LEARNING OUTCOMES

Rigorous treatment of Mathematical Analysis, focusing on differential and integral calculus of functions of one real variable.

TEACHING METHODS

Traditional: blackboard.

SYLLABUS/CONTENT

 1. Real numbers. The axioms of ordered fields. Absolute value. Natural and integer numbers. Rational numbers and their geometric representation. Completeness and its consequences. Real numbers and the straight line. Archimedean property. Decimal representations.

2. Functions. Relations, functions, domain, codomain, image and graph of a function. Composition of functions. Invertible functions. Operations on real functions. Monotone functions. Polynomials and rational functions. Trigonometric functions. The exponential function on rational numbers.

3. Limits. Metric and e topological properties of R. Continuity. Operations with continuous functions. Limits and their properties. The algebra of limits. Comparison theorems. Limits of monotone functions. Limits of compositions and change of variables. Sequences and their limits. Sunsequences.  Bolzano-Weierstrass' theorem. Cauchy sequences. Sequences defined by recurrence and their limits. Neper's number e.

4. Global properties of continuous functions. Weierstrass' theorem. Zeroes of continuous functions. Intermediate value theorem.  Continuity and monotonicity. Continuity of the inverse function. Uniform continuity. Heine-Cantor's theorem. The exponential funcion on real numbers.

5. Differential calculus. The derivative: definition and elementary properties. Differentiability and the properties of the differential.  Derivative of compositions and inverse functions. Derivatives of elementary functions. Higher order derivatives. The classical theorems by Rolle, Lagrange and  Cauchy and their consequences. The theorem of de l'Hopital. Local comparison of functions. Vanishing and diverging functions. Taylor's formula. Convexity. Study of monotonicity and convexity by means of first and second derivatives.  Newton's method. Iterative procedures for the solution of equations.

6.The  indefinite integral. Integration techniques.  Integration of elementary functions.  Integration by parts and by substitution.  Integration of rational functions. 

7. The Riemann integral. Definition and properties of the definite integral. Integrability of continuous and monotonic functions. The oriented integral. The integral mean theorem. Relations between derivation and integration: integral functions, the fundamental theorem of calculus and its consequences. Improper integrals. Convergence criteria. 

8. Series.  Geometric and telescopic series. Convergence. Series with non negative terms: comparison, root and ratio criteria; condensation, order and integral tests.  Alternating series and Leibniz' theorem.

9. Differential equations. Separation of variables. Linear first order equations Second order linear equations with constant coefficients.

RECOMMENDED READING/BIBLIOGRAPHY

A.Bacciotti, F.Ricci - Analisi Matematica I - Liguori Editore

M. Baronti, F. De Mari, R. van der Putten, I. Venturi - Calculus Problems, Springer, 2016

Further readings will be posted on the web page (AULAWEB)