27991 - Mathematical Analysis T-1

Learning outcomes

Knowledge of basic notions of mathem,atics (limits, derivation, integration) for the qualitative analysis of functions and the solutions of applicative problems.

Course contents

• Introduction.

1. Properties of the real numbers (cardinality, total order) and of the relevant subsets (N, Z, Q), intervals.
2. Definition of function in one variable, injectivity, surjectivity, invertibility, inverse function, composition of function, monotone functions.
3. Elementary functions (powers, roots, exponentials, logarithms, hyperbolic functions, trigonometric functions and inverse trigonometric functions, absolute value functions).

• Complex numbers

1. Definiton of the field of the complex numbers. Algebraic form. Modulus and argument of a complex number. Exponential form of a complex number. De Moivre's formula.
2. Complex roots of a complex number. Algebraic equations in C.

• Limits

1. Accumulation point.
2. definitions of limit for real functions in one variable, one-sided limits.
3. Elementary properties of limits: unicity, locality. Algebraic properties of the limit, Sandwich theorem, limits of monotone functions.
4. Indeterminacy.
5. Landau symbols

• Continuity

1. Definition of a continuous function of one real variable.
2. The Weierstrass theorem, the Bolzano theorem and the intermediate value theorem.
3. Continuity of the composition of two continuous functions. Continuity of the inverse function.

• Differential calculus and applications

1. Definition of a differentiable function and of derivative of a function.
2. The algebra of derivatives. The mean value theorems and their application to study the monotonicity of a function.
3. Higher order derivatives.
4. Hospital theorem for indeterminate limits. Taylor's formula.
5. Asymptotes. Relative maxima and minima of a function: definitions, necessary conditions, sufficient conditions.
6. Convex functions.

• Integration

1. Definition of the Riemann integral.
2. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem.
3. The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts.
4. Integration rules for rational functions.
5. Improper integrals.

• Differential equations.

1. Linear differential equations.The general solutions of homogeneous and nonhomogeneous linear differential equations.The Cauchy problem.
2. Solution of linear differential equations (of order one , of order n with constant coefficients).
3. The sympathy method and the method of Lagrange for the search of a particular solution to a inhomogeneous linear differential equation
4. Differential equations solvable by separation.

Readings/Bibliography

Simonetta Abenda : Analisi Matematica (Esculapio)
Simonetta Abenda: Esercizi di Analisi Matematica (Esculapio)

Teaching methods

Traditional lessons and exercise classes

Assessment methods

Exam is in written form.

Students solve multiple choice exercises, answers to theoretical question and provides the complete solution to the exercises.

Marks of exercises will be published on the IOL page of the course in due time.

Exam calls are publshed on Almaesami and are visible on the degree programme website.

Students may partecipate in any exam call. They may refuse the mark only once. It is obligeatory to enrol on Almaesami list for the exam call.

Exams will be the same either on-line or in presence.

At the end of correction of written exams there will be office hours for students to check their proof and the Commission will sign the marks at the end of such office hours.

Exercises at exam difficulty level shall be distributed during lessons and published on IOL.

Teaching tools

In case lessons will be in mixed form, screen shots and registrations of the lessons will be published on IOL. These have to be considered as auxiliary teahing tools and they are not intended to substitute the active participation of students at lessons and their autonomous study.

Students are encouraged to take notes during lessons, to check and study them, to compare them to texts and t solve exercises autonomously.

Students are strongly encouraged to ask questions to professors during lessons and not just at office hours. Indeed the main teaching tool is the active participation of students to lessons both on-line and in presence.

Office hours

See the website of Simonetta Abenda

See the website of Cataldo Grammatico