Calculus

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Chapter 1. Sets of numbers.

Natural numbers. Integer numbers. Rational and irrational numbers. Real numbers. Subsets of real numbers. Methods of proof.

Chapter 2. Sequences and series of real numbers.

Sequences of real numbers. Concept and calculation of the limit of a sequence. Types of sequences (bounded, monotone, recursive). Series of real numbers. Series with positive terms and alternating series. Tests for convergence.

Chapter 3. Real functions: limits and continuity.

General concepts and elementary functions. Limit of a real function and its properties. Continuity of real functions. Theorems on continuous functions.

Chapter 4. Real functions: derivative.

Definition of derivative of a real function. Geometrical and physical interpretation of the derivative. Differentiable functions and derivative calculation. Theorems on differentiable functions. Power functions.

Chapter 5. The Newton-Raphson method.

Introduction and algorithm. Approximation of roots of a real function.

Chapter 6. Taylor polynomial.

Taylor polynomial. Lagrange form of the remainder. Approximation of functions and error upper bounds. Calculation of limits. Taylor series. 

Chapter 7. Local and global behavior of a real function.

Extrema of a real function. Local behavior and Taylor polynomial. Global maxima and minima. Concavity and inflection points.

Chapter 8. Integration: fundamental theorems and techniques.

Definite integral of a real function. Fundamental Theorem of Calculus. Indefinite integral. Techniques of integration (by parts, change of variable, integral of rational functions).

Chapter 9. Improper integrals.

Improper integrals of the first and second kind. Tests for convergence.