# 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.

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

## Teaching methods

Lessons and exercises at the blackboard

## Assessment methods

The examination is written and consists of two parts. It is obligatory to enrol in the Amaesami list of exams for both parts of the exam.

During the exam, it is forbidden to use any electronic equipment connected to internet; otherwise the exam will be cancelled.

The first part lasts for 2 and 1/2 hours and it consists of multiple choice and traditional exercises. Students may use their own books and notes. It is forbidden to use any electronic device. The highest rank of this part is 16. If the student achieves 6.5/16 or more in the A part, they may partecipate in the B part.

The B part lasts for 1 h and the students may take only the pen with themselves. In this part, the student must solve one of the multiple choice exercises of their A part and must answer to two theoretical questions following the proposed draft. The highest rank for this part of the exam is 21.

The final mark is obtained adding the marks of both parts. Final marks greater than 30 will correspond to 30/30 cum laude on Almaesami.

The student may check his work during a special office hours before the verbalization of all valid marks.

Further piece of information on exams is available in the web pages http://www.unibo.it/docenti/simonetta.abenda .

The dates of the exams are published on Almaesami.

Facsimiles of the part A written examination are avalaible in the Alma Campus collection.

## Teaching tools

Facsimiles of the part A written examination are available in the Alma Campus collection

## Office hours

See the website of Simonetta Abenda