Calculus and Statistics

Sets, operations with sets. Number sets. Intervals. Upper bound, lower bound, maximum, minimum, supremum, infimum of real sets. Absolute value, distance. Neighbourhoods. Left/right neighbourhoods.
Functions
The concept of function. Injective, surjective, bijective function. Composition of functions. Inverse function. Review of elementary functions: linear, quadratic, power, exponential, logarithmic, trigonometric and hyperbolic functions. Bounded functions. Monotonic functions, strictly monotonic functions. Maxima/minima, maximizers/minimizers: global and local. Concave/convex functions
Linear Algebra
Limits of functions and continuity
Limits of functions of one real. Asymptotes. Theorem on the uniqueness of the limit. Limits of elementary functions. Indeterminate forms. The symbols ~ and o. Calculation of limits. Change of variable. Fundamental limits. Theorem on the permanence of sign. Comparison criterion. Continuity for functions of one real variable. Points of discontinuity. Weierstrass's theorem, zero-value theorem (Bolzano's theorem)
One-variable differential calculus
Difference quotient, derivative at a point; geometric meaning, equation of the tangent line. Rules on derivatives. Derivative of the inverse function. Higher derivatives. Differentiability, differential. Relationship between derivability and differentiability.
Stationary points. Necessary condition for local maximizers/minimizers (Fermat's theorem). Rolle's theorem. Lagrange's mean value theorem. Monotonicity/strict monotonicity test on an interval.
Taylor's polynomial and Maclaurin's polynomial. Taylor's theorem; Taylor's formula and Maclaurin's formula of order n, with Peano's remainder. Second sufficient condition for local maximizers/minimizers. Determination of global and local maximizers/minimizers. Study of the graph of a function.
Antiderivatives of a function. Indefinite integral: definition. Elementary integrals. Integration methods: by decomposition, by parts, by substitution. Definite integral (according to Riemann): definition, geometric meaning. Fundamental theorems of calculus. Computation of definite integrals. Mean value. Generalized integral on unbounded intervals.

Statistics:
Descriptive statistics. Sample space, data analysis, dispersion measures. Quantitative and qualitative variables. Frequencies.
Probability: sample space. Random events, probability of an event, probability space. Probability axioms. Stochastically independent events, positively/negatively correlated events. Bayes's theorem. Discrete and continuous random numbers Distribution function. Expected value and variance .Random vectors: covariance and correlation. Binomial, Poisson distributions. Uniform, exponential, normal, t-student distributions (introductory notes). Central limit theorem. Hypothesis testing (introductory notes).