Mathematical Analysis 1

1. The real number system
Known sets of numbers: decimal representation of rational numbers. Real numbers: definition and fundamental properties. The least upper bound and greatest lower bound. Completeness of the real numbers. Euclidian spaces.

2. Elements of set theory and metric spaces
Functions and their elementary properties. Cardinality of sets: finite, countably infinity and uncountably infinite sets. The uncountability of the real numbers. Metric spaces: neighborhoods and classification of points. Types of sets in metric spaces (open, closed, bounded, compact) and their properties. Nested families of closed intervals and the theorem of Bolzano-Weierstrass.

3. Sequences
Convergence of sequences in metric spaces: definition and fundamental properties. Uniqueness of limits and boundedness of convergent sequences. The Cauchy criterion and the completeness of Euclidian spaces. Real valued sequences: operations, sign preservation under limits and comparison tests. Monotonicity and existence of limits. The number e of Napier and notable limit formulas. Subsequences: compactness and the limits of subsequences. Landau symbols for comparison of asymptotic behavior of sequences.

4. Numerical series
Definitions, examples and character of infinite series of real numbers. The Cauchy criterion and necessary conditions for convergence. Absolute convergence. Tests for convergene of series with non negative terms: comparison, limit comparison, ratio and roots tests. The condensation test. Alternating series and the Leibniz criterion.


5. Limits and continuity of functions
Mappings between metric spaces. Metric and sequence definitions for the existence of limits and their equivalence. Limits of elementary functions and the existence of limits for real valued functions of a real variable by monotonicity. Asymptotes to the graph of a function. Pointwise and global continuity. Continuity and the inverse image of open sets. Composition of continuous functions. Continuity, compactenss and the theorem of Weierstrass. Uniform continuity and the theorem of Heine-Cantor. Real valued functions: the theorems on zeros, intermediate values and Darboux's theorem. Monotonicity and continuity. Continuity of the inverse function.

6. Differential calculus for real valued functions of a real variable
The derivative: definition and geometric interpretation in terms of the tangent line. Continuity and differentiability. Derivative of elementary functions and differentiation rules for algebraic operations, compositions and inversion. Higher order derivatives. Local extrema: the theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. Calculus of indefinite form limits by the theorem of de l'Hôpital. Taylor's formula of second order with Peano and Lagrange remainders. Taylor expansions of elementary functions. Convexity, concavity and inflection points. Applications to local extrema of functions.

1. The real number system
Known sets of numbers: decimal representation of rational numbers. Real numbers: definition and fundamental properties. The least upper bound and greatest lower bound. Completeness of the real numbers. Euclidian spaces.

2. Elements of set theory and metric spaces
Functions and their elementary properties. Cardinality of sets: finite, countably infinity and uncountably infinite sets. The uncountability of the real numbers. Metric spaces: neighborhoods and classification of points. Types of sets in metric spaces (open, closed, bounded, compact) and their properties. Nested families of closed intervals and the theorem of Bolzano-Weierstrass.

3. Sequences
Convergence of sequences in metric spaces: definition and fundamental properties. Uniqueness of limits and boundedness of convergent sequences. The Cauchy criterion and the completeness of Euclidian spaces. Real valued sequences: operations, sign preservation under limits and comparison tests. Monotonicity and existence of limits. The number e of Napier and notable limit formulas. Subsequences: compactness and the limits of subsequences. Landau symbols for comparison of asymptotic behavior of sequences.

4. Numerical series
Definitions, examples and character of infinite series of real numbers. The Cauchy criterion and necessary conditions for convergence. Absolute convergence. Tests for convergene of series with non negative terms: comparison, limit comparison, ratio and roots tests. The condensation test. Alternating series and the Leibniz criterion.


5. Limits and continuity of functions
Mappings between metric spaces. Metric and sequence definitions for the existence of limits and their equivalence. Limits of elementary functions and the existence of limits for real valued functions of a real variable by monotonicity. Asymptotes to the graph of a function. Pointwise and global continuity. Continuity and the inverse image of open sets. Composition of continuous functions. Continuity, compactenss and the theorem of Weierstrass. Uniform continuity and the theorem of Heine-Cantor. Real valued functions: the theorems on zeros, intermediate values and Darboux's theorem. Monotonicity and continuity. Continuity of the inverse function.

6. Differential calculus for real valued functions of a real variable
The derivative: definition and geometric interpretation in terms of the tangent line. Continuity and differentiability. Derivative of elementary functions and differentiation rules for algebraic operations, compositions and inversion. Higher order derivatives. Local extrema: the theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. Calculus of indefinite form limits by the theorem of de l'Hôpital. Taylor's formula of second order with Peano and Lagrange remainders. Taylor expansions of elementary functions. Convexity, concavity and inflection points. Applications to local extrema of functions.