Lectures in course MATH-101(en): Analysis I (English)

Covers the basics of analysis, including real numbers, proofs, sets, and operations.

Explores continuous strictly monotone functions, differentiability, and linear function approximation.

Explains the definitions and properties of maximum, minimum, supremum, and infimum in real numbers.

Covers the computation of derivatives and the properties of differentiable functions.

Covers inf/sup, integral part of real numbers, and density of rational numbers.

Explores complex numbers, operations, absolute value, and polar form, along with the analysis and graphic representation of complex numbers.

Covers higher derivatives, local extrema, and the application of Rolle's and Mean Value Theorems.

Covers solving equations over complex numbers, sequences, induction, and properties of complex numbers.

Covers the Mean Value Theorem, l'Hopital rule, Taylor's approximation, and more.

Explores induction, Bernoulli's inequality, and algebraic limits in sequences with examples and computations.

Covers the application of Taylor's formula, including composition of functions and detecting local extrema.

Explores inflection points, convexity, concavity, and asymptotes in functions, with examples and applications.

Covers the Squeeze Theorem, Quotient criterion, and recursive sequences with their limits.

Explores limits of recursive sequences, convergence, boundedness, and algebraic operations related to limits at infinity.

Covers integrability, anti-derivatives, and integration by parts in calculus.

Explores integration by substitution with proofs and examples on anti-derivatives and function equivalence.

Covers the squeeze theorem, monotone sequences, subsequences, and Bolzano-Weierstrass theorem, emphasizing the importance of peaks in sequences.

Explores Cauchy sequences, series definition, and convergence criteria.

Covers examples of integration by substitution and rational functions.

Covers the squeeze theorem, examples, and the Leibniz criterion for series convergence.