Calculus and Statistics

Fundamental concepts:
Concept of sets and main operations. Numerical sets.
Upper bound, lower bound, maximum, minimum, supremum, and infimum of sets. Cardinality. Natural, integer, rational, and real numbers. Distance, neighborhoods, and intervals. Absolute value and the triangle inequality. First and second degree equations and inequalities.

Real functions of a real variable:
The concept of a function. Injective, surjective, and bijective functions. Composition of functions and operations on graphs. Inverse function. Examples of elementary (linear, quadratic, power, exponential, logarithmic, and trigonometric) and non-elementary functions. Bounded and monotone functions. Concept of global and local maxima and minima. Concave/convex functions.

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

Differential calculus with one variable:
Incremental ratio, derivative at a point, differentiability; geometric meaning, equation of the tangent line. Derivation rules. Derivative of the inverse function. Higher-order derivatives. Differentiability, differential. Relationships between differentiability and differentiability. Stationary points. Necessary condition for local maximum/minimum points (Fermat's theorem). L'Hôpital's rule. Taylor and Maclaurin polynomials (briefly). Determination of local and global maximum/minimum points. Study of the graph of a function.

Introduction to integral calculus:
Primitive of a function, Riemann integral, fundamental theorem of integral calculus. Calculation of the area under the curve. Integration methods: immediate integrals, substitution integration, integration by parts. Mean value of an integral. Generalized integrals, improper integrals over unbounded intervals.

Introduction to probability:
Elements of combinatorial analysis. Permutations, factorial, arrangements, combinations, and binomial coefficient. Concept of random events, frequency, probability, law of large numbers. Elements of probability calculus. Axioms of probability. Bayes' theorem.

Descriptive statistics:
Discrete and continuous random variables. Concept of observable. Expected value and variance. Covariance, correlation. Probability distributions: binomial, Poisson, exponential, and uniform. Normal distribution: Standardized z variable, use of the standard normal distribution table. Central limit theorem.