Elements of logic

Propositions

Propositional connectives

Quantifiers

Definitions, theorems and proofs

Functions and numerical sets

Sets and operations between sets

The concept of function

Injective functions, composition and inverse function

Natural, integer, rational and real numbers

Intervals

Extended system of real numbers

Correspondence between a function and its graph

The elementary functions

Increasing and decreasing functions

Concave and convex functions

Even and odd functions

Linear functions, straight lines, first-order equations and inequations

Quadratic functions, parabolas, second-order equations and inequations

The power function with integer exponent, polynomial equations and inequations

Hyperbole, rational equation and inequations

The power function with fractional exponent, roots, equations and inequations with radicals

The exponential function, equations and inequations with exponentials

The logarithmic function, equations and inequations with logarithms

The "integer part" functions and the absolute value, equations and inequations with absolute values

Almost-elementary functions

Sequences

Sequences defined by recurrence

Limits

Neighbourhood of a number and accumulation points

Definition of limit

One-sided limit.

Limits of sequences.

Existence of the limit for increasing or decreasing functions

Squeeze theorem

Uniqueness of the limit

Continuous functions

Definition of continuous functions

Properties of continuous functions: Weierstrass, Bolzano and intermediate values theorems

Calculation of limits and indeterminate forms

Infinites and infinitesimals

The "little-O" notation

Derivatives

Definition of derivative of a function

Geometric meaning of the derivative

Calculation of derivatives

Derivability and continuity

Higher derivatives

Maxima and minima of functions

Convexity, concavity and inflection points

Study of a function

Integrals

The anti-derivative (or primitive) of a function

Immediate and quasi-immediate anti-derivatives

The definite integral

Property of the definite integral

Fundamental theorem of calculus

Improper integrals

Vectors and matrices

Vectors of R^n

Operations on vectors: sum, scalar multiplication, internal product

Geometric interpretation of operations between vectors

Linear combination of vectors, linear independence, bases and generators

Distance between vectors, norm of a vector, neighborhood of a vector

Matrices: definition and special types

Operations between matrices: sum, scalar multiplication, matrix product, transposition

Determinant of a matrix in the cases 1x1, 2x2, 3x3

Definition of inverse of a square matrix

Rank of a matrix