MATHEMATICAL ANALYSIS I Cp - I

Main objective. Providing basic knowledge and tools regarding differential and integral calculus and number series.

The course is divided into two parts. The first one involves the construction and properties of the field of real numbers, basic topology notions, sequences and functions, ending with derivatives and applications. The second involves series and integrals. Both parts will be followed by a written partial exam.

1. Sets of numbers. Real numbers. The ordering of real numbers. Completeness of R. Factorials and binomial coefficients. Relations in the plane. Complex numbers. Algebraic operation. Cartesian coordinates. Trigonometric and exponential form. Powers and nth roots. Algebraic equations.
2. Limits. Neighbourhoods. Real functions. Limits of functions. Theorems on limits: uniqueness and sign of the limit, comparison theorems, algebra of limits. Indeterminate forms of algebraic and exponential type. Substitution theorem. Limits of monotone functions. Sequences. Limit of a sequence. Sequential characterization of a limit. Cauchy's criterion for convergent sequences. Infinitesimal and infinite functions. Local comparison of functions. Landau symbols and their applications.
3. Continuity. Continuous functions. Sequential characterization of the continuity. Points of discontinuity. Discontinuities for monotone functions. Properties of continuous functions (Weierstrass's theorem, Intermediate value theorem). Continuity of the composition and the inverse functions.
4. Differential Calculus. The derivative. Derivatives of the elementary functions. Rules of differentiation. Differentiability and continuity. Extrema and critical points. Theorems of Rolle, Lagrange and Cauchy. Consequences of Lagrange's Theorem. De L'Hôpital Rule. Monotone functions. Higher-order derivatives. Convexity and inflection points. Qualitative study of a function. Recurrences.
5. Numerical series. Round-up on sequences. Numerical series. Series with positive terms. Alternating series. The algebra of series. Absolute and Conditional Convergence.
6. Integrals. Areas and distances. The definite integral. The Fundamental Theorem of Calculus. Indefinite integrals and the Net Change Theorem. The substitution rule. Integration by parts. Trigonometric integrals. Trigonometric substitution. Integration of rational functions by partial fractions. Strategy for integration. Impropers integrals. Applications of integration.

1. C. Canuto, A. Tabacco – Mathematical Analysis I – Springer (2015).
2. J. Stewart – Calculus. Early Transcendentals – Eight Edition, Cengage Learning (2016).