Calculus and Statistics

Mathematical Analysis
Prerequisites (review):
Fractions: classification and main properties. Factoring of polynomials.
Exponents and their properties. Logarithms and their properties. Conditions for the existence of powers and logarithms. Exponential and logarithmic equations.
Trigonometric functions: sine, cosine, and tangent. The radian.
Trigonometric equations and inequalities.
Irrational equations and inequalities.
Area of a circular sector.
Solving systems of linear equations.

Number sets:
The concept of sets and main set operations. Cardinality of a set. Numerical sets. Natural numbers, integers, rational and real numbers.

Line and parabola in the Cartesian plane:
Cartesian coordinates in the plane. Distance between two points, midpoint of a segment.
Equation of a line: slope and y-intercept. Conditions for parallelism and perpendicularity between two lines.
Equation of a line through two points. Equation of a line through one point with a given slope.
Equation of a parabola: definition and properties.
Position of a line with respect to a parabola and conditions for tangency.

Real functions of a real variable:
The concept of function. Graph of a function. Composition of functions and operations on graphs.
Injective, surjective, and bijective functions.
Symmetric, bounded, and monotonic functions. Periodic functions.
Concept of global and local maximum and minimum. Concave and convex functions.

Limits of functions and continuity:
Asymptotes. Uniqueness theorem for limits. Sign permanence theorem.
Limit calculation. Indeterminate forms. Change of variable. Notable limits.
Continuity of functions of a real variable. Types of discontinuities.

Differential calculus in one variable:
Incremental ratio, derivative at a point, differentiability; geometric meaning, equation of the tangent line.
Differentiation rules. Higher-order derivatives. Stationary points.
Necessary condition for local maxima/minima (Fermat's Theorem).
L'Hôpital's Rule.
Identification of local and global maxima/minima.
Graph analysis of a function.

Introduction to integral calculus:
Antiderivative of a function, Riemann integral, Fundamental Theorem of Calculus.
Area under the curve.
Integration methods: immediate integrals, substitution, integration by parts.
Mean value of a function (integral average).

Probability and Statistics
Elements of combinatorics:
Permutations, arrangements, and combinations.

Elements of probability theory:
Random events and the classical and frequentist interpretations of probability.
Law of large numbers.
Complementary, compound, and total probability.
Bayes' Theorem.

Elements of descriptive statistics:
Discrete and continuous random variables.
Measures of central tendency and dispersion.
Covariance, correlation, and regression.
Probability distributions: binomial, uniform, Poisson, exponential, and normal.
Use of the standard normal distribution table.
The Central Limit Theorem.