Calculus

Course content

Propositional logic
1. Propositions and truth
2. Negation, conjunction, and disjunction
3. Implication
4. Compound propositions
5. Propositions as variables
6. Truth tables
Calculating with propositions
1. Equivalence
2. Rules of calculation for propositions
3. Order of operations
4. Verum and Falsum
Predicate logic
1. Logical quantifiers
2. Reasoning by induction
Logical argumentation
1. Methamtical proofs
2. Modus ponens
3. Substitution in logic
4. Proof by contradiction

Sets
1. The notion of set
2. Set builders
3. Subsets
4. Intervals
Operations of sets
1. Union and intersection of sets
2. Rules of calculation of sets
3. Difference and complement of sets
4. Rules of calculation with set difference
5. Cartesian product
Relations
1. The notion of relation
2. Equivalence relation
3. Graphs
4. Functions
Real numbers
1. Properties of the real numbers
2. Ordering of real numbers
3. The notion of real number
4. Ordering and decimal development
5. Decimal developments of real numbers
6. Calculating with real numbers

Functions
1. The notion of function
2. Domain
3. Functions and graphs
4. The range of a function
Operations for functions
1. Arithmetic operations on functions
2. Composition of functions
Range
1. Recognizing graphs
2. Transformations of a graph
3. Symmetry of functions
Injectivity
1. Injectivity
2. Monotonic functions
3. The inverse of a function
4. Properties of inverse functions
Applications
1. Power functions
2. Power functions & equations
3. Equations and functions
4. Substitution of equations

Quadratic polynomials
1. Linear functions
2. Quadratic functions
3. Quadratic equations
4. Quadratic inequalities
Polynomials
1. The notion of polynomial
2. Calculating with polynomials
3. Division with remainder for polynomials
4. Other representations of polynomials
Greatest common divisor
1. The notion of gcd for polynomials
2. Rules of calculation for gcd of polynomials
3. The Euclidean algorithm of polynomials
4. The notion of lcm for polynomials
5. The extended Euclidean algorithm for polynomials
Factorisation of polynomials
1. Factors and zeros
2. Factorisation of polynomials
3. The Fundamental Theorem of Algrebra
4. Factorisation techniques
Rational functions
1. The notion of rational function
2. Calculating with rational functions
3. Standard form for rational functions
4. Partial fraction decomposition for rational functions
Other
1. Polynomial interpolation

The functions sine and cosine
1. Unit circle and angles
2. Sine and cosine
3. Sinusoids
4. Right-angled triangles and trigonometric funcitions
5. Symmetry of trigonometric functions
Calculating with sine and cosine
1. Special values of trigonometric functions
2. Addition formulas for trigonometric functions
3. Triangles and trigonometric functions
More trigonometric functions
1. Tangent
2. Reciprocal trigonometric functions
3. Inverses of trigonometric 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

Definition of limit
1. The notion of limit
2. Limits and infinity
Calculating with limits
1. Limits of rational functions
2. Rules of arithmetic calculation with limits
3. Composition rule for limits
4. Sandwich rule for limits
Asymptotes
1. Vertical asymptotes
2. Horizontal asymptotes
3. Oblique asymptotes
Limits of some non-rational functions
1. Limits of exponential functions
2. Limits of trigonometric functions
3. Limits of inverse functions

Definition of sequence and series
1. The notions of sequence and series
2. Arithmetic sequence and series
3. Geometric series
Limits of sequences
1. Convergence of sequences
2. Divergence of sequences
3. Rules for limits of sequences
4. Monotonic sequences
Convergence of series
1. Absolute convergence and ratio test for series
2. Alternating and comparison test for series
3. Consideration test for series
Power series
1. The notion of power series
2. The natural exponential function
3. Limits involving exponential functions
Completeness of the real numbers
1. Infima and superma for sets of real numbers
2. Limits and superma
3. Cauchy sequences

Definition of continuity
1. The notion of continuity
2. Standard continuous functions
3. Continuous extension
Three theorems on continuous functions
1. The Min-Max Theorem
2. The Intermediate Value Theorem
3. Uniform continuity
Limits & continuity
1. Limits and continuous functions
2. Rules for continuity
Continuity in geometry
1. Curves in the plane
2. Length of a curve
3. Regions
4. Area of a region

Definition of differentiation
1. The notion of difference quotient
2. The notion of differentiation
3. The derivative and the tangent
Calculating derivatives and tangent lines
1. Derivative of a power function
2. Sum rule for differentiation
3. Product rule for differentiation
4. Chain rule for differentiation
5. Quotient rule for differentiation
6. Calculating tangent lines
Derivatives of special functions
1. Derivatives of trigonometric functions
2. Derivatives of exponential functions
3. Derivatives of inverse functions
4. Derivatives of logarithmic functions
Other
1. The De L’Hôpital rule

Minima and maxima
1. Local minima and maxima
2. The Mean Value Theorem
3. Monotonocity
Higher derivatives
1. Higher derivatives
2. Applications of higher derivatives
Implicit derivatives
1. Implicit derivatives
2. Derivatives of bivariate functions
3. Implicit function theorem
4. Tangent line to a curve
Approximation with polynomials
1. Linear approximation
2. Taylor series
3. Taylor series of some known functions

Antiderivatives
1. The notion of an antiderivative
2. Antiderivatives of some known functions
Definite integrals
1. Riemann sums
2. The definite integral of a function
3. Rules of calculation for integrals
Properties of definite integrals
1. Estimates of definite integrals
2. The Mean Value Theorem for integrals
3. The Fundamental Theorem of Calculus
4. Improper integrals
Calculating with definite integrals
1. Area between graphs
2. Length of a curve revisited
3. Volume in space
4. Series and integrals
Finding antiderivatives
1. Substitution method
2. Trigonometric integrals
3. Inverse substitution
4. Integration by parts
5. Integration by parts, advanced usage
Antiderivatives of rational functions
1. Known antiderivatives of rational functions
2. Fraction decomposition for integration
3. Existence of antiderivatives of rational functions
4. Finding antiderivatives of rational functions

All our courses