oblique asymptotes
  4. Squeeze theorem for limits
• Exp and gonio
  1. Limits of exponential functions
  2. Trigonometric limits
• Applications
  1. Applications of limits

Chapter 6: Sequences & series

• Definition
  1. The notions of sequence and series
  2. Arithmetic series
  3. Geometric series
• Convergence
  1. Convergence
  2. Monotonic sequences
  3. Divergence
• Rules
  1. Rules for limits of sequences
• Power series
  1. Power series
  2. Convergence criteria
• Length
  1. Length
• Applications
  1. Applications of sequences and series

Chapter 7: Continuity

• Definition of continuity
  1. The notion of continuity
  2. Global minimum and maximum
  3. Continuous extension
• Min-max and Intermediate Value Theorem
  1. The Min-Max Theorem
  2. Intermediate Value Theorem
• Limits
  1. Limits of continuous functions
  2. Rules for continuity
• Applications
  1. Applications of continuity

Chapter 8: Differentiation

• Definition
  1. The notion of difference quotient
  2. The notion of differentiation
  3. A simple derivative
• Simple rules
  1. The derivative of a sum function
  2. The derivative of a polynomial
  3. The product rule for differentiation
  4. Tangent lines
• More rules
  1. The chain rule for differentiation
  2. Derivatives of trigonometric functions
  3. The quotient rule for differentiation
  4. Derivatives of inverse functions
• Exp and log
  1. The natural logarithm
  2. Derivatives of exponential and logarithmic functions
• Applications
  1. Applications of differentiation

Chapter 9: Analysis of functions

• Minima and maxima
  1. Local minima and maxima
  2. The Mean Value Theorem
  3. Monotonocity
• Higher derivatives
  1. Higher derivatives
• Implicit derivatives
  1. Implicit derivatives
• Approximation with polynomials
  1. Linear approximation
  2. Taylor series
  3. Taylor series of some known functions
• De L’Hôpital
  1. The De L’Hôpital rule
• Applications
  1. Applications of analysis of functions

Chapter 10: intergration

• Antiderivation
  1. The notion of an antiderivative
  2. Antiderivatives of some known functions
  3. Integration by parts
• Area
  1. Area
• Integral
  1. Riemann sums
  2. The integral of a function
  3. Rules of calculation for integrals
• Estimates
  1. Estimates of integrals
  2. Mean Value Theorem for Integrals
• The Fundamental Theorem of Calculus
  1. The fundamental theorem of calculus
• Applications
  1. Applications of integration